Optimal. Leaf size=60 \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.0706185, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3090, 3767, 2606, 30} \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3767
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int \left (a \sec ^6(c+d x)+b \sec ^6(c+d x) \tan (c+d x)\right ) \, dx\\ &=a \int \sec ^6(c+d x) \, dx+b \int \sec ^6(c+d x) \tan (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{b \operatorname{Subst}\left (\int x^5 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{b \sec ^6(c+d x)}{6 d}+\frac{a \tan (c+d x)}{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 = 0.147887, size = 53, normalized size = 0.88 \[ \frac{a \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{b \sec ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 48, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -a \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) +{\frac{b}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14306, size = 72, normalized size = 1.2 \begin{align*} \frac{2 \,{\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 - \frac{5 \, b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.472191, size = 147, normalized size = 2.45 \begin{align*} \frac{2 \,{\left (8 \, a \cos \left (d x + c\right )^{5} + 4 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 5 \, b}{30 \, d \cos \left (d x + c\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 = 1.1932, size = 95, normalized size = 1.58 \begin{align*} \frac{5 \, b \tan \left (d x + c\right )^{6} + 6 \, a \tan \left (d x + c\right )^{5} + 15 \, b \tan \left (d x + c\right )^{4} + 20 \, a \tan \left (d x + c\right )^{3} + 15 \, b \tan \left (d x + c\right )^{2} + 30 \, a \tan \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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