3.592 \(\int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} (\sinh ^2(x)+\sinh (2 x))} \, dx\)

Optimal. Leaf size=69 \[ \frac {1}{6} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {\sinh (2 x)}}\right )+\frac {\cosh (x)}{\sqrt {\sinh (2 x)}}+\sqrt {2} \tan ^{-1}\left (\text {sech}(x) \sqrt {\sinh (x) \cosh (x)}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\text {sech}(x) \sqrt {\sinh (x) \cosh (x)}\right ) \]

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Rubi [A]  time = 0.97, antiderivative size = 102, normalized size of antiderivative = 1.48, number of steps used = 8, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4390, 6725, 207, 203} \[ -\frac {2 \sinh (x) \tanh ^{-1}\left (\sqrt {\tanh (x)}\right )}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\cosh (x)}{\sqrt {\sinh (2 x)}}+\frac {2 \sinh (x) \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\sinh (x) \tan ^{-1}\left (\frac {\sqrt {\tanh (x)}}{\sqrt {2}}\right )}{3 \sqrt {2} \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]*(-Cosh[2*x] + Tanh[x]))/(Sqrt[Sinh[2*x]]*(Sinh[x]^2 + Sinh[2*x])),x]

[Out]

Cosh[x]/Sqrt[Sinh[2*x]] + (2*ArcTan[Sqrt[Tanh[x]]]*Sinh[x])/(Sqrt[Sinh[2*x]]*Sqrt[Tanh[x]]) + (ArcTan[Sqrt[Tan
h[x]]/Sqrt[2]]*Sinh[x])/(3*Sqrt[2]*Sqrt[Sinh[2*x]]*Sqrt[Tanh[x]]) - (2*ArcTanh[Sqrt[Tanh[x]]]*Sinh[x])/(3*Sqrt
[Sinh[2*x]]*Sqrt[Tanh[x]])

Rule 203

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

Rule 207

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

Rule 4390

Int[(u_)*((c_.)*sin[v_])^(m_), x_Symbol] :> With[{w = FunctionOfTrig[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]}, D
ist[((c*Sin[v])^m*(c*Tan[v/2])^m)/Sin[v/2]^(2*m), Int[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x], x] /;  !FalseQ[w]
&& FunctionOfQ[NonfreeFactors[Tan[w], x], (u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]] /; FreeQ[c, x] && LinearQ[v,
x] && IntegerQ[m + 1/2] &&  !SumQ[u] && InverseFunctionFreeQ[u, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx &=\frac {\sinh (x) \int \frac {-\cosh (2 x)+\tanh (x)}{\left (\sinh ^2(x)+\sinh (2 x)\right ) \sqrt {\tanh (x)}} \, dx}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ &=\frac {\sinh (x) \operatorname {Subst}\left (\int \frac {-1+x-x^2-x^3}{x^{3/2} (2+x) \left (1-x^2\right )} \, dx,x,\tanh (x)\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ &=\frac {(2 \sinh (x)) \operatorname {Subst}\left (\int \frac {1-x^2+x^4+x^6}{x^2 \left (2+x^2\right ) \left (-1+x^4\right )} \, dx,x,\sqrt {\tanh (x)}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ &=\frac {(2 \sinh (x)) \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^2}+\frac {1}{3 \left (-1+x^2\right )}+\frac {1}{1+x^2}+\frac {1}{6 \left (2+x^2\right )}\right ) \, dx,x,\sqrt {\tanh (x)}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ &=\frac {\cosh (x)}{\sqrt {\sinh (2 x)}}+\frac {\sinh (x) \operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {(2 \sinh (x)) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {(2 \sinh (x)) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ &=\frac {\cosh (x)}{\sqrt {\sinh (2 x)}}+\frac {2 \tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sinh (x)}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\tanh (x)}}{\sqrt {2}}\right ) \sinh (x)}{3 \sqrt {2} \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}-\frac {2 \tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sinh (x)}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\\ \end {align*}

