Optimal. Leaf size=30 \[ \frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]
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Rubi [A] time = 0.0477584, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^5}{x^7} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^6 \tan ^6(c+d x)}{6 b d}\\ \end{align*}
Mathematica [B] time = 0.479837, size = 89, normalized size = 2.97 \[ \frac{\tan (c+d x) \left (20 a^3 b^2 \tan ^2(c+d x)+15 a^2 b^3 \tan ^3(c+d x)+15 a^4 b \tan (c+d x)+6 a^5+6 a b^4 \tan ^4(c+d x)+b^5 \tan ^5(c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.257, size = 120, normalized size = 4. \begin{align*}{\frac{1}{d} \left ({a}^{5}\tan \left ( dx+c \right ) +{\frac{5\,{a}^{4}b}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{10\,{a}^{3}{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\,{a}^{2}{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{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 [B] time = 1.20596, size = 224, normalized size = 7.47 \begin{align*} \frac{6 \, a b^{4} \tan \left (d x + c\right )^{5} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{5} \tan \left (d x + c\right ) + \frac{15 \,{\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac{{\left (3 \, \sin \left (d x + c\right )^{4} - 3 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - \frac{15 \, a^{4} b}{\sin \left (d x + c\right )^{2} - 1}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.507109, size = 329, normalized size = 10.97 \begin{align*} \frac{b^{5} + 3 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a b^{4} \cos \left (d x + c\right ) +{\left (3 \, a^{5} - 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (5 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{6 \, 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 [B] time = 1.31267, size = 120, normalized size = 4. \begin{align*} \frac{b^{5} \tan \left (d x + c\right )^{6} + 6 \, a b^{4} \tan \left (d x + c\right )^{5} + 15 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 15 \, a^{4} b \tan \left (d x + c\right )^{2} + 6 \, a^{5} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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