Optimal. Leaf size=64 \[ -\frac {a \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}-\frac {a \cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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} \[ -\frac {a \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}-\frac {a \cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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 )^{3/2} \, dx &=\int \left (-a \tan ^2(c+d x)\right )^{3/2} \, dx\\ &=-\left (\left (a \cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\right )\\ &=-\frac {a \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}+\left (a \cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac {a \cot (c+d x) \log (\cos (c+d x)) \sqrt {-a \tan ^2(c+d x)}}{d}-\frac {a \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 48, normalized size = 0.75 \[ \frac {\cot ^3(c+d x) \left (-a \tan ^2(c+d x)\right )^{3/2} \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 68, normalized size = 1.06 \[ -\frac {{\left (2 \, a \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + a\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 137, normalized size = 2.14 \[ -\frac {\sqrt {-a} a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + 2\right ) - \sqrt {-a} a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2\right ) + \frac {{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )} \sqrt {-a} a - 6 \, \sqrt {-a} a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.34, size = 148, normalized size = 2.31 \[ \frac {\left (2 \left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-\sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+2 \left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-2 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-\left (\cos ^{2}\left (d x +c \right )\right )+1\right ) \cos \left (d x +c \right ) \left (-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}\right )^{\frac {3}{2}}}{2 d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 40, normalized size = 0.62 \[ -\frac {\sqrt {-a} a \tan \left (d x + c\right )^{2} - \sqrt {-a} a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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