Optimal. Leaf size=74 \[ \frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0808342, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3090, 3768, 3770, 2606, 30} \[ \frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3768
Rule 3770
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int \left (a \sec ^5(c+d x)+b \sec ^5(c+d x) \tan (c+d x)\right ) \, dx\\ &=a \int \sec ^5(c+d x) \, dx+b \int \sec ^5(c+d x) \tan (c+d x) \, dx\\ &=\frac{a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} (3 a) \int \sec ^3(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{b \sec ^5(c+d x)}{5 d}+\frac{3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (3 a) \int \sec (c+d x) \, dx\\ &=\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \sec ^5(c+d x)}{5 d}+\frac{3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.197467, size = 68, normalized size = 0.92 \[ \frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d}+\frac{b \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 74, normalized size = 1. \begin{align*}{\frac{a \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,a\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{b}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16705, size = 116, normalized size = 1.57 \begin{align*} -\frac{5 \, a{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{16 \, b}{\cos \left (d x + c\right )^{5}}}{80 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.513423, size = 242, normalized size = 3.27 \begin{align*} \frac{15 \, a \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, a \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 10 \,{\left (3 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 16 \, b}{80 \, d \cos \left (d x + c\right )^{5}} \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 [B] time = 1.19875, size = 190, normalized size = 2.57 \begin{align*} \frac{15 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (25 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 40 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 10 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 80 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 10 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 25 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5}}}{40 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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