Optimal. Leaf size=117 \[ \frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac{2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac{\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac{7 \cos \left (a+b x^2\right )}{9 b^3}-\frac{x^4 \cos \left (a+b x^2\right )}{3 b}-\frac{x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.130316, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3379, 3311, 3296, 2638, 2633} \[ \frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac{2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac{\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac{7 \cos \left (a+b x^2\right )}{9 b^3}-\frac{x^4 \cos \left (a+b x^2\right )}{3 b}-\frac{x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3379
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int x^5 \sin ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sin ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac{1}{3} \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,x^2\right )-\frac{\operatorname{Subst}\left (\int \sin ^3(a+b x) \, dx,x,x^2\right )}{9 b^2}\\ &=-\frac{x^4 \cos \left (a+b x^2\right )}{3 b}-\frac{x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (a+b x^2\right )\right )}{9 b^3}+\frac{2 \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=\frac{\cos \left (a+b x^2\right )}{9 b^3}-\frac{x^4 \cos \left (a+b x^2\right )}{3 b}-\frac{\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac{2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac{x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac{2 \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{3 b^2}\\ &=\frac{7 \cos \left (a+b x^2\right )}{9 b^3}-\frac{x^4 \cos \left (a+b x^2\right )}{3 b}-\frac{\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac{2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac{x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac{x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.266078, size = 75, normalized size = 0.64 \[ \frac{-81 \left (b^2 x^4-2\right ) \cos \left (a+b x^2\right )+\left (9 b^2 x^4-2\right ) \cos \left (3 \left (a+b x^2\right )\right )-6 b x^2 \left (\sin \left (3 \left (a+b x^2\right )\right )-27 \sin \left (a+b x^2\right )\right )}{216 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 113, normalized size = 1. \begin{align*} -{\frac{3\,{x}^{4}\cos \left ( b{x}^{2}+a \right ) }{8\,b}}+{\frac{3}{2\,b} \left ({\frac{{x}^{2}\sin \left ( b{x}^{2}+a \right ) }{2\,b}}+{\frac{\cos \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \right ) }+{\frac{{x}^{4}\cos \left ( 3\,b{x}^{2}+3\,a \right ) }{24\,b}}-{\frac{1}{6\,b} \left ({\frac{{x}^{2}\sin \left ( 3\,b{x}^{2}+3\,a \right ) }{6\,b}}+{\frac{\cos \left ( 3\,b{x}^{2}+3\,a \right ) }{18\,{b}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.990429, size = 107, normalized size = 0.91 \begin{align*} -\frac{6 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) - 162 \, b x^{2} \sin \left (b x^{2} + a\right ) -{\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right ) + 81 \,{\left (b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )}{216 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.23135, size = 182, normalized size = 1.56 \begin{align*} \frac{{\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )^{3} - 3 \,{\left (9 \, b^{2} x^{4} - 14\right )} \cos \left (b x^{2} + a\right ) - 6 \,{\left (b x^{2} \cos \left (b x^{2} + a\right )^{2} - 7 \, b x^{2}\right )} \sin \left (b x^{2} + a\right )}{54 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.958, size = 143, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{x^{4} \sin ^{2}{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{2 b} - \frac{x^{4} \cos ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac{7 x^{2} \sin ^{3}{\left (a + b x^{2} \right )}}{9 b^{2}} + \frac{2 x^{2} \sin{\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{3 b^{2}} + \frac{7 \sin ^{2}{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{9 b^{3}} + \frac{20 \cos ^{3}{\left (a + b x^{2} \right )}}{27 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{6} \sin ^{3}{\left (a \right )}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12512, size = 165, normalized size = 1.41 \begin{align*} -\frac{\frac{6 \, x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right )}{b} - \frac{162 \, x^{2} \sin \left (b x^{2} + a\right )}{b} - \frac{{\left (9 \,{\left (b x^{2} + a\right )}^{2} - 18 \,{\left (b x^{2} + a\right )} a + 9 \, a^{2} - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right )}{b^{2}} + \frac{81 \,{\left ({\left (b x^{2} + a\right )}^{2} - 2 \,{\left (b x^{2} + a\right )} a + a^{2} - 2\right )} \cos \left (b x^{2} + a\right )}{b^{2}}}{216 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]