\(\int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx\) [77]

Optimal result

Integrand size = 25, antiderivative size = 590 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\frac {9 a b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}-\frac {9 a b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}-\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}+\frac {9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{5/2}}-\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \sin (c+d x)}}+\frac {9 a^2 b^3 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}+\frac {9 a^2 b^3 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^3 d e^4 \sqrt {\sin (c+d x)}} \] Output:

9/2*a*b^(7/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2) 
)/(-a^2+b^2)^(13/4)/d/e^(7/2)-9/2*a*b^(7/2)*arctanh(b^(1/2)*(e*sin(d*x+c)) 
^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(13/4)/d/e^(7/2)-b/(a^2-b^2)/d 
/e/(a+b*cos(d*x+c))/(e*sin(d*x+c))^(5/2)+1/5*(9*a*b-(2*a^2+7*b^2)*cos(d*x+ 
c))/(a^2-b^2)^2/d/e/(e*sin(d*x+c))^(5/2)-3/5*(15*a*b^3+(2*a^4-10*a^2*b^2-7 
*b^4)*cos(d*x+c))/(a^2-b^2)^3/d/e^3/(e*sin(d*x+c))^(1/2)-9/2*a^2*b^3*Ellip 
ticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+ 
c)^(1/2)/(a^2-b^2)^3/(b-(-a^2+b^2)^(1/2))/d/e^3/(e*sin(d*x+c))^(1/2)-9/2*a 
^2*b^3*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/ 
2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/(b+(-a^2+b^2)^(1/2))/d/e^3/(e*sin(d*x+c)) 
^(1/2)+3/5*(2*a^4-10*a^2*b^2-7*b^4)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^ 
(1/2))*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/d/e^4/sin(d*x+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 7.38 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2)),x]
 

Output:

(Sin[c + d*x]^4*((-2*(20*a*b^3 + 3*a^4*Cos[c + d*x] - 15*a^2*b^2*Cos[c + d 
*x] - 8*b^4*Cos[c + d*x])*Csc[c + d*x])/(5*(a^2 - b^2)^3) - (2*(-2*a*b + a 
^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^3)/(5*(a^2 - b^2)^2) - (b 
^5*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x]))))/(d*(e*Sin[c + d*x] 
)^(7/2)) - (3*Sin[c + d*x]^(7/2)*(((2*a^4*b - 10*a^2*b^3 - 7*b^5)*Cos[c + 
d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[ 
Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin 
[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[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] 
+ Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) 
+ 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^ 
2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12 
*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a 
^5 - 10*a^3*b^2 - 22*a*b^4)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + 
 I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I 
)*Sqrt[b]*Sqrt[Sin[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[Sin[c + d*x]] + I*b*Sin[c + d*x]] 
 + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + 
d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4 
, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c...
 

Rubi [A] (warning: unable to verify)

Time = 2.81 (sec) , antiderivative size = 583, normalized size of antiderivative = 0.99, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3173, 27, 3042, 3345, 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.

\(\displaystyle \int \frac {1}{(e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 3173

\(\displaystyle -\frac {\int -\frac {2 a-7 b \cos (c+d x)}{2 (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}dx}{a^2-b^2}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 a-7 b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}dx}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {2 a+7 b \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}-\frac {2 \int -\frac {3 \left (2 a \left (a^2-4 b^2\right )+b \left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{2 (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{5 e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {2 a \left (a^2-4 b^2\right )+b \left (2 a^2+7 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \int \frac {2 a \left (a^2-4 b^2\right )-b \left (2 a^2+7 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {\frac {3 \left (-\frac {2 \int \frac {\left (2 a \left (a^4-5 b^2 a^2-11 b^4\right )+b \left (2 a^4-10 b^2 a^2-7 b^4\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{2 (a+b \cos (c+d x))}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {\left (2 a \left (a^4-5 b^2 a^2-11 b^4\right )+b \left (2 a^4-10 b^2 a^2-7 b^4\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (2 a \left (a^4-5 b^2 a^2-11 b^4\right )+b \left (2 a^4-10 b^2 a^2-7 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3346

\(\displaystyle \frac {\frac {3 \left (-\frac {\left (2 a^4-10 a^2 b^2-7 b^4\right ) \int \sqrt {e \sin (c+d x)}dx-15 a b^4 \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (-\frac {\left (2 a^4-10 a^2 b^2-7 b^4\right ) \int \sqrt {e \sin (c+d x)}dx-15 a b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}-15 a b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}-15 a b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3180

