Integrand size = 37, antiderivative size = 880 \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {a^2 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\left (\sqrt {g}+\sqrt {g} \cot (e+f x)\right ) \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\left (\sqrt {g}+\sqrt {g} \cot (e+f x)\right ) \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}} \] Output:
1/2*a^2*d^(5/2)*arctan(-1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d* sin(f*x+e))^(1/2))*2^(1/2)/b/(a^2-b^2)/f/g^(3/2)-1/2*b*d^(5/2)*arctan(-1+2 ^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/2))*2^(1/2)/ (a^2-b^2)/f/g^(3/2)+1/2*a^2*d^(5/2)*arctan(1+2^(1/2)*d^(1/2)*(g*cos(f*x+e) )^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/2))*2^(1/2)/b/(a^2-b^2)/f/g^(3/2)-1/2*b* d^(5/2)*arctan(1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e ))^(1/2))*2^(1/2)/(a^2-b^2)/f/g^(3/2)-1/2*a^2*d^(5/2)*arctanh(2^(1/2)*d^(1 /2)*(g*cos(f*x+e))^(1/2)/(g^(1/2)+g^(1/2)*cot(f*x+e))/(d*sin(f*x+e))^(1/2) )*2^(1/2)/b/(a^2-b^2)/f/g^(3/2)+1/2*b*d^(5/2)*arctanh(2^(1/2)*d^(1/2)*(g*c os(f*x+e))^(1/2)/(g^(1/2)+g^(1/2)*cot(f*x+e))/(d*sin(f*x+e))^(1/2))*2^(1/2 )/(a^2-b^2)/f/g^(3/2)-2*2^(1/2)*a^3*d^3*EllipticPi((g*cos(f*x+e))^(1/2)/g^ (1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*sin(f*x+e)^(1/2)/b /(-a+b)^(3/2)/(a+b)^(3/2)/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+2*2^(1/2)*a^3*d^3 *EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),(-a+b)^(1/2) /(a+b)^(1/2),I)*sin(f*x+e)^(1/2)/b/(-a+b)^(3/2)/(a+b)^(3/2)/f/g^(3/2)/(d*s in(f*x+e))^(1/2)-2*b*d^2*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/g/(g*cos(f*x+e)) ^(1/2)+2*a*d*(d*sin(f*x+e))^(3/2)/(a^2-b^2)/f/g/(g*cos(f*x+e))^(1/2)+2*a*d ^2*(g*cos(f*x+e))^(1/2)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(d*sin(f*x+e) )^(1/2)/(a^2-b^2)/f/g^2/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 28.73 (sec) , antiderivative size = 1290, normalized size of antiderivative = 1.47 \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:
Integrate[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f* x])),x]
Output:
(2*Cot[e + f*x]*Csc[e + f*x]*(d*Sin[e + f*x])^(5/2)*(-b + a*Sin[e + f*x])) /((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*(d*Sin[e + f *x])^(5/2)*((-2*(3*a^2 - b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x ]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Co s[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x ]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*Appe llF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan [e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f *x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*S qrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)* ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2 ]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*S qrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[ a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1 /4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTa n[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2) ^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f* x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}-\frac {b d \int \frac {(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}-\frac {b d \int \frac {(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}+\frac {a d^2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}+\frac {a d^2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3051 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{a^2-b^2}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}-\frac {b d \left (\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}-\frac {d^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{g^2}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \int \frac {g \cot (e+f x)}{d \left (\cot ^2(e+f x) g^2+g^2\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\int \frac {\cot (e+f x) g+g}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}+\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} d \sqrt {g}}+\frac {\int \frac {\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}-\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {a^2 d^2 \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)}dx}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle -\frac {a^2 d^2 \left (\frac {d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 d^2 \left (\frac {d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle -\frac {a^2 d^2 \left (-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \int \frac {g \cot (e+f x)}{d \left (\cot ^2(e+f x) g^2+g^2\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{b f}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {a^2 d^2 \left (-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\int \frac {\cot (e+f x) g+g}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {a^2 d^2 \left (-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}+\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {a^2 d^2 \left (-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {a^2 d^2 \left (-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\right )}{g^2 \left (a^2-b^2\right )}-\frac {b d \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f g}+\frac {2 d \sqrt {d \sin (e+f x)}}{f g \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {a d^2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {a \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right ) d^2}{a^2-b^2}-\frac {a^2 \left (-\frac {2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^2}{b f}-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx d}{b}\right ) d^2}{\left (a^2-b^2\right ) g^2}-\frac {b \left (\frac {2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\cot (e+f x) g+g+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\cot (e+f x) g+g-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^3}{f g}+\frac {2 \sqrt {d \sin (e+f x)} d}{f g \sqrt {g \cos (e+f x)}}\right ) d}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right ) d^2}{a^2-b^2}-\frac {a^2 \left (-\frac {2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^2}{b f}-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx d}{b}\right ) d^2}{\left (a^2-b^2\right ) g^2}-\frac {b \left (\frac {2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\cot (e+f x) g+g+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\cot (e+f x) g+g-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^3}{f g}+\frac {2 \sqrt {d \sin (e+f x)} d}{f g \sqrt {g \cos (e+f x)}}\right ) d}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right ) d^2}{a^2-b^2}-\frac {a^2 \left (-\frac {2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} d \sqrt {g}}+\frac {\int \frac {\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d \sqrt {g}}}{2 d}\right ) d^2}{b f}-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx d}{b}\right ) d^2}{\left (a^2-b^2\right ) g^2}-\frac {b \left (\frac {2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\cot (e+f x) g+g+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\cot (e+f x) g+g-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^3}{f g}+\frac {2 \sqrt {d \sin (e+f x)} d}{f g \sqrt {g \cos (e+f x)}}\right ) d}{a^2-b^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right ) d^2}{a^2-b^2}-\frac {a^2 \left (-\frac {2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\cot (e+f x) g+g+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\cot (e+f x) g+g-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^2}{b f}-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx d}{b}\right ) d^2}{\left (a^2-b^2\right ) g^2}-\frac {b \left (\frac {2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\cot (e+f x) g+g+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\cot (e+f x) g+g-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right ) d^3}{f g}+\frac {2 \sqrt {d \sin (e+f x)} d}{f g \sqrt {g \cos (e+f x)}}\right ) d}{a^2-b^2}\) |
Input:
Int[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x ]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) /(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Sin[e + f*x ])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m , -1] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> With[{k = Denominator[m]}, Simp[(-k)*a*(b/f) Subst[Int[x ^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[ e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a*(d^2/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Simp[ b*(d/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a^2*(d^2/(g^2*(a^2 - b^2))) Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[ e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g }, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1 ]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int[(g *Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a*(d/b) Int[(g*C os[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; F reeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2012 vs. \(2 (724 ) = 1448\).
Time = 5.44 (sec) , antiderivative size = 2013, normalized size of antiderivative = 2.29
Input:
int((d*sin(f*x+e))^(5/2)/(g*cos(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x,method=_R ETURNVERBOSE)
Output:
-1/f*csc(f*x+e)*(I*(1+cos(f*x+e))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc( f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-a^2+b^2)^(1/ 2)*a^2*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+I *(-1-cos(f*x+e))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+ e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2*EllipticPi ((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*(csc(f*x+e)-cot( f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+ e))^(1/2)*(1+cos(f*x+e))*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/ 2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+I*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(- 2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-1-cos( f*x+e))*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))* (-a^2+b^2)^(1/2)*b^2-2*(1+cos(f*x+e))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2* csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-a^2+b^2) ^(1/2)*a*b*EllipticF((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))+4*(csc(f *x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e )+cot(f*x+e))^(1/2)*(1+cos(f*x+e))*a*b*EllipticE((csc(f*x+e)-cot(f*x+e)+1) ^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)-4*cos(f*x+e)*(-a^2+b^2)^(1/2)*a*b+(cs c(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f* x+e)+cot(f*x+e))^(1/2)*(1+cos(f*x+e))*b*EllipticPi((csc(f*x+e)-cot(f*x+e)+ 1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2+(csc(f*x+e)-cot(f*x...
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((d*sin(f*x+e))^(5/2)/(g*cos(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((d*sin(f*x+e))**(5/2)/(g*cos(f*x+e))**(3/2)/(a+b*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \] Input:
integrate((d*sin(f*x+e))^(5/2)/(g*cos(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="maxima")
Output:
integrate((d*sin(f*x + e))^(5/2)/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)), x)
\[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \] Input:
integrate((d*sin(f*x+e))^(5/2)/(g*cos(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="giac")
Output:
integrate((d*sin(f*x + e))^(5/2)/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)), x)
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:
int((d*sin(e + f*x))^(5/2)/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))),x )
Output:
int((d*sin(e + f*x))^(5/2)/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x)
\[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}\, \sin \left (f x +e \right )^{2}}{\cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) b +\cos \left (f x +e \right )^{2} a}d x \right ) d^{2}}{g^{2}} \] Input:
int((d*sin(f*x+e))^(5/2)/(g*cos(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x)
Output:
(sqrt(g)*sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x))*sin(e + f*x)** 2)/(cos(e + f*x)**2*sin(e + f*x)*b + cos(e + f*x)**2*a),x)*d**2)/g**2