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

[Out]

Sqrt[2]*ArcTan[Sech[x]*Sqrt[Cosh[x]*Sinh[x]]] + ArcTan[Sinh[x]/Sqrt[Sinh[2*x]]]/6 - (Sqrt[2]*ArcTanh[Sech[x]*S
qrt[Cosh[x]*Sinh[x]]])/3 + Cosh[x]/Sqrt[Sinh[2*x]]

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Rubi [A]  time = 0.973021, 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 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]

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 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])

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*}

Mathematica [C]  time = 29.6156, size = 392, normalized size = 5.68 \[ \frac{\sqrt{\sinh (2 x)} (\tanh (x)-\cosh (2 x)) \left (-3 \coth (x)+\frac{\sqrt [4]{-1} \cosh (x) \sqrt{\tanh ^3\left (\frac{x}{2}\right )+\tanh \left (\frac{x}{2}\right )} \left (\frac{8 \sqrt [6]{-1} \left (2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+\left (3-3 i \sqrt{3}\right ) \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+i \left (\sqrt{3}+i\right ) \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )\right )}{\left (\sqrt{3}-i\right ) \sqrt{\tanh ^2\left (\frac{x}{2}\right )+1}}-\frac{9 \coth \left (\frac{x}{2}\right ) \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right ),-1\right )-\Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right )\right |-1\right )-\Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right )\right |-1\right )\right )}{\sqrt{\coth ^2\left (\frac{x}{2}\right )+1}}\right )}{(\cosh (x)+1) \sqrt{\tanh \left (\frac{x}{2}\right )} \sqrt{\frac{\sinh (2 x)}{(\cosh (x)+1)^2}}}\right )}{3 (-2 \sinh (x)+\cosh (x)+\cosh (3 x))} \]

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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Maple [C]  time = 0.217, size = 609, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

1/12*((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)*(-12*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)+9*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)+4*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)+12*(-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)+4*(-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*2^(1/2)*sum(_alpha*(I*_alpha+1+I)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)
*(I*tanh(1/2*x))^(1/2)/((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),_alpha+1-I,
1/2*2^(1/2)),_alpha=RootOf(_Z^2+_Z+1))*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)
-6*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)*tanh(1/2*x)^2-6*(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., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt{\sinh \left (2 \, x\right )}}\,{d x} \end{align*}

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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Fricas [B]  time = 2.47632, size = 1337, normalized size = 19.38 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt{\sinh \left (2 \, x\right )}}\,{d x} \end{align*}

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]

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