\(\int x^3 (a+a \sin (e+f x))^{3/2} \, dx\) [128]

Optimal result

Integrand size = 18, antiderivative size = 337 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=-\frac {1280 a \sqrt {a+a \sin (e+f x)}}{9 f^4}+\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {4 a x^3 \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2} \] Output:

-1280/9*a*(a+a*sin(f*x+e))^(1/2)/f^4+16*a*x^2*(a+a*sin(f*x+e))^(1/2)/f^2+6 
40/9*a*x*cot(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/f^3-8/3*a*x^3*co 
t(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/f+32/9*a*x*cos(1/2*e+1/4*Pi 
+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/f^3-4/3*a*x^3*c 
os(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/ 
f-64/27*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f*x+e))^(1/2)/f^4+8/3*a*x^2 
*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f*x+e))^(1/2)/f^2
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.69 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\frac {2 a \left (-\frac {2 \left (\left (968-480 f x-117 f^2 x^2+18 f^3 x^3\right ) \cos \left (\frac {e}{2}\right )+\left (968+480 f x-117 f^2 x^2-18 f^3 x^3\right ) \sin \left (\frac {e}{2}\right )\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}-\cos (f x) \left (3 f x \left (-8+3 f^2 x^2\right ) \cos (e)+2 \left (8-9 f^2 x^2\right ) \sin (e)\right )+\left (2 \left (-8+9 f^2 x^2\right ) \cos (e)+3 f x \left (-8+3 f^2 x^2\right ) \sin (e)\right ) \sin (f x)+\frac {24 f x \left (-80+3 f^2 x^2\right ) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) \sqrt {a (1+\sin (e+f x))}}{27 f^4} \] Input:

Integrate[x^3*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

(2*a*((-2*((968 - 480*f*x - 117*f^2*x^2 + 18*f^3*x^3)*Cos[e/2] + (968 + 48 
0*f*x - 117*f^2*x^2 - 18*f^3*x^3)*Sin[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f 
*x]*(3*f*x*(-8 + 3*f^2*x^2)*Cos[e] + 2*(8 - 9*f^2*x^2)*Sin[e]) + (2*(-8 + 
9*f^2*x^2)*Cos[e] + 3*f*x*(-8 + 3*f^2*x^2)*Sin[e])*Sin[f*x] + (24*f*x*(-80 
 + 3*f^2*x^2)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin 
[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^4)
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.08, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3800, 3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3118, 3791, 3042, 3777, 3042, 3118}

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.

\(\displaystyle \int x^3 (a \sin (e+f x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 (a \sin (e+f x)+a)^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x^3 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x^3 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{3 f^2}+\frac {2}{3} \int x^3 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \int x^3 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \int x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {4 \int -x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}+\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {2 \int \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {2 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \left (\frac {2}{3} \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \left (\frac {2}{3} \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \left (\frac {2}{3} \left (\frac {2 \int \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \left (\frac {2}{3} \left (\frac {2 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {8 \left (\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

Input:

Int[x^3*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

2*a*Csc[e/2 + Pi/4 + (f*x)/2]*((-2*x^3*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + 
 Pi/4 + (f*x)/2]^2)/(3*f) + (4*x^2*Sin[e/2 + Pi/4 + (f*x)/2]^3)/(3*f^2) - 
(8*((-2*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]^2)/(3*f) + ( 
4*Sin[e/2 + Pi/4 + (f*x)/2]^3)/(9*f^2) + (2*((-2*x*Cos[e/2 + Pi/4 + (f*x)/ 
2])/f + (4*Sin[e/2 + Pi/4 + (f*x)/2])/f^2))/3))/(3*f^2) + (2*((-2*x^3*Cos[ 
e/2 + Pi/4 + (f*x)/2])/f + (6*((2*x^2*Sin[e/2 + Pi/4 + (f*x)/2])/f - (4*(( 
-2*x*Cos[e/2 + Pi/4 + (f*x)/2])/f + (4*Sin[e/2 + Pi/4 + (f*x)/2])/f^2))/f) 
)/f))/3)*Sqrt[a + a*Sin[e + f*x]]
 

Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int x^{3} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\int x^{3} \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \] Input:

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

Output:

Integral(x**3*(a*(sin(e + f*x) + 1))**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((a*sin(f*x + e) + a)^(3/2)*x^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (259) = 518\).

Time = 0.47 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.93 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/216*sqrt(2)*sqrt(a)*(972*(pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2 
*pi*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + (pi - 2*f*x 
 - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*pi*a*e*sgn(cos(-1/4*pi 
 + 1/2*f*x + 1/2*e)) + 4*(pi - 2*f*x - 2*e)*a*e*sgn(cos(-1/4*pi + 1/2*f*x 
+ 1/2*e)) + 4*a*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(-1/ 
4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)/f^3 + 4*(9*pi^2*a 
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18*pi*(pi - 2*f*x - 2*e)*a*sgn(cos( 
-1/4*pi + 1/2*f*x + 1/2*e)) + 9*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1 
/2*f*x + 1/2*e)) - 36*pi*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*(pi 
- 2*f*x - 2*e)*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*a*e^2*sgn(cos( 
-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*co 
s(-3/4*pi + 3/2*f*x + 3/2*e)/f^3 + 81*(pi^3*a*sgn(cos(-1/4*pi + 1/2*f*x + 
1/2*e)) - 3*pi^2*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) 
+ 3*pi*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 
2*f*x - 2*e)^3*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 6*pi^2*a*e*sgn(cos( 
-1/4*pi + 1/2*f*x + 1/2*e)) + 12*pi*(pi - 2*f*x - 2*e)*a*e*sgn(cos(-1/4*pi 
 + 1/2*f*x + 1/2*e)) - 6*(pi - 2*f*x - 2*e)^2*a*e*sgn(cos(-1/4*pi + 1/2*f* 
x + 1/2*e)) + 12*pi*a*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 12*(pi - 2 
*f*x - 2*e)*a*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 8*a*e^3*sgn(cos(-1 
/4*pi + 1/2*f*x + 1/2*e)) - 96*pi*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sin \left (f x +e \right )+1}\, x^{3}d x +\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right ) x^{3}d x \right ) \] Input:

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

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1)*x**3,x) + int(sqrt(sin(e + f*x) + 1) 
*sin(e + f*x)*x**3,x))