Optimal. Leaf size=49 \[ -\frac {2}{3} \sin ^{-1}\left (\frac {\cosh (x)}{\sqrt {2}}\right )+\frac {1}{6} \text {sech}(x) \sqrt {1-\sinh ^2(x)}+\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3852, 4928, 12,
4442, 462, 222} \begin {gather*} \frac {1}{6} \sqrt {2-\cosh ^2(x)} \text {sech}(x)-\frac {2}{3} \sin ^{-1}\left (\frac {\cosh (x)}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^3(x) \sin ^{-1}(\sinh (x))+\tanh (x) \sin ^{-1}(\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 222
Rule 462
Rule 3852
Rule 4442
Rule 4928
Rubi steps
\begin {align*} \int \sin ^{-1}(\sinh (x)) \text {sech}^4(x) \, dx &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\int \frac {(2+\cosh (2 x)) \text {sech}(x) \tanh (x)}{3 \sqrt {1-\sinh ^2(x)}} \, dx\\ &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac {1}{3} \int \frac {(2+\cosh (2 x)) \text {sech}(x) \tanh (x)}{\sqrt {1-\sinh ^2(x)}} \, dx\\ &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac {1}{3} \text {Subst}\left (\int \frac {1+2 x^2}{x^2 \sqrt {2-x^2}} \, dx,x,\cosh (x)\right )\\ &=\frac {1}{6} \sqrt {2-\cosh ^2(x)} \text {sech}(x)+\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac {2}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2}} \, dx,x,\cosh (x)\right )\\ &=-\frac {2}{3} \sin ^{-1}\left (\frac {\cosh (x)}{\sqrt {2}}\right )+\frac {1}{6} \sqrt {2-\cosh ^2(x)} \text {sech}(x)+\sin ^{-1}(\sinh (x)) \tanh (x)-\frac {1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 66, normalized size = 1.35 \begin {gather*} \frac {1}{12} \left (8 i \log \left (i \sqrt {2} \cosh (x)+\sqrt {3-\cosh (2 x)}\right )+\sqrt {6-2 \cosh (2 x)} \text {sech}(x)+4 \sin ^{-1}(\sinh (x)) (2+\cosh (2 x)) \text {sech}^2(x) \tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \arcsin \left (\sinh \left (x \right )\right ) \mathrm {sech}\left (x \right )^{4}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 519 vs.
\(2 (40) = 80\).
time = 0.34, size = 519, normalized size = 10.59
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.06, size = 269, normalized size = 5.49 \begin {gather*} 16 \left (\frac {8 \sqrt {2}-\frac {8 \left (4 \sqrt {2}-2 \sqrt {-\left (\mathrm {e}^{x}\right )^{4}+6 \left (\mathrm {e}^{x}\right )^{2}-1}\right )}{-2 \left (\mathrm {e}^{x}\right )^{2}+6}}{768 \left (\left (\frac {4 \sqrt {2}-2 \sqrt {-\left (\mathrm {e}^{x}\right )^{4}+6 \left (\mathrm {e}^{x}\right )^{2}-1}}{-2 \left (\mathrm {e}^{x}\right )^{2}+6}\right )^{2}+1-\frac {\sqrt {2} \left (4 \sqrt {2}-2 \sqrt {-\left (\mathrm {e}^{x}\right )^{4}+6 \left (\mathrm {e}^{x}\right )^{2}-1}\right )}{-2 \left (\mathrm {e}^{x}\right )^{2}+6}\right )}-\frac {\arctan \left (-2 \sqrt {2}+\frac {3 \left (4 \sqrt {2}-2 \sqrt {-\left (\mathrm {e}^{x}\right )^{4}+6 \left (\mathrm {e}^{x}\right )^{2}-1}\right )}{-2 \left (\mathrm {e}^{x}\right )^{2}+6}\right )}{12}-\frac {-8 \mathrm {i} \sqrt {2} \arctan \left (-\mathrm {i}\right )-3 \sqrt {2}+32 \arctan \left (-\mathrm {i}\right )-3 \mathrm {i}}{96 \mathrm {i} \sqrt {2}-384}+\frac {\left (-3 \left (\mathrm {e}^{x}\right )^{2}-1\right ) \arcsin \left (\frac {\left (\mathrm {e}^{x}\right )^{2}-1}{2 \mathrm {e}^{x}}\right )}{2\cdot 6 \left (\left (\mathrm {e}^{x}\right )^{2}+1\right )^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {asin}\left (\mathrm {sinh}\left (x\right )\right )}{{\mathrm {cosh}\left (x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________