Optimal. Leaf size=101 \[ -\frac {a^2 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}+\frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}-\frac {a^2 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4121, 3658, 3473, 3475} \[ \frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}-\frac {a^2 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}-\frac {a^2 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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 )^{5/2} \, dx &=\int \left (-a \tan ^2(c+d x)\right )^{5/2} \, dx\\ &=\left (a^2 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan ^5(c+d x) \, dx\\ &=\frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}-\left (a^2 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=-\frac {a^2 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}+\frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}+\left (a^2 \cot (c+d x) \sqrt {-a \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac {a^2 \cot (c+d x) \log (\cos (c+d x)) \sqrt {-a \tan ^2(c+d x)}}{d}-\frac {a^2 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}+\frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 60, normalized size = 0.59 \[ -\frac {\cot ^5(c+d x) \left (-a \tan ^2(c+d x)\right )^{5/2} \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 87, normalized size = 0.86 \[ -\frac {{\left (4 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) + 4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{4 \, d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 182, normalized size = 1.80 \[ -\frac {2 \, \sqrt {-a} a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + 2\right ) - 2 \, \sqrt {-a} a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2\right ) + \frac {3 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )}^{2} \sqrt {-a} a^{2} - 20 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )} \sqrt {-a} a^{2} + 44 \, \sqrt {-a} a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - 2\right )}^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.99, size = 158, normalized size = 1.56 \[ -\frac {\left (4 \left (\cos ^{4}\left (d x +c \right )\right ) \ln \left (-\frac {-\sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+4 \left (\cos ^{4}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-4 \ln \left (\frac {2}{1+\cos \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right )-3 \left (\cos ^{4}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )-1\right ) \cos \left (d x +c \right ) \left (-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{2}}\right )^{\frac {5}{2}}}{4 d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 62, normalized size = 0.61 \[ \frac {\sqrt {-a} a^{2} \tan \left (d x + c\right )^{4} - 2 \, \sqrt {-a} a^{2} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {-a} a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{4 \, d} \]
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 \[ \int {\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \]
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 \[ \int \left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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