Optimal. Leaf size=98 \[ \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a+a \text {sech}(c+d x)}}+\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3860, 4000,
3859, 209, 3877} \begin {gather*} \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a \text {sech}(c+d x)+a}}+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3859
Rule 3860
Rule 3877
Rule 4000
Rubi steps
\begin {align*} \int (a+a \text {sech}(c+d x))^{5/2} \, dx &=\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d}+\frac {1}{3} (2 a) \int \sqrt {a+a \text {sech}(c+d x)} \left (\frac {3 a}{2}+\frac {7}{2} a \text {sech}(c+d x)\right ) \, dx\\ &=\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d}+a^2 \int \sqrt {a+a \text {sech}(c+d x)} \, dx+\frac {1}{3} \left (7 a^2\right ) \int \text {sech}(c+d x) \sqrt {a+a \text {sech}(c+d x)} \, dx\\ &=\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a+a \text {sech}(c+d x)}}+\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d}+\frac {\left (2 i a^3\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}\\ &=\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a+a \text {sech}(c+d x)}}+\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 99, normalized size = 1.01 \begin {gather*} \frac {a^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \sqrt {a (1+\text {sech}(c+d x))} \left (3 \sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh ^{\frac {3}{2}}(c+d x)-6 \sinh \left (\frac {1}{2} (c+d x)\right )+8 \sinh \left (\frac {3}{2} (c+d x)\right )\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.74, size = 0, normalized size = 0.00 \[\int \left (a +a \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 924 vs.
\(2 (84) = 168\).
time = 0.53, size = 924, normalized size = 9.43 \begin {gather*} \frac {3 \, {\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2}\right )} \sqrt {a} \log \left (-\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) - 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} - 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} - 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} - 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) - 7\right )} \sinh \left (d x + c\right )^{2} - 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} - 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} - 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) - 4\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} - 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} - 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) - 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + 3 \, {\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2}\right )} \sqrt {a} \log \left (\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) + 1\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} + a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right ) + 8 \, {\left (4 \, a^{2} \cosh \left (d x + c\right )^{3} + 4 \, a^{2} \sinh \left (d x + c\right )^{3} - 3 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2} \cosh \left (d x + c\right ) + 3 \, {\left (4 \, a^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \, a^{2} + 3 \, {\left (4 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{6 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d\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 \operatorname {sech}{\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 = 0.46, size = 151, normalized size = 1.54 \begin {gather*} \frac {\frac {6 \, a^{3} \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - 3 \, a^{\frac {5}{2}} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) - \frac {4 \, {\left (4 \, a^{4} - {\left (3 \, a^{4} e^{c} + {\left (4 \, a^{4} e^{\left (d x + 3 \, c\right )} - 3 \, a^{4} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )}\right )} e^{\left (d x\right )}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{\frac {3}{2}}}}{3 \, 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}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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