\(\int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx\) [1398]

Optimal result

Integrand size = 33, antiderivative size = 601 \[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=-\frac {a^4 \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{7/4} f g^{5/2}}-\frac {a^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{7/4} f g^{5/2}}-\frac {2 b}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac {2 a^2 \sqrt {g \cos (e+f x)}}{b \left (a^2-b^2\right ) f g^3}-\frac {2 b \sqrt {g \cos (e+f x)}}{\left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {2 a^3 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b^2 \left (a^2-b^2\right ) f g^2 \sqrt {g \cos (e+f x)}}-\frac {a^5 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^2 \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) f g^2 \sqrt {g \cos (e+f x)}}-\frac {a^5 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^2 \left (a^2-b^2\right ) \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {2 a \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}} \] Output:

-a^4*arctan(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^(1/4)/g^(1/2))/b^(3/2) 
/(-a^2+b^2)^(7/4)/f/g^(5/2)-a^4*arctanh(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2 
+b^2)^(1/4)/g^(1/2))/b^(3/2)/(-a^2+b^2)^(7/4)/f/g^(5/2)-2/3*b/(a^2-b^2)/f/ 
g/(g*cos(f*x+e))^(3/2)+2*a^2*(g*cos(f*x+e))^(1/2)/b/(a^2-b^2)/f/g^3-2*b*(g 
*cos(f*x+e))^(1/2)/(a^2-b^2)/f/g^3-4/3*a*cos(f*x+e)^(1/2)*InverseJacobiAM( 
1/2*f*x+1/2*e,2^(1/2))/(a^2-b^2)/f/g^2/(g*cos(f*x+e))^(1/2)+2*a^3*cos(f*x+ 
e)^(1/2)*InverseJacobiAM(1/2*f*x+1/2*e,2^(1/2))/b^2/(a^2-b^2)/f/g^2/(g*cos 
(f*x+e))^(1/2)-a^5*cos(f*x+e)^(1/2)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(b-( 
-a^2+b^2)^(1/2)),2^(1/2))/b^2/(a^2-b^2)/(a^2-b*(b-(-a^2+b^2)^(1/2)))/f/g^2 
/(g*cos(f*x+e))^(1/2)-a^5*cos(f*x+e)^(1/2)*EllipticPi(sin(1/2*f*x+1/2*e),2 
*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))/b^2/(a^2-b^2)/(a^2-b*(b+(-a^2+b^2)^(1/2)) 
)/f/g^2/(g*cos(f*x+e))^(1/2)+2/3*a*sin(f*x+e)/(a^2-b^2)/f/g/(g*cos(f*x+e)) 
^(3/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 26.85 (sec) , antiderivative size = 1958, normalized size of antiderivative = 3.26 \[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:

Integrate[Sin[e + f*x]^4/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]
 

Output:

(2*Cos[e + f*x]*(-b + a*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^( 
5/2)) + (Cos[e + f*x]^(5/2)*((-2*(-7*a^2 + 3*b^2)*(a + b*Sqrt[1 - Cos[e + 
f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2 
*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^ 
2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + 
f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2 
, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 
9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*( 
a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 
 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + 
 I)*Sqrt[b]*Sqrt[Cos[e + f*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[e + f*x]] + I*b*Cos[e + f*x 
]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e 
+ f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - 
Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + ((3*a^2 - 3*b^2)*(a + b*Sqrt[1 - C 
os[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ( 
(1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + 
b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt 
[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqr 
t[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b...
 

Rubi [A] (warning: unable to verify)

Time = 3.18 (sec) , antiderivative size = 542, normalized size of antiderivative = 0.90, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {3042, 3381, 3042, 3045, 27, 244, 2009, 3046, 3042, 3121, 3042, 3120, 3388, 3042, 3045, 15, 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.

\(\displaystyle \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (e+f x)^4}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))}dx\)

\(\Big \downarrow \) 3381

\(\displaystyle -\frac {a^2 \int \frac {\sin ^2(e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}-\frac {b \int \frac {\sin ^3(e+f x)}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}+\frac {a \int \frac {\sin ^2(e+f x)}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \int \frac {\sin (e+f x)^2}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}-\frac {b \int \frac {\sin (e+f x)^3}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}\)

\(\Big \downarrow \) 3045

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {b \int \frac {g^2-g^2 \cos ^2(e+f x)}{(g \cos (e+f x))^{5/2}}d(g \cos (e+f x))}{f g^3 \left (a^2-b^2\right )}+\frac {a \int \frac {\sin (e+f x)^2}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}\)

\(\Big \downarrow \) 244

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {b \int \left (\frac {g^2}{(g \cos (e+f x))^{5/2}}-\frac {1}{\sqrt {g \cos (e+f x)}}\right )d(g \cos (e+f x))}{f g^3 \left (a^2-b^2\right )}+\frac {a \int \frac {\sin (e+f x)^2}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \int \frac {\sin (e+f x)^2}{(g \cos (e+f x))^{5/2}}dx}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3046