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Mathematica [C]  time = 30.38, size = 392, normalized size = 5.68 \[ \frac {\sqrt {\sinh (2 x)} (\tanh (x)-\cosh (2 x)) \left (-3 \coth (x)+\frac {\sqrt [4]{-1} \cosh (x) \sqrt {\tanh ^3\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )} \left (\frac {8 \sqrt [6]{-1} \left (2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} \sqrt {\tanh \left (\frac {x}{2}\right )}\right )\right |-1\right )+\left (3-3 i \sqrt {3}\right ) \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tanh \left (\frac {x}{2}\right )}\right )\right |-1\right )+i \left (\sqrt {3}+i\right ) \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tanh \left (\frac {x}{2}\right )}\right )\right |-1\right )+2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tanh \left (\frac {x}{2}\right )}\right )\right |-1\right )\right )}{\left (\sqrt {3}-i\right ) \sqrt {\tanh ^2\left (\frac {x}{2}\right )+1}}-\frac {9 \coth \left (\frac {x}{2}\right ) \left (\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right ),-1\right )-\Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right )\right |-1\right )-\Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right )\right |-1\right )\right )}{\sqrt {\coth ^2\left (\frac {x}{2}\right )+1}}\right )}{(\cosh (x)+1) \sqrt {\tanh \left (\frac {x}{2}\right )} \sqrt {\frac {\sinh (2 x)}{(\cosh (x)+1)^2}}}\right )}{3 (-2 \sinh (x)+\cosh (x)+\cosh (3 x))} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cosh[x]*(-Cosh[2*x] + Tanh[x]))/(Sqrt[Sinh[2*x]]*(Sinh[x]^2 + Sinh[2*x])),x]

[Out]

(Sqrt[Sinh[2*x]]*(-3*Coth[x] + ((-1)^(1/4)*Cosh[x]*Sqrt[Tanh[x/2] + Tanh[x/2]^3]*((-9*Coth[x/2]*(EllipticF[I*A
rcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1] - EllipticPi[-(-1)^(1/6), I*ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1] -
EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1]))/Sqrt[1 + Coth[x/2]^2] + (8*(-1)^(1/6)*((
3 - (3*I)*Sqrt[3])*EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*Sqrt[Tanh[x/2]]], -1] + 2*(-1 + (-1)^(1/3))*EllipticPi[
I, ArcSin[(-1)^(3/4)*Sqrt[Tanh[x/2]]], -1] + I*(I + Sqrt[3])*EllipticPi[-(-1)^(1/6), I*ArcSinh[(-1)^(1/4)*Sqrt
[Tanh[x/2]]], -1] + 2*(-1 + (-1)^(1/3))*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(1/4)*Sqrt[Tanh[x/2]]], -1]))/(
(-I + Sqrt[3])*Sqrt[1 + Tanh[x/2]^2])))/((1 + Cosh[x])*Sqrt[Sinh[2*x]/(1 + Cosh[x])^2]*Sqrt[Tanh[x/2]]))*(-Cos
h[2*x] + Tanh[x]))/(3*(Cosh[x] + Cosh[3*x] - 2*Sinh[x]))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(Cosh[x]*(-Cosh[2*x] + Tanh[x]))/(Sqrt[Sinh[2*x]]*(Sinh[x]^2 + Sinh[2*x])),x]

[Out]

Could not integrate

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fricas [B]  time = 1.15, size = 376, normalized size = 5.45 \[ -\frac {{\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \arctan \left (\frac {{\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + 3 \, \sqrt {2}\right )} \sqrt {\frac {\cosh \relax (x) \sinh \relax (x)}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )}}\right ) + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} - \sqrt {2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {\cosh \relax (x) \sinh \relax (x)}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1}\right ) - {\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} - \sqrt {2}\right )} \log \left (2 \, \cosh \relax (x)^{4} + 8 \, \cosh \relax (x)^{3} \sinh \relax (x) + 12 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 8 \, \cosh \relax (x) \sinh \relax (x)^{3} + 2 \, \sinh \relax (x)^{4} - 4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \sqrt {\frac {\cosh \relax (x) \sinh \relax (x)}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 1\right ) - 12 \, \sqrt {2} \sqrt {\frac {\cosh \relax (x) \sinh \relax (x)}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{12 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/12*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*arctan(1/2*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(
x) + sqrt(2)*sinh(x)^2 + 3*sqrt(2))*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)
^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)) + 6*(sqrt(2)*cosh(x)^
2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*arctan(2*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh
(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sin
h(x)^4 - 1)) - (sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*log(2*cosh(x)^4 +
8*cosh(x)^3*sinh(x) + 12*cosh(x)^2*sinh(x)^2 + 8*cosh(x)*sinh(x)^3 + 2*sinh(x)^4 - 4*(cosh(x)^2 + 2*cosh(x)*s
inh(x) + sinh(x)^2)*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1) - 12*sqrt(2)*sqrt(c
osh(x)*sinh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -1/2*(((-i)*pi+sqrt(2)*atan((1+2*i)/(1-i
)/sqrt(2))+12*atan(i)+6-6*i)/3/sqrt(2)+2*(-sqrt(2)/(-exp(x)^2+sqrt(exp(x)^4-1)+1)-1/6*atan(1/2*(3*(-exp(x)^2+s
qrt(exp(x)^4-1))-1)/sqrt(2))-1/3*ln(exp(x)^2-sqrt(exp(x)^4-1))/sqrt(2)-sqrt(2)*atan(-exp(x)^2+sqrt(exp(x)^4-1)
)))