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {b e \int \frac {\sqrt {e \sin (c+d x)}}{b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \sin (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {2 b e \int \frac {e^2 \sin ^2(c+d x)}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}+\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {2 \left (9 a b-\left (2 a^2+7 b^2\right ) \cos (c+d x)\right )}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}+\frac {3 \left (-\frac {\frac {2 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-15 a b^4 \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \cos (c+d x)+15 a b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}\right )}{5 e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}\)

Input:

Int[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2)),x]
 

Output:

-(b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))) + ((2*( 
9*a*b - (2*a^2 + 7*b^2)*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x]) 
^(5/2)) + (3*((-2*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Cos[c + d*x]))/ 
((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - ((2*(2*a^4 - 10*a^2*b^2 - 7*b^4)* 
EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x 
]]) - 15*a*b^4*((-2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b^2 
)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sqrt[e] 
*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e])) 
)/d + (a*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2] 
*Sqrt[Sin[c + d*x]])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + ( 
a*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[S 
in[c + d*x]])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])))/((a^2 - 
b^2)*e^2)))/(5*(a^2 - b^2)*e^2))/(2*(a^2 - b^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 221

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]
 

rule 266

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]
 

rule 827

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 3121

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]
 

rule 3173

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]
 

rule 3180

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]
 

rule 3284

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]
 

rule 3286

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]
 

rule 3345

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]
 

rule 3346

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1744\) vs. \(2(522)=1044\).

Time = 3.40 (sec) , antiderivative size = 1745, normalized size of antiderivative = 2.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1745\)

Input:

int(1/(a+cos(d*x+c)*b)^2/(e*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

(-2*e^3*a*b*(2*b^4/e^6/(a-b)^3/(a+b)^3*(1/4*(e*sin(d*x+c))^(3/2)/(-b^2*cos 
(d*x+c)^2*e^2+a^2*e^2)+9/32/b^2/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*(ln((e*s 
in(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2 
-b^2)/b^2)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^( 
1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/(e^2*(a^2-b^2)/b 
^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+2*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/ 
4)*(e*sin(d*x+c))^(1/2)-1)))+4/e^6/(a-b)^3/(a+b)^3*b^2/(e*sin(d*x+c))^(1/2 
)-2/5/e^4/(a+b)^2/(a-b)^2/(e*sin(d*x+c))^(5/2))-(cos(d*x+c)^2*e*sin(d*x+c) 
)^(1/2)/e^3*(1/5*(a^2+b^2)/(a^2-b^2)^2/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/s 
in(d*x+c)/(cos(d*x+c)^2-1)*(6*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)* 
sin(d*x+c)^(7/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-3*(1-sin(d*x+ 
c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(7/2)*EllipticF((1-sin(d*x+c)) 
^(1/2),1/2*2^(1/2))+6*cos(d*x+c)^4*sin(d*x+c)-8*cos(d*x+c)^2*sin(d*x+c))+b 
^2*(3*a^2+b^2)/(a^2-b^2)^3*(2*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)* 
sin(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-(1-sin(d*x+c) 
)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^( 
1/2),1/2*2^(1/2))-2*cos(d*x+c)^2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)-2*a^2* 
b^4/(a-b)^2/(a+b)^2*(1/2*b^2/e/a^2/(a^2-b^2)*sin(d*x+c)*(cos(d*x+c)^2*e*si 
n(d*x+c))^(1/2)/(-cos(d*x+c)^2*b^2+a^2)-1/2/a^2/(a^2-b^2)*(1-sin(d*x+c))^( 
1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))**2/(e*sin(d*x+c))**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

integrate(1/((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \] Input:

int(1/((e*sin(c + d*x))^(7/2)*(a + b*cos(c + d*x))^2),x)
 

Output:

int(1/((e*sin(c + d*x))^(7/2)*(a + b*cos(c + d*x))^2), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )}}{\cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{4} b^{2}+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a b +\sin \left (d x +c \right )^{4} a^{2}}d x \right )}{e^{4}} \] Input:

int(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(7/2),x)
 

Output:

(sqrt(e)*int(sqrt(sin(c + d*x))/(cos(c + d*x)**2*sin(c + d*x)**4*b**2 + 2* 
cos(c + d*x)*sin(c + d*x)**4*a*b + sin(c + d*x)**4*a**2),x))/e**4