\(\int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} (\sinh ^2(x)+\sinh (2 x))} \, dx\) [592]

Optimal result

Integrand size = 31, antiderivative size = 69 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\sqrt {2} \arctan \left (\text {sech}(x) \sqrt {\cosh (x) \sinh (x)}\right )+\frac {1}{6} \arctan \left (\frac {\sinh (x)}{\sqrt {\sinh (2 x)}}\right )-\frac {1}{3} \sqrt {2} \text {arctanh}\left (\text {sech}(x) \sqrt {\cosh (x) \sinh (x)}\right )+\frac {\cosh (x)}{\sqrt {\sinh (2 x)}} \]

[Out]

1/6*arctan(sinh(x)/sinh(2*x)^(1/2))+arctan(sech(x)*(cosh(x)*sinh(x))^(1/2))*2^(1/2)-1/3*arctanh(sech(x)*(cosh(
x)*sinh(x))^(1/2))*2^(1/2)+cosh(x)/sinh(2*x)^(1/2)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4475, 6857, 213, 209} \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\frac {2 \sinh (x) \arctan \left (\sqrt {\tanh (x)}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\sinh (x) \arctan \left (\frac {\sqrt {\tanh (x)}}{\sqrt {2}}\right )}{3 \sqrt {2} \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}-\frac {2 \sinh (x) \text {arctanh}\left (\sqrt {\tanh (x)}\right )}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\cosh (x)}{\sqrt {\sinh (2 x)}} \]

[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 209

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

Rule 213

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

Rule 4475

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 6857

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*} \text {integral}& = \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) \text {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)) \text {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)) \text {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) \text {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)) \text {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)) \text {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 \arctan \left (\sqrt {\tanh (x)}\right ) \sinh (x)}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}+\frac {\arctan \left (\frac {\sqrt {\tanh (x)}}{\sqrt {2}}\right ) \sinh (x)}{3 \sqrt {2} \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}-\frac {2 \text {arctanh}\left (\sqrt {\tanh (x)}\right ) \sinh (x)}{3 \sqrt {\sinh (2 x)} \sqrt {\tanh (x)}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(69)=138\).

Time = 18.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.32 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\frac {\sqrt {\sinh (2 x)} \left (6 \sqrt {2} \arctan \left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {\frac {\cosh (x)}{1+\cosh (x)}}}\right )+\arctan \left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {1+\tanh ^2\left (\frac {x}{2}\right )}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {\frac {\cosh (x)}{1+\cosh (x)}}}\right )+\frac {3 \sqrt {\cosh (x) \text {sech}^2\left (\frac {x}{2}\right )}}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right )}{6 (1+\cosh (x)) \sqrt {\tanh \left (\frac {x}{2}\right )} \sqrt {1+\tanh ^2\left (\frac {x}{2}\right )}} \]

[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]]*(6*Sqrt[2]*ArcTan[Sqrt[Tanh[x/2]]/Sqrt[Cosh[x]/(1 + Cosh[x])]] + ArcTan[Sqrt[Tanh[x/2]]/Sqrt[
1 + Tanh[x/2]^2]] - 2*Sqrt[2]*ArcTanh[Sqrt[Tanh[x/2]]/Sqrt[Cosh[x]/(1 + Cosh[x])]] + (3*Sqrt[Cosh[x]*Sech[x/2]
^2])/Sqrt[Tanh[x/2]]))/(6*(1 + Cosh[x])*Sqrt[Tanh[x/2]]*Sqrt[1 + Tanh[x/2]^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(53)=106\).

Time = 1.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.42

method	result	size
default	\(\frac {\sqrt {\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right )^{2}}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right ) \left (2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\, \sqrt {2}}{2 \tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )+6 \sqrt {2}\, \arctan \left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\, \sqrt {2}}{2 \tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )+\arctan \left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}}{\tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )-3 \sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\right )}{6 \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \tanh \left (\frac {x}{2}\right )}\)	\(167\)

[In]

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

[Out]

1/6*((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)*(2*2^(1/2)*arctanh(1/2/tanh(1/
2*x)*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)*2^(1/2))*tanh(1/2*x)+6*2^(1/2)*arctan(1/2/tanh(1/2*x)*(tanh(1/2*x)^3+ta
nh(1/2*x))^(1/2)*2^(1/2))*tanh(1/2*x)+arctan(1/tanh(1/2*x)*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2))*tanh(1/2*x)-3*(t
anh(1/2*x)^3+tanh(1/2*x))^(1/2))/((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)/tanh(1/2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (53) = 106\).

Time = 0.26 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.45 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=-\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \arctan \left (\frac {{\left (\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} + 3 \, \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )}}\right ) + 6 \, {\left (\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}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {\cosh \left (x\right ) \sinh \left (x\right )}{\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 )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1}\right ) - {\left (\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}\right )} \log \left (2 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, \sinh \left (x\right )^{4} - 4 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {\frac {\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 1\right ) - 12 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{12 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=- \int \frac {\cosh {\left (x \right )} \cosh {\left (2 x \right )}}{\sinh ^{2}{\left (x \right )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\, dx - \int \left (- \frac {\cosh {\left (x \right )} \tanh {\left (x \right )}}{\sinh ^{2}{\left (x \right )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\right )\, dx \]

[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)

Maxima [F]

\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\int { -\frac {{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt {\sinh \left (2 \, x\right )}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\sqrt {2} \arctan \left (\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {e^{\left (4 \, x\right )} - 1} + e^{\left (2 \, x\right )}\right ) + \frac {\sqrt {2}}{\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )} + 1} + \frac {1}{6} \, \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, \sqrt {e^{\left (4 \, x\right )} - 1} - 3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) \]

[In]

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

[Out]

sqrt(2)*arctan(sqrt(e^(4*x) - 1) - e^(2*x)) + 1/6*sqrt(2)*log(-sqrt(e^(4*x) - 1) + e^(2*x)) + sqrt(2)/(sqrt(e^
(4*x) - 1) - e^(2*x) + 1) + 1/6*arctan(1/4*sqrt(2)*(3*sqrt(e^(4*x) - 1) - 3*e^(2*x) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=-\int \frac {\mathrm {cosh}\left (x\right )\,\left (\mathrm {cosh}\left (2\,x\right )-\mathrm {tanh}\left (x\right )\right )}{\sqrt {\mathrm {sinh}\left (2\,x\right )}\,\left ({\mathrm {sinh}\left (x\right )}^2+\mathrm {sinh}\left (2\,x\right )\right )} \,d x \]

[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)