Optimal. Leaf size=66 \[ \frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3775, 21, 3774, 203} \[ \frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 203
Rule 3774
Rule 3775
Rubi steps
\begin {align*} \int (a+a \text {sech}(c+d x))^{3/2} \, dx &=\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}}+(2 a) \int \frac {\frac {a}{2}+\frac {1}{2} a \text {sech}(c+d x)}{\sqrt {a+a \text {sech}(c+d x)}} \, dx\\ &=\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}}+a \int \sqrt {a+a \text {sech}(c+d x)} \, dx\\ &=\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}}+\frac {\left (2 i a^2\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^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 75, normalized size = 1.14 \[ \frac {a \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\text {sech}(c+d x)+1)} \left (2 \sinh \left (\frac {1}{2} (c+d x)\right )+\sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cosh (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 697, normalized size = 10.56 \[ \frac {a^{\frac {3}{2}} \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 ) + a^{\frac {3}{2}} \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 ) + 4 \, {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - a\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}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 118, normalized size = 1.79 \[ \frac {\frac {2 \, a^{2} \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - a^{\frac {3}{2}} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) + \frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} - a^{2}\right )}}{\sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (a +a \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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