Optimal. Leaf size=101 \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.0701793, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 3767, 3768, 3770} \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \sec ^6(c+d x) \, dx &=a \int \sec ^6(c+d x) \, dx+b \int \sec ^5(c+d x) \, dx\\ &=\frac{b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} (3 b) \int \sec ^3(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a \tan (c+d x)}{d}+\frac{3 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{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^5(c+d x)}{5 d}+\frac{1}{8} (3 b) \int \sec (c+d x) \, dx\\ &=\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x)}{d}+\frac{3 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{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.324824, size = 88, normalized size = 0.87 \[ \frac{a \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 112, normalized size = 1.1 \begin{align*}{\frac{8\,a\tan \left ( dx+c \right ) }{15\,d}}+{\frac{a\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{4\,a\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{b \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,b\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,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 = 0.987564, size = 144, normalized size = 1.43 \begin{align*} \frac{16 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a - 15 \, b{\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 )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98316, size = 304, normalized size = 3.01 \begin{align*} \frac{45 \, b \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, b \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (64 \, a \cos \left (d x + c\right )^{4} + 45 \, b \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} + 30 \, b \cos \left (d x + c\right ) + 24 \, a\right )} \sin \left (d x + c\right )}{240 \, 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 [A] time = 1.43671, size = 240, normalized size = 2.38 \begin{align*} \frac{45 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (120 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 160 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 160 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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