\(\int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx\) [506]

Optimal result

Integrand size = 27, antiderivative size = 318 \[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

-4/105*a^3*(4*c^2-21*c*d+65*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f+8 
/35*a^3*(c-4*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f-2/7*cos(f*x+e)*(a^ 
3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/d/f-4/105*a^3*(4*c^3-21*c^2*d+62* 
c*d^2+147*d^3)*EllipticE(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)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-4/105*a^3*(c 
^2-d^2)*(4*c^2-21*c*d+65*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)/d^3/f/(c+d*sin(f*x+e))^(1 
/2)
 

Mathematica [A] (verified)

Time = 3.09 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.84 \[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {a^3 \left (16 \left (4 c^4-17 c^3 d+41 c^2 d^2+209 c d^3+147 d^4\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-16 \left (4 c^4-21 c^3 d+61 c^2 d^2+21 c d^3-65 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-2 d \cos (e+f x) \left (16 c^3-84 c^2 d-556 c d^2-126 d^3+18 d^2 (2 c+7 d) \cos (2 (e+f x))+d \left (4 c^2-336 c d-565 d^2\right ) \sin (e+f x)+15 d^3 \sin (3 (e+f x))\right )\right )}{420 d^3 f \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]
 

Output:

-1/420*(a^3*(16*(4*c^4 - 17*c^3*d + 41*c^2*d^2 + 209*c*d^3 + 147*d^4)*Elli 
pticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + 
 d)] - 16*(4*c^4 - 21*c^3*d + 61*c^2*d^2 + 21*c*d^3 - 65*d^4)*EllipticF[(- 
2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2 
*d*Cos[e + f*x]*(16*c^3 - 84*c^2*d - 556*c*d^2 - 126*d^3 + 18*d^2*(2*c + 7 
*d)*Cos[2*(e + f*x)] + d*(4*c^2 - 336*c*d - 565*d^2)*Sin[e + f*x] + 15*d^3 
*Sin[3*(e + f*x)])))/(d^3*f*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 3242, 3042, 3447, 3042, 3502, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}dx\)

\(\Big \downarrow \) 3242

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3447

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

\(\displaystyle \frac {2 \left (\frac {2 \int -\frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}+\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2}{3} \int -\frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

Input:

Int[(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]
 

Output:

(-2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d 
*f) + (2*((4*a^3*(c - 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f 
) - ((2*a^3*(4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x] 
])/(3*f) + ((-2*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*EllipticE[(e - 
 Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d* 
Sin[e + f*x])/(c + d)]) + (2*a^3*(c^2 - d^2)*(4*c^2 - 21*c*d + 65*d^2)*Ell 
ipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d 
)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/3)/(5*d)))/(7*d)
 

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 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*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]
 

rule 3232

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

rule 3242

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 - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* 
(m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 
 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c 
 - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[ 
n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ 
c, 0]))
 

rule 3447

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

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1315\) vs. \(2(299)=598\).

Time = 5.49 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.14

method	result	size
default	\(\text {Expression too large to display}\)	\(1316\)
parts	\(\text {Expression too large to display}\)	\(2443\)

Input:

int((a+sin(f*x+e)*a)^3*(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/105*a^3*(18*sin(f*x+e)^4*c*d^4+115*d^5*sin(f*x+e)^3-63*d^5*sin(f*x+e)^2- 
130*d^5*sin(f*x+e)-4*sin(f*x+e)^2*c^3*d^2-sin(f*x+e)^3*c^2*d^3-8*((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)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)) 
*c^5+294*((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)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),(( 
c-d)/(c+d))^(1/2))*d^5-424*((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)*EllipticF(((c+d*sin(f*x+e 
))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^5+8*((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)*EllipticF 
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-48*((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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^ 
3*d^2+416*((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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),( 
(c-d)/(c+d))^(1/2))*c^2*d^3+48*((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)*EllipticF(((c+d*sin(f 
*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^4+42*((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)*El 
lipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-116*(...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.93 \[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-2/315*(2*(8*a^3*c^4 - 42*a^3*c^3*d + 121*a^3*c^2*d^2 - 84*a^3*c*d^3 - 195 
*a^3*d^4)*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 
7*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 
2*I*c)/d) + 2*(8*a^3*c^4 - 42*a^3*c^3*d + 121*a^3*c^2*d^2 - 84*a^3*c*d^3 - 
 195*a^3*d^4)*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, 
 -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + 
 e) + 2*I*c)/d) + 6*(4*I*a^3*c^3*d - 21*I*a^3*c^2*d^2 + 62*I*a^3*c*d^3 + 1 
47*I*a^3*d^4)*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 
7*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, 
 -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + 
e) - 2*I*c)/d)) + 6*(-4*I*a^3*c^3*d + 21*I*a^3*c^2*d^2 - 62*I*a^3*c*d^3 - 
147*I*a^3*d^4)*sqrt(-1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8 
/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d 
^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f* 
x + e) + 2*I*c)/d)) - 3*(15*a^3*d^4*cos(f*x + e)^3 - 3*(a^3*c*d^3 + 21*a^3 
*d^4)*cos(f*x + e)*sin(f*x + e) + (4*a^3*c^2*d^2 - 21*a^3*c*d^3 - 145*a^3* 
d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)
 

Sympy [F]

\[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=a^{3} \left (\int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}}\, dx\right ) \] Input:

integrate((a+a*sin(f*x+e))**3*(c+d*sin(f*x+e))**(1/2),x)
 

Output:

a**3*(Integral(3*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(3*sq 
rt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(sqrt(c + d*sin(e + f 
*x))*sin(e + f*x)**3, x) + Integral(sqrt(c + d*sin(e + f*x)), x))
 

Maxima [F]

\[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)^3*sqrt(d*sin(f*x + e) + c), x)
 

Giac [F]

\[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate((a*sin(f*x + e) + a)^3*sqrt(d*sin(f*x + e) + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \] Input:

int((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2),x)
 

Output:

int((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=a^{3} \left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x +\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right )\right ) \] Input:

int((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

a**3*(int(sqrt(sin(e + f*x)*d + c),x) + int(sqrt(sin(e + f*x)*d + c)*sin(e 
 + f*x)**3,x) + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x) + 3*int( 
sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x))