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maple [C]  time = 0.48, size = 987, normalized size = 14.30

method	result	size
default	\(-\frac {\sqrt {\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right )^{2}}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right ) \left (\sqrt {3}\, \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \sqrt {2}\, \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )-\sqrt {3}\, \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \sqrt {2}\, \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )+i \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}+24 i \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}-8 i \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}+i \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}-18 i \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}-2 \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \sqrt {2}\, \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )-24 \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}-8 \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}-2 \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \sqrt {2}\, \sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {-i \left (-\tanh \left (\frac {x}{2}\right )+i\right )}\, \sqrt {i \tanh \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {-i \left (\tanh \left (\frac {x}{2}\right )+i\right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )+12 \sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+12 \sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\right )}{24 \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right ) \sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}}\)	\(987\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*((tanh(1/2*x)^2+1)*tanh(1/2*x)/(tanh(1/2*x)^2-1)^2)^(1/2)*(tanh(1/2*x)^2-1)*(3^(1/2)*((tanh(1/2*x)^2+1)*
tanh(1/2*x))^(1/2)*2^(1/2)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*Ellipt
icPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-I+1/2*I*3^(1/2),1/2*2^(1/2))-3^(1/2)*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)
*2^(1/2)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2
*x)+I))^(1/2),1/2-I-1/2*I*3^(1/2),1/2*2^(1/2))+I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2
)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-I+1/2*I*3^(1/2),1/2*2^(1/2))*((tanh(1/2*x)^2
+1)*tanh(1/2*x))^(1/2)+24*I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/
2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-8*I*(-I*
(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I
))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)+I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I
*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-I-1/2*I*3^(1/2),1/2*2
^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-18*I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1
/2)*(I*tanh(1/2*x))^(1/2)*EllipticF((-I*(tanh(1/2*x)+I))^(1/2),1/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1
/2)-2*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)*2^(1/2)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*
tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-I+1/2*I*3^(1/2),1/2*2^(1/2))-24*(-I*(tanh(1/2*x)+
I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-
1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-8*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x
)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^2
+1)*tanh(1/2*x))^(1/2)-2*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)*2^(1/2)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1
/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-I-1/2*I*3^(1/2),1/2*2^(1/2))+1
2*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)*tanh(1/2*x)^2+12*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2))/(tanh(1/2*x)^2+1)/tanh
(1/2*x)/(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (\cosh \left (2 \, x\right ) - \tanh \relax (x)\right )} \cosh \relax (x)}{{\left (\sinh \relax (x)^{2} + \sinh \left (2 \, x\right )\right )} \sqrt {\sinh \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x, algorithm="maxima")

[Out]

-integrate((cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\mathrm {cosh}\relax (x)\,\left (\mathrm {cosh}\left (2\,x\right )-\mathrm {tanh}\relax (x)\right )}{\sqrt {\mathrm {sinh}\left (2\,x\right )}\,\left ({\mathrm {sinh}\relax (x)}^2+\mathrm {sinh}\left (2\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(cosh(x)*(cosh(2*x) - tanh(x)))/(sinh(2*x)^(1/2)*(sinh(2*x) + sinh(x)^2)),x)

[Out]

-int((cosh(x)*(cosh(2*x) - tanh(x)))/(sinh(2*x)^(1/2)*(sinh(2*x) + sinh(x)^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\cosh {\relax (x )} \cosh {\left (2 x \right )}}{\sinh ^{2}{\relax (x )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\, dx - \int \left (- \frac {\cosh {\relax (x )} \tanh {\relax (x )}}{\sinh ^{2}{\relax (x )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)**2+sinh(2*x))/sinh(2*x)**(1/2),x)

[Out]

-Integral(cosh(x)*cosh(2*x)/(sinh(x)**2*sqrt(sinh(2*x)) + sinh(2*x)**(3/2)), x) - Integral(-cosh(x)*tanh(x)/(s
inh(x)**2*sqrt(sinh(2*x)) + sinh(2*x)**(3/2)), x)

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