\(\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx\) [79]

Optimal result

Integrand size = 25, antiderivative size = 604 \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=-\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}-\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2} \] Output:

-9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/ 
(-a^2+b^2)^(1/4)/e^(1/2))/b^(11/2)/(-a^2+b^2)^(3/4)/d-9/8*(7*a^4-9*a^2*b^2 
+2*b^4)*e^(11/2)*arctanh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^( 
1/2))/b^(11/2)/(-a^2+b^2)^(3/4)/d+3/4*a*(21*a^2-13*b^2)*e^6*InverseJacobiA 
M(1/2*c-1/4*Pi+1/2*d*x,2^(1/2))*sin(d*x+c)^(1/2)/b^6/d/(e*sin(d*x+c))^(1/2 
)+9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*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)/b^6/(a^2-b*(b-(-a^2+b^2) 
^(1/2)))/d/(e*sin(d*x+c))^(1/2)+9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*Elliptic 
Pi(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)/b^6/(a^2-b*(b+(-a^2+b^2)^(1/2)))/d/(e*sin(d*x+c))^(1/2)-3/4*e^5*(21* 
a^2-6*b^2-7*a*b*cos(d*x+c))*(e*sin(d*x+c))^(1/2)/b^5/d+9/20*e^3*(7*a+2*b*c 
os(d*x+c))*(e*sin(d*x+c))^(5/2)/b^3/d/(a+b*cos(d*x+c))+1/2*e*(e*sin(d*x+c) 
)^(9/2)/b/d/(a+b*cos(d*x+c))^2
 

Mathematica [C] (warning: unable to verify)

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

Time = 17.64 (sec) , antiderivative size = 2024, normalized size of antiderivative = 3.35 \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]
 

Output:

(((2*a*Cos[c + d*x])/b^4 + (-a^2 + b^2)^2/(2*b^5*(a + b*Cos[c + d*x])^2) - 
 (17*a*(a^2 - b^2))/(4*b^5*(a + b*Cos[c + d*x])) - Cos[2*(c + d*x)]/(5*b^3 
))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d + (3*(e*Sin[c + d*x])^(11/2)* 
((2*(25*a^3 - 37*a*b^2)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(( 
a*(-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)] - Lo 
g[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]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/ 
4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Si 
n[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/( 
(-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d 
*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2 
, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9 
/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a 
^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x] 
^2)) + (2*(30*a^2*b - 16*b^3)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2] 
)*(((-1/8 + I/8)*Sqrt[b]*(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^...
 

Rubi [A] (warning: unable to verify)

Time = 2.60 (sec) , antiderivative size = 562, normalized size of antiderivative = 0.93, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3172, 25, 3042, 3342, 27, 3042, 3344, 27, 3042, 3346, 3042, 3121, 3042, 3120, 3181, 266, 756, 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.

\(\displaystyle \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{11/2}}{\left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3342

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3344

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e^2 \int -\frac {2 b \left (7 a^2-3 b^2\right )+a \left (21 a^2-13 b^2\right ) \cos (c+d x)}{2 (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 b^2}+\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \int \frac {2 b \left (7 a^2-3 b^2\right )+a \left (21 a^2-13 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \int \frac {2 b \left (7 a^2-3 b^2\right )-a \left (21 a^2-13 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3346

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^2-13 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{b}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^2-13 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{b}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{b \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{b \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3181

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2\right )}d(e \sin (c+d x))}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {2 b e \int \frac {1}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {3 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \left (-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}+\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}-\frac {e (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{5 b^2 d (a+b \cos (c+d x))}\right )}{4 b}\)

Input:

Int[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]
 

Output:

