Integrand size = 25, antiderivative size = 492 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=-\frac {5 a b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}+\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a^2 b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{2 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a^2 b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{2 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}} \]
-5/2*a*b^(3/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2 ))/(-a^2+b^2)^(9/4)/d/e^(3/2)+5/2*a*b^(3/2)*arctanh(b^(1/2)*(e*cos(d*x+c)) ^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(9/4)/d/e^(3/2)+b/(a^2-b^2)/d/ e/(a+b*sin(d*x+c))/(e*cos(d*x+c))^(1/2)+(-5*a*b+(2*a^2+3*b^2)*sin(d*x+c))/ (a^2-b^2)^2/d/e/(e*cos(d*x+c))^(1/2)-5/2*a^2*b*(cos(1/2*d*x+1/2*c)^2)^(1/2 )/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2) ),2^(1/2))*cos(d*x+c)^(1/2)/(a^2-b^2)^2/d/e/(b-(-a^2+b^2)^(1/2))/(e*cos(d* x+c))^(1/2)-5/2*a^2*b*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticPi(sin(1/2*d*x+1/2*c),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/ 2)/(a^2-b^2)^2/d/e/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-(2*a^2+3*b^2) *(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2 *c),2^(1/2))*(e*cos(d*x+c))^(1/2)/(a^2-b^2)^2/d/e^2/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 4.41 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {12 \left (-6 a^2 b+b^3-\left (2 a^2 b+3 b^3\right ) \cos (2 (c+d x))+4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x) \left (-\frac {\left (2 a^2+3 b^2\right ) \csc (c+d x) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right )}{\sqrt {b} \left (-a^2+b^2\right )}-\frac {48 a \left (a^2+4 b^2\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right )}{\sqrt {\sin ^2(c+d x)}}\right ) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{(a-b)^2 (a+b)^2}\right )}{24 d (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \]
(Cos[c + d*x]^(3/2)*((12*(-6*a^2*b + b^3 - (2*a^2*b + 3*b^3)*Cos[2*(c + d* x)] + 4*a*(a^2 - b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*Sqrt[Cos[c + d*x]]) - (Sin[c + d*x]*(-(((2*a^2 + 3*b^2)*Csc[c + d*x]*(8*b^(5/2)*AppellF1[3/4, -1 /2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x ]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqr t[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[C os[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*( a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2 ] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] )))/(Sqrt[b]*(-a^2 + b^2))) - (48*a*(a^2 + 4*b^2)*((a*AppellF1[3/4, 1/2, 1 , 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/ 2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos [c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a ^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b ^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/Sqrt[Sin[c + d*x]^2])*(a + b*Sqrt[S in[c + d*x]^2]))/((a - b)^2*(a + b)^2)))/(24*d*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))
Time = 2.37 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 3173, 27, 3042, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 218, 221, 3042, 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.
| \(\displaystyle \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {\int -\frac {2 a-3 b \sin (c+d x)}{2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a-3 b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a-3 b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {-\frac {2 \int \frac {\sqrt {e \cos (c+d x)} \left (2 a \left (a^2+4 b^2\right )+b \left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{2 (a+b \sin (c+d x))}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 a \left (a^2+4 b^2\right )+b \left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 a \left (a^2+4 b^2\right )+b \left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {-\frac {\left (2 a^2+3 b^2\right ) \int \sqrt {e \cos (c+d x)}dx+5 a b^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (2 a^2+3 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a b^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {-\frac {\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}+5 a b^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}+5 a b^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {5 a b^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (\frac {b e \int \frac {\sqrt {e \cos (c+d x)}}{b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \cos (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (\frac {2 b e \int \frac {e^2 \cos ^2(c+d x)}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {-\frac {5 a b^2 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )+\frac {2 \left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}\) |
b/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) + (-(((2*(2* a^2 + 3*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + 5*a*b^2*((2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sqr t[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[ e])))/d + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*e *Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])))/((a^2 - b^2)*e^2)) - (2*(5*a*b - (2*a^2 + 3*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Cos[ c + d*x]]))/(2*(a^2 - b^2))
3.6.91.3.1 Defintions of rubi rules used
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 8.68 (sec) , antiderivative size = 2128, normalized size of antiderivative = 4.33
(8/e*a*b*(-1/8/(a^2-b^2)^2/e*2^(1/2)/(cos(1/2*d*x+1/2*c)-1/2*2^(1/2))*(-2* sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)+1/8/(a^2-b^2)^2/e*2^(1/2)/(cos(1/2*d*x+1/2 *c)+1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)-1/64/(a-b)^2/(a+b)^2/ (e^2*(a^2-b^2)/b^2)^(1/4)*(16*(e^2*(a^2-b^2)/b^2)^(1/4)*(cos(1/2*d*x+1/2*c )^2-1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*b^2+(4*cos(1/2*d*x+1/2*c)^4*b^ 2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)*e*2^(1/2)*(ln((2*e*cos(1/2*d*x+1/2*c)^2- e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^ 2*(a^2-b^2)/b^2)^(1/2))/(2*e*cos(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1 /4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))) +2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^2)^ (1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c )^2-e)^(1/2)-(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))))/e/(4* cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)-1/16/(a-b)^2/(a+b )^2/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*(ln((2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2 *(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2 -b^2)/b^2)^(1/2))/(2*e*cos(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2 *e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arc tan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/4)) /(e^2*(a^2-b^2)/b^2)^(1/4))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e) ^(1/2)-(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))))+2*(e*(2*...
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]