Optimal. Leaf size=44 \[ \frac{1}{6} \sin ^3(5 x)+\frac{1}{2} \sin (5 x)+\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \tanh ^{-1}(\sin (5 x)) \]
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Rubi [A] time = 0.0380145, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2592, 288, 302, 206} \[ \frac{1}{6} \sin ^3(5 x)+\frac{1}{2} \sin (5 x)+\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \tanh ^{-1}(\sin (5 x)) \]
Antiderivative was successfully verified.
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Rule 2592
Rule 288
Rule 302
Rule 206
Rubi steps
\begin{align*} \int \sin ^3(5 x) \tan ^3(5 x) \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (5 x)\right )\\ &=\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\sin (5 x)\right )\\ &=\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\sin (5 x)\right )\\ &=\frac{1}{2} \sin (5 x)+\frac{1}{6} \sin ^3(5 x)+\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (5 x)\right )\\ &=-\frac{1}{2} \tanh ^{-1}(\sin (5 x))+\frac{1}{2} \sin (5 x)+\frac{1}{6} \sin ^3(5 x)+\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)\\ \end{align*}
Mathematica [A] time = 0.0448652, size = 52, normalized size = 1.18 \[ -\frac{1}{15} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{3} \sin (5 x) \tan ^2(5 x)-\frac{1}{2} \tanh ^{-1}(\sin (5 x))+\frac{1}{2} \tan (5 x) \sec (5 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 50, normalized size = 1.1 \begin{align*}{\frac{ \left ( \sin \left ( 5\,x \right ) \right ) ^{7}}{10\, \left ( \cos \left ( 5\,x \right ) \right ) ^{2}}}+{\frac{ \left ( \sin \left ( 5\,x \right ) \right ) ^{5}}{10}}+{\frac{ \left ( \sin \left ( 5\,x \right ) \right ) ^{3}}{6}}+{\frac{\sin \left ( 5\,x \right ) }{2}}-{\frac{\ln \left ( \sec \left ( 5\,x \right ) +\tan \left ( 5\,x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943373, size = 66, normalized size = 1.5 \begin{align*} \frac{1}{15} \, \sin \left (5 \, x\right )^{3} - \frac{\sin \left (5 \, x\right )}{10 \,{\left (\sin \left (5 \, x\right )^{2} - 1\right )}} - \frac{1}{4} \, \log \left (\sin \left (5 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\sin \left (5 \, x\right ) - 1\right ) + \frac{2}{5} \, \sin \left (5 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51175, size = 182, normalized size = 4.14 \begin{align*} -\frac{15 \, \cos \left (5 \, x\right )^{2} \log \left (\sin \left (5 \, x\right ) + 1\right ) - 15 \, \cos \left (5 \, x\right )^{2} \log \left (-\sin \left (5 \, x\right ) + 1\right ) + 2 \,{\left (2 \, \cos \left (5 \, x\right )^{4} - 14 \, \cos \left (5 \, x\right )^{2} - 3\right )} \sin \left (5 \, x\right )}{60 \, \cos \left (5 \, x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.115596, size = 51, normalized size = 1.16 \begin{align*} \frac{\log{\left (\sin{\left (5 x \right )} - 1 \right )}}{4} - \frac{\log{\left (\sin{\left (5 x \right )} + 1 \right )}}{4} + \frac{\sin ^{3}{\left (5 x \right )}}{15} + \frac{2 \sin{\left (5 x \right )}}{5} - \frac{\sin{\left (5 x \right )}}{5 \left (2 \sin ^{2}{\left (5 x \right )} - 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.06945, size = 657, normalized size = 14.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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