Integrand size = 18, antiderivative size = 337 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=-\frac {1280 a \sqrt {a+a \sin (e+f x)}}{9 f^4}+\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {4 a x^3 \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2} \]
-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
Time = 0.77 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.69 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\frac {2 a \left (-\frac {2 \left (\left (968-480 f x-117 f^2 x^2+18 f^3 x^3\right ) \cos \left (\frac {e}{2}\right )+\left (968+480 f x-117 f^2 x^2-18 f^3 x^3\right ) \sin \left (\frac {e}{2}\right )\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}-\cos (f x) \left (3 f x \left (-8+3 f^2 x^2\right ) \cos (e)+2 \left (8-9 f^2 x^2\right ) \sin (e)\right )+\left (2 \left (-8+9 f^2 x^2\right ) \cos (e)+3 f x \left (-8+3 f^2 x^2\right ) \sin (e)\right ) \sin (f x)+\frac {24 f x \left (-80+3 f^2 x^2\right ) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) \sqrt {a (1+\sin (e+f x))}}{27 f^4} \]
(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)
Time = 1.39 (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 |
\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {4 x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {4 \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}\right )}{f}-\frac {2 x^3 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {8 \left (\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {2 x^3 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\) |
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]]
3.2.28.3.1 Defintions of rubi rules used
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]
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]
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]
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])
\[\int x^{3} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\int x^{3} \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \]
\[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (259) = 518\).
Time = 0.44 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.93 \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \]
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))...
Timed out. \[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx=\int x^3\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]