Optimal. Leaf size=74 \[ \frac{a \sec ^4(c+d x)}{4 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.105266, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2834, 2606, 30, 2611, 3768, 3770} \[ \frac{a \sec ^4(c+d x)}{4 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2606
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec ^4(c+d x) \tan (c+d x) \, dx+b \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{b \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{4} b \int \sec ^3(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int x^3 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a \sec ^4(c+d x)}{4 d}-\frac{b \sec (c+d x) \tan (c+d x)}{8 d}+\frac{b \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{8} b \int \sec (c+d x) \, dx\\ &=-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{b \sec (c+d x) \tan (c+d x)}{8 d}+\frac{b \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.023758, size = 74, normalized size = 1. \[ \frac{a \sec ^4(c+d x)}{4 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 92, normalized size = 1.2 \begin{align*}{\frac{a}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b\sin \left ( dx+c \right ) }{8\,d}}-{\frac{b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00289, size = 101, normalized size = 1.36 \begin{align*} -\frac{b \log \left (\sin \left (d x + c\right ) + 1\right ) - b \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (b \sin \left (d x + c\right )^{3} + b \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98192, size = 212, normalized size = 2.86 \begin{align*} -\frac{b \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - b \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (b \cos \left (d x + c\right )^{2} - 2 \, b\right )} \sin \left (d x + c\right ) - 4 \, a}{16 \, d \cos \left (d x + c\right )^{4}} \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 = 1.23533, size = 90, normalized size = 1.22 \begin{align*} -\frac{b \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - b \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (b \sin \left (d x + c\right )^{3} + b \sin \left (d x + c\right ) + 2 \, a\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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