Optimal. Leaf size=50 \[ -\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\frac {5}{2} \sinh ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3661, 416, 523, 215, 377, 206} \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)+1}-\frac {5}{2} \sinh ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 377
Rule 416
Rule 523
Rule 3661
Rubi steps
\begin {align*} \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-3-5 x^2}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tanh (x)\right )+4 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {5}{2} \sinh ^{-1}(\tanh (x))-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}+4 \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )\\ &=-\frac {5}{2} \sinh ^{-1}(\tanh (x))+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 74, normalized size = 1.48 \[ -\frac {\left (\tanh ^2(x)+1\right )^{3/2} \left (-4 \sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh (x)\right ) \cosh ^3(x)+\sinh (x) \sqrt {\cosh (2 x)} \cosh (x)+5 \cosh ^3(x) \tanh ^{-1}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right )}{2 \cosh ^{\frac {3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 1027, normalized size = 20.54 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 202, normalized size = 4.04 \[ -\frac {1}{4} \, \sqrt {2} {\left (5 \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} - {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 1\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - 4 \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 158, normalized size = 3.16 \[ -\frac {\left (\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \relax (x ) \sqrt {\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )}}{4}-\frac {5 \arcsinh \left (\tanh \relax (x )\right )}{2}-\sqrt {\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )}+\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \relax (x )+2\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \relax (x )-1\right )^{2}+2 \tanh \relax (x )}}\right )+\frac {\left (\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \relax (x ) \sqrt {\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )}}{4}+\sqrt {\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )}-\sqrt {2}\, \arctanh \left (\frac {\left (2-2 \tanh \relax (x )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \relax (x )\right )^{2}-2 \tanh \relax (x )}}\right ) \]
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 \[ \int {\left (\tanh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 78, normalized size = 1.56 \[ \sqrt {2}\,\left (\ln \left (\mathrm {tanh}\relax (x)+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\relax (x)}^2+1}-\mathrm {tanh}\relax (x)+1\right )\right )-\frac {5\,\mathrm {asinh}\left (\mathrm {tanh}\relax (x)\right )}{2}-\frac {\mathrm {tanh}\relax (x)\,\sqrt {{\mathrm {tanh}\relax (x)}^2+1}}{2}+\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\relax (x)+\sqrt {2}\,\sqrt {{\mathrm {tanh}\relax (x)}^2+1}+1\right )-\ln \left (\mathrm {tanh}\relax (x)-1\right )\right ) \]
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 \[ \int \left (\tanh ^{2}{\relax (x )} + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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