Optimal. Leaf size=67 \[ \frac {\cot (c+d x)}{2 a d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt {-a \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4121, 3658, 3473, 3475} \[ \frac {\cot (c+d x)}{2 a d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt {-a \tan ^2(c+d x)}} \]
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 )^{3/2}} \, dx &=\int \frac {1}{\left (-a \tan ^2(c+d x)\right )^{3/2}} \, dx\\ &=-\frac {\tan (c+d x) \int \cot ^3(c+d x) \, dx}{a \sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\cot (c+d x)}{2 a d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \int \cot (c+d x) \, dx}{a \sqrt {-a \tan ^2(c+d x)}}\\ &=\frac {\cot (c+d x)}{2 a d \sqrt {-a \tan ^2(c+d x)}}+\frac {\log (\sin (c+d x)) \tan (c+d x)}{a d \sqrt {-a \tan ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 57, normalized size = 0.85 \[ -\frac {\tan ^3(c+d x) \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d \left (-a \tan ^2(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 94, normalized size = 1.40 \[ -\frac {{\left (2 \, {\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{2 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 212, normalized size = 3.16 \[ \frac {\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\sqrt {-a} a \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {8 \, \sqrt {-a} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{2} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {4 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )}{\sqrt {-a} a \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}{\sqrt {-a} a \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.66, size = 141, normalized size = 2.10 \[ \frac {\left (4 \left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-4 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-\left (\cos ^{2}\left (d x +c \right )\right )-4 \ln \left (-\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+4 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right )-1\right ) \sin \left (d x +c \right )}{4 d \cos \left (d x +c \right )^{3} \left (-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 60, normalized size = 0.90 \[ -\frac {\frac {\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a} a} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a} a} + \frac {\sqrt {-a}}{a^{2} \tan \left (d x + c\right )^{2}}}{2 \, d} \]
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 \[ \int \frac {1}{{\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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