Optimal. Leaf size=134 \[ -\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0623407, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4121, 3658, 3473, 3475} \[ -\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (a-a \sec ^2(c+d x)\right )^{7/2} \, dx &=\int \left (-a \tan ^2(c+d x)\right )^{7/2} \, dx\\ &=-\left (\left (a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^7(c+d x) \, dx\right )\\ &=-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\left (a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^5(c+d x) \, dx\\ &=\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}-\left (a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\left (a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x) \log (\cos (c+d x)) \sqrt{-a \tan ^2(c+d x)}}{d}-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}\\ \end{align*}
Mathematica [A] time = 2.20579, size = 70, normalized size = 0.52 \[ \frac{\cot ^7(c+d x) \left (-a \tan ^2(c+d x)\right )^{7/2} \left (2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.337, size = 167, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( dx+c \right ) }{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}} \left ( 12\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\ln \left ({\frac{1-\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +12\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -12\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) -11\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+18\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-9\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \left ( -{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46337, size = 109, normalized size = 0.81 \begin{align*} -\frac{2 \, \sqrt{-a} a^{3} \tan \left (d x + c\right )^{6} - 3 \, \sqrt{-a} a^{3} \tan \left (d x + c\right )^{4} + 6 \, \sqrt{-a} a^{3} \tan \left (d x + c\right )^{2} - 6 \, \sqrt{-a} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.528424, size = 244, normalized size = 1.82 \begin{align*} -\frac{{\left (12 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) + 18 \, a^{3} \cos \left (d x + c\right )^{4} - 9 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3}\right )} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{12 \, d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right )} \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 = 1.70824, size = 293, normalized size = 2.19 \begin{align*} -\frac{6 \, \sqrt{-a} a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + 2\right ) - 6 \, \sqrt{-a} a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - 2\right ) + \frac{11 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )}^{3} \sqrt{-a} a^{3} - 90 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )}^{2} \sqrt{-a} a^{3} + 276 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )} \sqrt{-a} a^{3} - 408 \, \sqrt{-a} a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - 2\right )}^{3}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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