Optimal. Leaf size=101 \[ -\frac {a^2 \cot (c+d x) \log (\cos (c+d x)) \sqrt {-a \tan ^2(c+d x)}}{d}-\frac {a^2 \tan (c+d x) \sqrt {-a \tan ^2(c+d x)}}{2 d}+\frac {a^2 \tan ^3(c+d x) \sqrt {-a \tan ^2(c+d x)}}{4 d} \]
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Rubi [A]
time = 0.03, 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 = {4206, 3739,
3554, 3556} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
Rule 4206
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 = 0.61, size = 60, normalized size = 0.59 \begin {gather*} -\frac {\cot ^5(c+d x) \left (-a \tan ^2(c+d x)\right )^{5/2} \left (4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 157, normalized size = 1.55
method | result | size |
default | \(-\frac {\left (4 \ln \left (\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right )+4 \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right )-4 \ln \left (\frac {2}{\cos \left (d x +c \right )+1}\right ) \left (\cos ^{4}\left (d x +c \right )\right )-3 \left (\cos ^{4}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )-1\right ) \cos \left (d x +c \right ) \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a}{\cos \left (d x +c \right )^{2}}\right )^{\frac {5}{2}}}{4 d \sin \left (d x +c \right )^{5}}\) | \(157\) |
risch | \(\frac {a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, x}{{\mathrm e}^{2 i \left (d x +c \right )}-1}-\frac {2 a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left (d x +c \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) d}-\frac {4 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{4 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} d}-\frac {i a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) d}\) | \(296\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 62, normalized size = 0.61 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.21, size = 87, normalized size = 0.86 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \int \left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 182, normalized size = 1.80 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int {\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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