Optimal. Leaf size=134 \[ -\frac {a^3 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}-\frac {a^3 \tan ^5(c+d x) \sqrt {-a \tan ^2(c+d x)}}{6 d}+\frac {a^3 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}-\frac {a^3 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 3473
Rule 3475
Rule 3658
Rule 4121
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*}
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Mathematica [A] time = 2.16, 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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fricas [A] time = 0.45, size = 100, normalized size = 0.75 \[ -\frac {{\left (12 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) + 18 \, a^{3} \cos \left (d x + c\right )^{4} - 9 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3}\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{12 \, d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 217, normalized size = 1.62 \[ -\frac {6 \, \sqrt {-a} a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + 2\right ) - 6 \, \sqrt {-a} a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2\right ) + \frac {11 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )}^{3} \sqrt {-a} a^{3} - 90 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )}^{2} \sqrt {-a} a^{3} + 276 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )} \sqrt {-a} a^{3} - 408 \, \sqrt {-a} a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2\right )}^{3}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.33, size = 168, normalized size = 1.25 \[ \frac {\left (12 \left (\cos ^{6}\left (d x +c \right )\right ) \ln \left (-\frac {-\sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-12 \left (\cos ^{6}\left (d x +c \right )\right ) \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )+12 \left (\cos ^{6}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-11 \left (\cos ^{6}\left (d x +c \right )\right )+18 \left (\cos ^{4}\left (d x +c \right )\right )-9 \left (\cos ^{2}\left (d x +c \right )\right )+2\right ) \cos \left (d x +c \right ) \left (-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}\right )^{\frac {7}{2}}}{12 d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 81, normalized size = 0.60 \[ -\frac {2 \, \sqrt {-a} a^{3} \tan \left (d x + c\right )^{6} - 3 \, \sqrt {-a} a^{3} \tan \left (d x + c\right )^{4} + 6 \, \sqrt {-a} a^{3} \tan \left (d x + c\right )^{2} - 6 \, \sqrt {-a} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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