3.8.3 \(\int \sin ^{-1}(\sinh (x)) \text {sech}^4(x) \, dx\) [703]

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]

-2/3*arcsin(1/2*cosh(x)*2^(1/2))+1/6*sech(x)*(1-sinh(x)^2)^(1/2)+arcsin(sinh(x))*tanh(x)-1/3*arcsin(sinh(x))*t
anh(x)^3

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Rubi [A]
time = 0.10, 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 {2}{3} \text {ArcSin}\left (\frac {\cosh (x)}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^3(x) \text {ArcSin}(\sinh (x))+\tanh (x) \text {ArcSin}(\sinh (x))+\frac {1}{6} \sqrt {2-\cosh ^2(x)} \text {sech}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sinh[x]]*Sech[x]^4,x]

[Out]

(-2*ArcSin[Cosh[x]/Sqrt[2]])/3 + (Sqrt[2 - Cosh[x]^2]*Sech[x])/6 + ArcSin[Sinh[x]]*Tanh[x] - (ArcSin[Sinh[x]]*
Tanh[x]^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 462

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4442

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist[-d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 4928

Int[((a_.) + ArcSin[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcSin[u], w, x] - Dist
[b, Int[SimplifyIntegrand[w*(D[u, x]/Sqrt[1 - u^2]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}
, x] && InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.16, 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]

Integrate[ArcSin[Sinh[x]]*Sech[x]^4,x]

[Out]

((8*I)*Log[I*Sqrt[2]*Cosh[x] + Sqrt[3 - Cosh[2*x]]] + Sqrt[6 - 2*Cosh[2*x]]*Sech[x] + 4*ArcSin[Sinh[x]]*(2 + C
osh[2*x])*Sech[x]^2*Tanh[x])/12

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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]

int(arcsin(sinh(x))*sech(x)^4,x)

[Out]

int(arcsin(sinh(x))*sech(x)^4,x)

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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]

integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="maxima")

[Out]

-1/3*(4*(3*e^(2*x) + 1)*arctan2(e^(2*x) - 1, sqrt(e^(2*x) + 2*e^x - 1)*sqrt(-e^(2*x) + 2*e^x + 1)) + 3*(e^(6*x
) + 3*e^(4*x) + 3*e^(2*x) + 1)*integrate(16/3*(3*e^(4*x) + e^(2*x))*e^(1/2*log(e^(2*x) + 2*e^x - 1) + 1/2*log(
-e^(2*x) + 2*e^x + 1))/((e^(8*x) - 4*e^(6*x) - 10*e^(4*x) - 4*e^(2*x) + 1)*e^(log(e^(2*x) + 2*e^x - 1) + log(-
e^(2*x) + 2*e^x + 1)) + e^(12*x) - 6*e^(10*x) - e^(8*x) + 12*e^(6*x) - e^(4*x) - 6*e^(2*x) + 1), x))/(e^(6*x)
+ 3*e^(4*x) + 3*e^(2*x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (40) = 80\).
time = 0.62, size = 519, normalized size = 10.59 \begin {gather*} \frac {\sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\frac {\sqrt {2} {\left (3 \, \cosh \left (x\right )^{2} + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) + 8 \, {\left (3 \, \cosh \left (x\right )^{2} + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{6 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="fricas")

[Out]

1/6*(sqrt(2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(c
osh(x)^3 + cosh(x))*sinh(x) + 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)
) - 4*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x
)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh
(x)^3 + cosh(x))*sinh(x) + 1)*arctan(sqrt(2)*(3*cosh(x)^2 + 6*cosh(x)*sinh(x) + 3*sinh(x)^2 - 1)*sqrt(-(cosh(x
)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4
+ 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)) + 8*(3*cosh(x)^2 + 6*co
sh(x)*sinh(x) + 3*sinh(x)^2 + 1)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-(cosh(x)
^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4
+ 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)))/(cosh(x)^6 + 6*cosh(x)*
sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 +
3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)

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Sympy [F(-1)] Timed out
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]

integrate(asin(sinh(x))*sech(x)**4,x)

[Out]

Timed out

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Giac [C] Result contains complex when optimal does not.
time = 1.01, size = 218, normalized size = 4.45 \begin {gather*} -\frac {16 \, {\left (-8 i \, \sqrt {2} \arctan \left (-i\right ) - 3 \, \sqrt {2} + 32 \, \arctan \left (-i\right ) - 3 i\right )}}{96 i \, \sqrt {2} - 384} + \frac {\sqrt {2} + \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3}}{6 \, {\left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3} + \frac {{\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{2}}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{2}} + 1\right )}} - \frac {4 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )} \arcsin \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} - \frac {4}{3} \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="giac")

[Out]

-16*(-8*I*sqrt(2)*arctan(-I) - 3*sqrt(2) + 32*arctan(-I) - 3*I)/(96*I*sqrt(2) - 384) + 1/6*(sqrt(2) + (2*sqrt(
2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3))/(sqrt(2)*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(
2*x) - 3) + (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))^2/(e^(2*x) - 3)^2 + 1) - 4/3*(3*e^(2*x) + 1)*arcsin(1
/2*(e^(2*x) - 1)*e^(-x))/(e^(2*x) + 1)^3 - 4/3*arctan(-2*sqrt(2) - 3*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) -
1))/(e^(2*x) - 3))

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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]

int(asin(sinh(x))/cosh(x)^4,x)

[Out]

int(asin(sinh(x))/cosh(x)^4, x)

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