Optimal. Leaf size=78 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan ^3(a+b x) \sec ^3(a+b x)}{6 b}-\frac{\tan (a+b x) \sec ^3(a+b x)}{8 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b} \]
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Rubi [A] time = 0.074756, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan ^3(a+b x) \sec ^3(a+b x)}{6 b}-\frac{\tan (a+b x) \sec ^3(a+b x)}{8 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^3(a+b x) \tan ^4(a+b x) \, dx &=\frac{\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}-\frac{1}{2} \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx\\ &=-\frac{\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac{\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}+\frac{1}{8} \int \sec ^3(a+b x) \, dx\\ &=\frac{\sec (a+b x) \tan (a+b x)}{16 b}-\frac{\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac{\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}+\frac{1}{16} \int \sec (a+b x) \, dx\\ &=\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\sec (a+b x) \tan (a+b x)}{16 b}-\frac{\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac{\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.028292, size = 99, normalized size = 1.27 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan ^3(a+b x) \sec ^3(a+b x)}{3 b}-\frac{\tan (a+b x) \sec ^5(a+b x)}{6 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{24 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 108, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{6\,b \left ( \cos \left ( bx+a \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{24\,b \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{48\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{48\,b}}-{\frac{\sin \left ( bx+a \right ) }{16\,b}}+{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{16\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14841, size = 123, normalized size = 1.58 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (b x + a\right )^{5} + 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75174, size = 227, normalized size = 2.91 \begin{align*} \frac{3 \, \cos \left (b x + a\right )^{6} \log \left (\sin \left (b x + a\right ) + 1\right ) - 3 \, \cos \left (b x + a\right )^{6} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \,{\left (3 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{96 \, b \cos \left (b x + a\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1874, size = 99, normalized size = 1.27 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (b x + a\right )^{5} + 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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