Optimal. Leaf size=98 \[ \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a \text {sech}(c+d x)+a}}+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.12, 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 = {3775, 3915, 3774, 203, 3792} \[ \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a \text {sech}(c+d x)+a}}+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 3775
Rule 3792
Rule 3915
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 ) \operatorname {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 = 0.34, size = 99, normalized size = 1.01 \[ \frac {a^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \sqrt {a (\text {sech}(c+d x)+1)} \left (-6 \sinh \left (\frac {1}{2} (c+d x)\right )+8 \sinh \left (\frac {3}{2} (c+d x)\right )+3 \sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh ^{\frac {3}{2}}(c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 924, normalized size = 9.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 151, normalized size = 1.54 \[ \frac {\frac {6 \, a^{3} \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - 3 \, a^{\frac {5}{2}} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) - \frac {4 \, {\left (4 \, a^{4} - {\left (3 \, a^{4} e^{c} + {\left (4 \, a^{4} e^{\left (d x + 3 \, c\right )} - 3 \, a^{4} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )}\right )} e^{\left (d x\right )}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{\frac {3}{2}}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, 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 \[ \int {\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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 \[ \int \left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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