Optimal. Leaf size=148 \[ -\frac{\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0583402, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6720, 3357, 3354, 3352, 3351} \[ -\frac{\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 3357
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin ^2\left (a+b x^2\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac{1}{2}-\frac{1}{2} \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\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 ) \int \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{1}{2} \left (\cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \cos \left (2 b x^2\right ) \, dx+\frac{1}{2} \left (\csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin \left (2 b x^2\right ) \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\sqrt{\pi } \cos (2 a) \csc ^2\left (a+b x^2\right ) C\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \csc ^2\left (a+b x^2\right ) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.102306, size = 93, normalized size = 0.63 \[ \frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+2 \sqrt{b} x\right )}{4 \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.087, size = 224, normalized size = 1.5 \begin{align*}{\frac{{{\rm e}^{2\,ib{x}^{2}}}\sqrt{\pi }\sqrt{2}}{16\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}{\it Erf} \left ( \sqrt{2}\sqrt{ib}x \right ){\frac{1}{\sqrt{ib}}}}+{\frac{\sqrt{\pi }{{\rm e}^{2\,i \left ( b{x}^{2}+2\,a \right ) }}}{8\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}{\it Erf} \left ( \sqrt{-2\,ib}x \right ){\frac{1}{\sqrt{-2\,ib}}}}-{\frac{x{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{2\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.72626, size = 483, normalized size = 3.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70179, size = 305, normalized size = 2.06 \begin{align*} \frac{4^{\frac{2}{3}}{\left (4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{C}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) - 4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 2 \cdot 4^{\frac{1}{3}} b x\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}}}{16 \,{\left (b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac{2}{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]