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {2 \int \frac {1}{\sqrt {g \cos (e+f x)}}dx}{3 g^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {2 \int \frac {1}{\sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{3 g^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3121

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {2 \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx}{3 g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {2 \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{3 g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {a^2 \int \frac {\sin (e+f x)^2}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3388

\(\displaystyle -\frac {a^2 \left (\frac {\int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a \int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \left (\frac {\int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a \int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3045

\(\displaystyle -\frac {a^2 \left (-\frac {a \int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {\int \frac {1}{\sqrt {g \cos (e+f x)}}d(g \cos (e+f x))}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {a^2 \left (-\frac {a \int \frac {\sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3346

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {\int \frac {1}{\sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3121

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {\sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx}{b \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {\sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{b \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3181

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (\frac {b g \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b^2 \cos ^2(e+f x) g^2+\left (a^2-b^2\right ) g^2\right )}d(g \cos (e+f x))}{f}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (\frac {2 b g \int \frac {1}{b^2 g^4 \cos ^4(e+f x)+\left (a^2-b^2\right ) g^2}d\sqrt {g \cos (e+f x)}}{f}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (\frac {2 b g \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} g-b g^2 \cos ^2(e+f x)}d\sqrt {g \cos (e+f x)}}{2 g \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b g^2 \cos ^2(e+f x)+\sqrt {b^2-a^2} g}d\sqrt {g \cos (e+f x)}}{2 g \sqrt {b^2-a^2}}\right )}{f}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (\frac {2 b g \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} g-b g^2 \cos ^2(e+f x)}d\sqrt {g \cos (e+f x)}}{2 g \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b g \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )} \left (b \sin \left (e+f x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b g \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (-\frac {a \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {b^2-a^2}-b \cos (e+f x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {g \cos (e+f x)}}-\frac {a \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \left (b \cos (e+f x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {g \cos (e+f x)}}+\frac {2 b g \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (-\frac {a \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {g \cos (e+f x)}}-\frac {a \sqrt {\cos (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )} \left (b \sin \left (e+f x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {g \cos (e+f x)}}+\frac {2 b g \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {a^2 \left (-\frac {a \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{b f \sqrt {g \cos (e+f x)}}-\frac {a \left (\frac {2 b g \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {g} \cos (e+f x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} g^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{f}+\frac {a \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{f \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}-\frac {a \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{f \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}\right )}{b}\right )}{b}-\frac {2 \sqrt {g \cos (e+f x)}}{b f g}\right )}{g^2 \left (a^2-b^2\right )}+\frac {a \left (\frac {2 \sin (e+f x)}{3 f g (g \cos (e+f x))^{3/2}}-\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{3 f g^2 \sqrt {g \cos (e+f x)}}\right )}{a^2-b^2}+\frac {b \left (-\frac {2 g^2}{3 (g \cos (e+f x))^{3/2}}-2 \sqrt {g \cos (e+f x)}\right )}{f g^3 \left (a^2-b^2\right )}\)

Input:

Int[Sin[e + f*x]^4/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]
 

Output:

(b*((-2*g^2)/(3*(g*Cos[e + f*x])^(3/2)) - 2*Sqrt[g*Cos[e + f*x]]))/((a^2 - 
 b^2)*f*g^3) - (a^2*((-2*Sqrt[g*Cos[e + f*x]])/(b*f*g) - (a*((2*Sqrt[Cos[e 
 + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*f*Sqrt[g*Cos[e + f*x]]) - (a*((2*b* 
g*(-1/2*ArcTan[(Sqrt[b]*Sqrt[g]*Cos[e + f*x])/(-a^2 + b^2)^(1/4)]/(Sqrt[b] 
*(-a^2 + b^2)^(3/4)*g^(3/2)) - ArcTanh[(Sqrt[b]*Sqrt[g]*Cos[e + f*x])/(-a^ 
2 + b^2)^(1/4)]/(2*Sqrt[b]*(-a^2 + b^2)^(3/4)*g^(3/2))))/f + (a*Sqrt[Cos[e 
 + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(Sqrt[- 
a^2 + b^2]*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a*Sqrt[Cos[e 
+ f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(Sqrt[-a 
^2 + b^2]*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])))/b))/b))/((a^2 - 
 b^2)*g^2) + (a*((-4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*f*g^ 
2*Sqrt[g*Cos[e + f*x]]) + (2*Sin[e + f*x])/(3*f*g*(g*Cos[e + f*x])^(3/2))) 
)/(a^2 - b^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3045

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ 
Symbol] :> Simp[-(a*f)^(-1)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], 
x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && 
 !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
 

rule 3046

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 
1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1)))   Int[(a*Sin[e + f 
*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && 
GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
 

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 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 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]
 

rule 3381

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a*(d^2/(a^2 
- b^2))   Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Simp[ 
b*(d/(a^2 - b^2))   Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] 
 - Simp[a^2*(d^2/(g^2*(a^2 - b^2)))   Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[ 
e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g 
}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1 
]
 

rule 3388

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b   Int[(g 
*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a*(d/b)   Int[(g*C 
os[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; F 
reeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && 
LtQ[-1, p, 1] && GtQ[n, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1138\) vs. \(2(541)=1082\).

