\(\int \frac {(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx\) [752]

Optimal result

Integrand size = 27, antiderivative size = 296 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {2 d (7 b c-3 a d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 d \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d)^3 \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b^3 (a+b) f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/3*d^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/f-2/3*d*(-3*a*d+7*b*c)*Ellipt 
icE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1 
/2)/b^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2/3*d*(6*a*b*c*d-3*a^2*d^2-b^2*(2 
*c^2+d^2))*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*( 
(c+d*sin(f*x+e))/(c+d))^(1/2)/b^3/f/(c+d*sin(f*x+e))^(1/2)-2*(-a*d+b*c)^3* 
EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*(( 
c+d*sin(f*x+e))/(c+d))^(1/2)/b^3/(a+b)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 29.51 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.05 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\frac {\frac {4 i \left (-2 a c d+b \left (9 c^2+d^2\right )\right ) \left ((-b c+a d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-a d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-a d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b \sqrt {-\frac {1}{c+d}} (b c-a d)}+\frac {2 i (-7 b c+3 a d) \left (-2 b (c-d) (b c-a d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (a+b) (-b c+a d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+\left (-2 a^2+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-a d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b^2 \sqrt {-\frac {1}{c+d}} (b c-a d)}-4 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}-\frac {2 \left (6 b c^3+7 b c d^2-a d^3\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) \sqrt {c+d \sin (e+f x)}}}{6 b f} \] Input:

Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]),x]
 

Output:

(((4*I)*(-2*a*c*d + b*(9*c^2 + d^2))*((-(b*c) + a*d)*EllipticF[I*ArcSinh[S 
qrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - a*d*Ellip 
ticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Si 
n[e + f*x]]], (c + d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]) 
)/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b*Sqrt[-(c + d)^(-1)]* 
(b*c - a*d)) + ((2*I)*(-7*b*c + 3*a*d)*(-2*b*(c - d)*(b*c - a*d)*EllipticE 
[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] 
 + d*(2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqr 
t[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (-2*a^2 + b^2)*d*EllipticPi[(b* 
(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x 
]]], (c + d)/(c - d)]))*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d 
))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b^2*Sqrt[-(c + d)^(-1)]*(b*c - 
 a*d)) - 4*d^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]] - (2*(6*b*c^3 + 7*b*c 
*d^2 - a*d^3)*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + 
d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) 
)/(6*b*f)
 

Rubi [A] (verified)

Time = 2.61 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 3272, 27, 3042, 3538, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

\(\displaystyle \frac {\frac {d (7 b c-3 a d) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {\left (-\left (\left (2 c^2+d^2\right ) b^2\right )+6 a c d b-3 a^2 d^2\right ) \sin (e+f x) d^2+\left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {d (7 b c-3 a d) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {\left (-\left (\left (2 c^2+d^2\right ) b^2\right )+6 a c d b-3 a^2 d^2\right ) \sin (e+f x) d^2+\left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {\left (-\left (\left (2 c^2+d^2\right ) b^2\right )+6 a c d b-3 a^2 d^2\right ) \sin (e+f x) d^2+\left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {\left (-\left (\left (2 c^2+d^2\right ) b^2\right )+6 a c d b-3 a^2 d^2\right ) \sin (e+f x) d^2+\left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {\left (-\left (\left (2 c^2+d^2\right ) b^2\right )+6 a c d b-3 a^2 d^2\right ) \sin (e+f x) d^2+\left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {3 d (b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {3 d (b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {2 d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {2 d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {2 d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {2 d^2 \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {6 d (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}\)

Input:

Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]),x]
 

Output:

(-2*d^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + ((2*d*(7*b*c - 3* 
a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]] 
)/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((2*d^2*(6*a*b*c*d - 3*a^2*d^ 
2 - b^2*(2*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[( 
c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]]) - (6*d*(b*c - 
 a*d)^3*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[ 
(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]]))/(b* 
d))/(3*b)
 

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 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 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 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1177\) vs. \(2(284)=568\).

Time = 4.93 (sec) , antiderivative size = 1178, normalized size of antiderivative = 3.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1178\)

Input:

int((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x,method=_RETURNVERBOSE)
 

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(d/b^3*(2*a^2*d^2*(c/d-1)*((c+d*si 
n(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/( 
c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f* 
x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-2*b*d*(a*d-3*b*c)*(c/d-1)*((c+d*si 
n(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/( 
c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE((( 
c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+ 
e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+d^2*b^2*(-2/3/d*(-(-c-d*sin(f*x+e)) 
*cos(f*x+e)^2)^(1/2)+2/3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin( 
f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*c 
os(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^ 
(1/2))-4/3*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c 
+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^ 
2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^ 
(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+6*b 
^2*c^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/ 
2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2) 
*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-6*a*b*c*d*( 
c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*( 
1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*Elli...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((c+d*sin(f*x+e))**(5/2)/(a+b*sin(f*x+e)),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(b*sin(f*x + e) + a), x)
 

Giac [F]

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

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

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(b*sin(f*x + e) + a), x)
 

Mupad [F(-1)]

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

int((c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x)),x)
 

Output:

int((c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x)), x)
 

Reduce [F]

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

int((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x)
 

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)*b + a),x)*c**2 + int((sqrt(sin( 
e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)*b + a),x)*d**2 + 2*int((sqr 
t(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)*b + a),x)*c*d