Optimal. Leaf size=19 \[ -\frac {\text {ArcTan}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}}+\sinh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 327, 209}
\begin {gather*} \sinh (x)-\frac {\text {ArcTan}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 327
Rubi steps
\begin {align*} \int \sinh (x) \tanh (2 x) \, dx &=-\text {Subst}\left (\int -\frac {2 x^2}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=2 \text {Subst}\left (\int \frac {x^2}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}}+\sinh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}}+\sinh (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 1.10, size = 54, normalized size = 2.84
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (15) = 30\).
time = 0.48, size = 53, normalized size = 2.79 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs.
\(2 (15) = 30\).
time = 0.35, size = 115, normalized size = 6.05 \begin {gather*} -\frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \cosh \left (x\right ) + \frac {1}{2} \, \sqrt {2} \sinh \left (x\right )\right ) - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \arctan \left (-\frac {\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}}{2 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + 1}{2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (x \right )} \tanh {\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (15) = 30\).
time = 0.40, size = 36, normalized size = 1.89 \begin {gather*} -\frac {1}{4} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.47, size = 47, normalized size = 2.47 \begin {gather*} \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {\sqrt {2}\,{\mathrm {e}}^{3\,x}}{2}\right )}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________