Time = 5.28 (sec) , antiderivative size = 1139, normalized size of antiderivative = 1.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1139\)

Input:

int(sin(f*x+e)^4/(g*cos(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x,method=_RETURNVER 
BOSE)
                                                                                    
                                                                                    
 

Output:

(64/g^2*b*(-1/384*2^(1/2)/(a^2-b^2)*(-2*g*sin(1/2*f*x+1/2*e)^2+g)^(1/2)*(- 
2^(1/2)+cos(1/2*f*x+1/2*e))/g/(2*2^(1/2)*cos(1/2*f*x+1/2*e)+2*sin(1/2*f*x+ 
1/2*e)^2-3)+1/32/b^2/g*(g*(-1+2*cos(1/2*f*x+1/2*e)^2))^(1/2)-1/384*2^(1/2) 
/(a^2-b^2)*(-2*g*sin(1/2*f*x+1/2*e)^2+g)^(1/2)*(2^(1/2)+cos(1/2*f*x+1/2*e) 
)/g/(2*2^(1/2)*cos(1/2*f*x+1/2*e)-2*sin(1/2*f*x+1/2*e)^2+3)+1/16*a^4/(a-b) 
/(a+b)/b^2*(g^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*(-ln((-(g^2*(a^2-b^2)/b^2)^(1 
/4)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)*2^(1/2)-2*g*cos(1/2*f*x+1/2*e)^2-(g 
^2*(a^2-b^2)/b^2)^(1/2)+g)/((g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2 
*e)^2-g)^(1/2)*2^(1/2)-2*g*cos(1/2*f*x+1/2*e)^2-(g^2*(a^2-b^2)/b^2)^(1/2)+ 
g))-2*arctan(2^(1/2)/(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^2-g 
)^(1/2)+1)+2*arctan(-2^(1/2)/(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/ 
2*e)^2-g)^(1/2)+1))/(16*a^2-16*b^2)/g)-32*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)*s 
in(1/2*f*x+1/2*e)^2)^(1/2)*a/g^2*(1/16/(a^2-b^2)*(-1/6*cos(1/2*f*x+1/2*e)/ 
g*(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f*x+1/2*e)^2))^(1/2)/(cos(1/2*f*x+1/ 
2*e)^2-1/2)^2+1/3*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^ 
(1/2)/(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f*x+1/2*e)^2))^(1/2)*EllipticF(c 
os(1/2*f*x+1/2*e),2^(1/2)))+1/16/b^2*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos 
(1/2*f*x+1/2*e)^2)^(1/2)/(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f*x+1/2*e)^2) 
)^(1/2)*EllipticF(cos(1/2*f*x+1/2*e),2^(1/2))-1/256*a^4/(a-b)/(a+b)/b^4*su 
m(1/_alpha/(2*_alpha^2-1)*(2^(1/2)/(g*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(sin(f*x+e)^4/(g*cos(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm= 
"fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(sin(f*x+e)**4/(g*cos(f*x+e))**(5/2)/(a+b*sin(f*x+e)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \] Input:

integrate(sin(f*x+e)^4/(g*cos(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm= 
"maxima")
 

Output:

integrate(sin(f*x + e)^4/((g*cos(f*x + e))^(5/2)*(b*sin(f*x + e) + a)), x)
 

Giac [F]

\[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \] Input:

integrate(sin(f*x+e)^4/(g*cos(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm= 
"giac")
 

Output:

integrate(sin(f*x + e)^4/((g*cos(f*x + e))^(5/2)*(b*sin(f*x + e) + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:

int(sin(e + f*x)^4/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))),x)
 

Output:

int(sin(e + f*x)^4/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x)
 

Reduce [F]

\[ \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \left (\int \frac {\sqrt {\cos \left (f x +e \right )}\, \sin \left (f x +e \right )^{4}}{\cos \left (f x +e \right )^{3} \sin \left (f x +e \right ) b +\cos \left (f x +e \right )^{3} a}d x \right )}{g^{3}} \] Input:

int(sin(f*x+e)^4/(g*cos(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x)
 

Output:

(sqrt(g)*int((sqrt(cos(e + f*x))*sin(e + f*x)**4)/(cos(e + f*x)**3*sin(e + 
 f*x)*b + cos(e + f*x)**3*a),x))/g**3