(e*(e*Sin[c + d*x])^(9/2))/(2*b*d*(a + b*Cos[c + d*x])^2) - (9*e^2*(-1/5*( 
e*(7*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(b^2*d*(a + b*Cos[c + d 
*x])) + (e^2*((2*e*(3*(7*a^2 - 2*b^2) - 7*a*b*Cos[c + d*x])*Sqrt[e*Sin[c + 
 d*x]])/(3*b^2*d) - (e^2*((2*a*(21*a^2 - 13*b^2)*EllipticF[(c - Pi/2 + d*x 
)/2, 2]*Sqrt[Sin[c + d*x]])/(b*d*Sqrt[e*Sin[c + d*x]]) - (3*(7*a^4 - 9*a^2 
*b^2 + 2*b^4)*((-2*b*e*(-1/2*ArcTan[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + 
 b^2)^(1/4)]/(Sqrt[b]*(-a^2 + b^2)^(3/4)*e^(3/2)) - ArcTanh[(Sqrt[b]*Sqrt[ 
e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*Sqrt[b]*(-a^2 + b^2)^(3/4)*e^(3/2) 
)))/d + (a*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2] 
*Sqrt[Sin[c + d*x]])/(Sqrt[-a^2 + b^2]*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin 
[c + d*x]]) - (a*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x) 
/2, 2]*Sqrt[Sin[c + d*x]])/(Sqrt[-a^2 + b^2]*(b + Sqrt[-a^2 + b^2])*d*Sqrt 
[e*Sin[c + d*x]])))/b))/(3*b^2)))/(2*b^2)))/(4*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   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 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 3172

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x 
])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1)))   Int[(g*Cos 
[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre 
eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I 
ntegersQ[2*m, 2*p]
 

rule 3181

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)* 
(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[-a/(2*q)   Int[1/( 
Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Simp[b*(g/f)   Subst[ 
Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - S 
imp[a/(2*q)   Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), 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 3342

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*C 
os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p 
 + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( 
p - 1)/(b^2*(m + 1)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin 
[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x 
], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
 

rule 3344

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* 
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* 
p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( 
(p - 1)/(b^2*(m + p)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si 
n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 
2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 
, 0] && 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. \(2753\) vs. \(2(530)=1060\).

Time = 101.66 (sec) , antiderivative size = 2754, normalized size of antiderivative = 4.56

method	result	size
default	\(\text {Expression too large to display}\)	\(2754\)

Input:

int((e*sin(d*x+c))^(11/2)/(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
 

Output:

(2*e^3*b*(-1/5/b^6*(e*sin(d*x+c))^(1/2)*e^2*(cos(d*x+c)^2*b^2+30*a^2-11*b^ 
2)+e^4/b^6*(-1/8*(e*sin(d*x+c))^(1/2)*e^2*(-19*cos(d*x+c)^2*a^4*b^2+21*a^2 
*b^4*cos(d*x+c)^2-2*b^6*cos(d*x+c)^2+15*a^6-13*a^4*b^2-2*b^4*a^2)/(-b^2*co 
s(d*x+c)^2*e^2+a^2*e^2)^2+9/64*(7*a^4-9*a^2*b^2+2*b^4)*(e^2*(a^2-b^2)/b^2) 
^(1/4)/(a^2*e^2-b^2*e^2)*2^(1/2)*(ln((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))/(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))))-(co 
s(d*x+c)^2*e*sin(d*x+c))^(1/2)*e^6*a*(1/b^6/(cos(d*x+c)^2*e*sin(d*x+c))^(1 
/2)*(10*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*Ellip 
ticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2-7*(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) 
)*b^2-2*b^2*cos(d*x+c)^2*sin(d*x+c))+3/b^6*(7*a^4-10*a^2*b^2+3*b^4)*(-1/2/ 
b/(-a^2+b^2)^(1/2)*(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))^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi( 
(1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/2/b/(-a^2+b^2 
)^(1/2)*(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))^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((1-sin(d*x+ 
c))^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2)))-3/b^6*(5*a^6-11*a^4*b^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((e*sin(d*x+c))**(11/2)/(a+b*cos(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate((e*sin(d*x + c))^(11/2)/(b*cos(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{11/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int((e*sin(c + d*x))^(11/2)/(a + b*cos(c + d*x))^3,x)
 

Output:

int((e*sin(c + d*x))^(11/2)/(a + b*cos(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) e^{5} \] Input:

int((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x)
 

Output:

sqrt(e)*int((sqrt(sin(c + d*x))*sin(c + d*x)**5)/(cos(c + d*x)**3*b**3 + 3 
*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*e**5