Optimal. Leaf size=133 \[ \frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot (c+d x)}{2 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \log (\sin (c+d x))}{a^3 d \sqrt {-a \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 133, 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} \[ \frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot (c+d x)}{2 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \log (\sin (c+d x))}{a^3 d \sqrt {-a \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx &=\int \frac {1}{\left (-a \tan ^2(c+d x)\right )^{7/2}} \, dx\\ &=-\frac {\tan (c+d x) \int \cot ^7(c+d x) \, dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \int \cot ^5(c+d x) \, dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\\ &=-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\tan (c+d x) \int \cot ^3(c+d x) \, dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\cot (c+d x)}{2 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \int \cot (c+d x) \, dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\cot (c+d x)}{2 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\log (\sin (c+d x)) \tan (c+d x)}{a^3 d \sqrt {-a \tan ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 79, normalized size = 0.59 \[ -\frac {\tan ^7(c+d x) \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d \left (-a \tan ^2(c+d x)\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 162, normalized size = 1.22 \[ \frac {{\left (18 \, \cos \left (d x + c\right )^{5} - 27 \, \cos \left (d x + c\right )^{3} - 12 \, {\left (\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 11 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{12 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.70, size = 275, normalized size = 2.07 \[ -\frac {\frac {384 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{\sqrt {-a} a^{3} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {192 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )}{\sqrt {-a} a^{3} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {352 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 87 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}{\sqrt {-a} a^{3} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 87 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\sqrt {-a} a^{10} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.83, size = 265, normalized size = 1.99 \[ -\frac {\left (48 \left (\cos ^{6}\left (d x +c \right )\right ) \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )-48 \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{6}\left (d x +c \right )\right )+25 \left (\cos ^{6}\left (d x +c \right )\right )-144 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right )+144 \left (\cos ^{4}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-3 \left (\cos ^{4}\left (d x +c \right )\right )+144 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-144 \left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-33 \left (\cos ^{2}\left (d x +c \right )\right )-48 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )+48 \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+19\right ) \sin \left (d x +c \right )}{48 d \left (-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}\right )^{\frac {7}{2}} \cos \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 94, normalized size = 0.71 \[ -\frac {\frac {6 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a} a^{3}} - \frac {12 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a} a^{3}} + \frac {6 \, \sqrt {-a} \tan \left (d x + c\right )^{4} - 3 \, \sqrt {-a} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {-a}}{a^{4} \tan \left (d x + c\right )^{6}}}{12 \, d} \]
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 \[ \int \frac {1}{{\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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