Optimal. Leaf size=65 \[ \frac {1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6715, 3207, 2635, 8} \[ \frac {1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 3207
Rule 6715
Rubi steps
\begin {align*} \int x \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (c \sin ^3(a+b x)\right )^{2/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname {Subst}\left (\int \sin ^2(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac {1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=-\frac {\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac {1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 55, normalized size = 0.85 \[ \frac {\left (2 \left (a+b x^2\right )-\sin \left (2 \left (a+b x^2\right )\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 72, normalized size = 1.11 \[ -\frac {{\left (b x^{2} - \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {2}{3}}}{4 \, {\left (b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {2}{3}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 182, normalized size = 2.80 \[ -\frac {x^{2} \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{4 i \left (b \,x^{2}+a \right )}}{16 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}}}{16 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 28, normalized size = 0.43 \[ -\frac {{\left (2 \, b x^{2} - \sin \left (2 \, b x^{2} + 2 \, a\right )\right )} c^{\frac {2}{3}}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________