Integrand size = 23, antiderivative size = 458 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac {b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-105 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{8 a^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2-35 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {15 b \left (4 a^2-7 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^5 d \sqrt {a+b \sin (c+d x)}} \] Output:
1/3*(2*a^2-3*b^2)*cot(d*x+c)*csc(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(3/2)-1/3 *cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))^(3/2)+1/8*b*(32*a^2-105*b^2) *cos(d*x+c)/a^5/d/(a+b*sin(d*x+c))^(1/2)+1/8*(16*a^2-35*b^2)*cot(d*x+c)/a^ 4/d/(a+b*sin(d*x+c))^(1/2)-1/12*(8*a^2-21*b^2)*cot(d*x+c)*csc(d*x+c)/a^3/b /d/(a+b*sin(d*x+c))^(1/2)-1/8*(32*a^2-105*b^2)*EllipticE(cos(1/2*c+1/4*Pi+ 1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/a^5/d/((a+b*sin(d *x+c))/(a+b))^(1/2)-1/8*(16*a^2-35*b^2)*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d *x,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^4/d/(a+b*sin( d*x+c))^(1/2)-15/8*b*(4*a^2-7*b^2)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2, 2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^5/d/(a+b*sin(d*x +c))^(1/2)
Result contains complex when optimal does not.
Time = 6.99 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\left (32 a^2 \cos (c+d x)-123 b^2 \cos (c+d x)\right ) \csc (c+d x)}{24 a^5}+\frac {17 b \cot (c+d x) \csc (c+d x)}{12 a^4}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a^3}+\frac {2 \left (a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right )}{3 a^4 (a+b \sin (c+d x))^2}+\frac {8 \left (a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right )}{3 a^5 (a+b \sin (c+d x))}\right )}{d}+\frac {-\frac {2 \left (32 a^3-140 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (152 a^2 b-315 b^3\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-32 a^2 b+105 b^3\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{32 a^5 d} \] Input:
Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]
Output:
(Sqrt[a + b*Sin[c + d*x]]*(((32*a^2*Cos[c + d*x] - 123*b^2*Cos[c + d*x])*C sc[c + d*x])/(24*a^5) + (17*b*Cot[c + d*x]*Csc[c + d*x])/(12*a^4) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^3) + (2*(a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x ]))/(3*a^4*(a + b*Sin[c + d*x])^2) + (8*(a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x]))/(3*a^5*(a + b*Sin[c + d*x]))))/d + ((-2*(32*a^3 - 140*a*b^2)*Elli pticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b )])/Sqrt[a + b*Sin[c + d*x]] - (2*(152*a^2*b - 315*b^3)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-32*a^2*b + 105*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[ c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(- 1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]) )*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))] )/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(32*a^5*d)
Time = 4.08 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.13, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.174, Rules used = {3042, 3203, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3203 |
| \(\displaystyle \frac {2 \int \frac {3 \csc ^3(c+d x) \left (8 a^2-2 b \sin (c+d x) a-21 b^2-\left (4 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{4 (a+b \sin (c+d x))^{3/2}}dx}{9 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (8 a^2-2 b \sin (c+d x) a-21 b^2-\left (4 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}}dx}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {8 a^2-2 b \sin (c+d x) a-21 b^2-\left (4 a^2-15 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^{3/2}}dx}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \csc ^2(c+d x) \left (-6 a \sin (c+d x) b^2-\left (8 a^2-21 b^2\right ) \sin ^2(c+d x) b+\left (16 a^2-35 b^2\right ) b\right )}{2 (a+b \sin (c+d x))^{3/2}}dx}{2 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \frac {\csc ^2(c+d x) \left (-6 a \sin (c+d x) b^2-\left (8 a^2-21 b^2\right ) \sin ^2(c+d x) b+\left (16 a^2-35 b^2\right ) b\right )}{(a+b \sin (c+d x))^{3/2}}dx}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \int \frac {-6 a \sin (c+d x) b^2-\left (8 a^2-21 b^2\right ) \sin (c+d x)^2 b+\left (16 a^2-35 b^2\right ) b}{\sin (c+d x)^2 (a+b \sin (c+d x))^{3/2}}dx}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\int -\frac {\csc (c+d x) \left (-\left (\left (16 a^2-35 b^2\right ) \sin ^2(c+d x) b^2\right )+15 \left (4 a^2-7 b^2\right ) b^2+2 a \left (8 a^2-21 b^2\right ) \sin (c+d x) b\right )}{2 (a+b \sin (c+d x))^{3/2}}dx}{a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {\csc (c+d x) \left (-\left (\left (16 a^2-35 b^2\right ) \sin ^2(c+d x) b^2\right )+15 \left (4 a^2-7 b^2\right ) b^2+2 a \left (8 a^2-21 b^2\right ) \sin (c+d x) b\right )}{(a+b \sin (c+d x))^{3/2}}dx}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {-\left (\left (16 a^2-35 b^2\right ) \sin (c+d x)^2 b^2\right )+15 \left (4 a^2-7 b^2\right ) b^2+2 a \left (8 a^2-21 b^2\right ) \sin (c+d x) b}{\sin (c+d x) (a+b \sin (c+d x))^{3/2}}dx}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {2 \int \frac {\csc (c+d x) \left (\left (32 a^4-137 b^2 a^2+105 b^4\right ) \sin ^2(c+d x) b^2+15 \left (4 a^4-11 b^2 a^2+7 b^4\right ) b^2+2 a \left (8 a^4-43 b^2 a^2+35 b^4\right ) \sin (c+d x) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\int \frac {\csc (c+d x) \left (\left (32 a^4-137 b^2 a^2+105 b^4\right ) \sin ^2(c+d x) b^2+15 \left (4 a^4-11 b^2 a^2+7 b^4\right ) b^2+2 a \left (8 a^4-43 b^2 a^2+35 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\int \frac {\left (32 a^4-137 b^2 a^2+105 b^4\right ) \sin (c+d x)^2 b^2+15 \left (4 a^4-11 b^2 a^2+7 b^4\right ) b^2+2 a \left (8 a^4-43 b^2 a^2+35 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {\int \frac {15 b^3 \left (4 a^4-11 b^2 a^2+7 b^4\right )-a b^2 \left (16 a^4-51 b^2 a^2+35 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\frac {\frac {15 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{2 a}-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}\right )}{4 a}-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac {-\frac {\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (-\frac {b \left (16 a^2-35 b^2\right ) \cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}-\frac {\frac {2 b^2 \left (32 a^2-105 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}+\frac {\frac {2 b \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {30 b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{a \left (a^2-b^2\right )}}{2 a}\right )}{4 a}}{6 a^2 b}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}\) |
Input:
Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]
Output:
((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x] )^(3/2)) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^(3/2) ) + (-1/2*((8*a^2 - 21*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d*Sqrt[a + b*Sin [c + d*x]]) - (3*(-((b*(16*a^2 - 35*b^2)*Cot[c + d*x])/(a*d*Sqrt[a + b*Sin [c + d*x]])) - ((2*b^2*(32*a^2 - 105*b^2)*Cos[c + d*x])/(a*d*Sqrt[a + b*Si n[c + d*x]]) + ((2*b*(32*a^4 - 137*a^2*b^2 + 105*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((-2*a*b^2*(16*a^4 - 51*a^2*b^2 + 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt [a + b*Sin[c + d*x]]) + (30*b^3*(4*a^4 - 11*a^2*b^2 + 7*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d *Sqrt[a + b*Sin[c + d*x]]))/b)/(a*(a^2 - b^2)))/(2*a)))/(4*a))/(6*a^2*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2)), x] - Simp[1/(3*a^2*b*( m + 1)) Int[((a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^ 2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[ e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && L tQ[m, -1] && IntegerQ[2*m]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2869\) vs. \(2(428)=856\).
Time = 2.47 (sec) , antiderivative size = 2870, normalized size of antiderivative = 6.27
Input:
int(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24*(-315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)* (-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2), (a-b)/a,((a-b)/(a+b))^(1/2))*b^6*sin(d*x+c)^4+48*((a+b*sin(d*x+c))/(a-b))^ (1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Elli pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*sin(d*x+c)^3 -96*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+ sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/( a+b))^(1/2))*a^6*sin(d*x+c)^3+48*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x +c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin( d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b*sin(d*x+c)^4+48*((a+b*sin( d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a -b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a ^4*b^2*sin(d*x+c)^4-306*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/ (a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/( a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3*sin(d*x+c)^4-105*((a+b*sin(d*x+c) )/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^( 1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4 *sin(d*x+c)^4+315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b)) ^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^ (1/2),((a-b)/(a+b))^(1/2))*a*b^5*sin(d*x+c)^4-96*((a+b*sin(d*x+c))/(a-b...
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**4/(a+b*sin(d*x+c))**(5/2),x)
Output:
Integral(cot(c + d*x)**4/(a + b*sin(c + d*x))**(5/2), x)
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")
Output:
Timed out
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Hanged} \] Input:
int(cot(c + d*x)^4/(a + b*sin(c + d*x))^(5/2),x)
Output:
\text{Hanged}
\[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4}}{\sin \left (d x +c \right )^{3} b^{3}+3 \sin \left (d x +c \right )^{2} a \,b^{2}+3 \sin \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:
int(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x)
Output:
int((sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4)/(sin(c + d*x)**3*b**3 + 3*s in(c + d*x)**2*a*b**2 + 3*sin(c + d*x)*a**2*b + a**3),x)