3.1044 \(\int \frac {1+\sinh ^2(x)}{1+\cosh (x)+\sinh (x)} \, dx\)

Optimal. Leaf size=69 \[ \frac {1}{2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{4} \log \left (1-\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{4} \log \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

[Out]

1/4*ln(1-tanh(1/2*x))+3/4*ln(1+tanh(1/2*x))+1/2/(1-tanh(1/2*x))-1/2/(1+tanh(1/2*x))^2+1/(1+tanh(1/2*x))

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Rubi [A]  time = 0.20, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4397, 12, 894} \[ \frac {1}{2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{4} \log \left (1-\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{4} \log \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]),x]

[Out]

Log[1 - Tanh[x/2]]/4 + (3*Log[1 + Tanh[x/2]])/4 + 1/(2*(1 - Tanh[x/2])) - 1/(2*(1 + Tanh[x/2])^2) + (1 + Tanh[
x/2])^(-1)

Rule 12

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

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 37, normalized size = 0.54 \[ \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {\cosh (x)}{2}-\frac {1}{8} \cosh (2 x)-\log \left (\cosh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]),x]

[Out]

x/4 + Cosh[x]/2 - Cosh[2*x]/8 - Log[Cosh[x/2]] + Sinh[2*x]/8

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fricas [A]  time = 0.43, size = 95, normalized size = 1.38 \[ \frac {6 \, x \cosh \relax (x)^{2} + 2 \, \cosh \relax (x)^{3} + 6 \, {\left (x + \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 2 \, \sinh \relax (x)^{3} - 8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (6 \, x \cosh \relax (x) + 3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + 2 \, \cosh \relax (x) - 1}{8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(6*x*cosh(x)^2 + 2*cosh(x)^3 + 6*(x + cosh(x))*sinh(x)^2 + 2*sinh(x)^3 - 8*(cosh(x)^2 + 2*cosh(x)*sinh(x)
+ sinh(x)^2)*log(cosh(x) + sinh(x) + 1) + 2*(6*x*cosh(x) + 3*cosh(x)^2 + 1)*sinh(x) + 2*cosh(x) - 1)/(cosh(x)^
2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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giac [A]  time = 0.13, size = 27, normalized size = 0.39 \[ \frac {1}{8} \, {\left (2 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{4} \, x + \frac {1}{4} \, e^{x} - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*e^x - 1)*e^(-2*x) + 3/4*x + 1/4*e^x - log(e^x + 1)

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maple [A]  time = 0.21, size = 48, normalized size = 0.70 \[ -\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(tanh(1/2*x)-1)+1/4*ln(tanh(1/2*x)-1)-1/2/(tanh(1/2*x)+1)^2+1/(tanh(1/2*x)+1)+3/4*ln(tanh(1/2*x)+1)

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maxima [A]  time = 0.31, size = 29, normalized size = 0.42 \[ -\frac {1}{4} \, x + \frac {1}{4} \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{4} \, e^{x} - \log \left (e^{\left (-x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x + 1/4*e^(-x) - 1/8*e^(-2*x) + 1/4*e^x - log(e^(-x) + 1)

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mupad [B]  time = 1.72, size = 27, normalized size = 0.39 \[ \frac {3\,x}{4}+\frac {{\mathrm {e}}^{-x}}{4}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\ln \left ({\mathrm {e}}^x+1\right )+\frac {{\mathrm {e}}^x}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x)/4 + exp(-x)/4 - exp(-2*x)/8 - log(exp(x) + 1) + exp(x)/4

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sympy [B]  time = 1.18, size = 381, normalized size = 5.52 \[ - \frac {x \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {x \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {x \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {x}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {2 \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {6 \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*tanh(x/2)**3/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) - x*tanh(x/2)**2/(4*tanh(x/2)**3 + 4*tanh(
x/2)**2 - 4*tanh(x/2) - 4) + x*tanh(x/2)/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) + x/(4*tanh(x/2)*
*3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) + 4*log(tanh(x/2) + 1)*tanh(x/2)**3/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 -
4*tanh(x/2) - 4) + 4*log(tanh(x/2) + 1)*tanh(x/2)**2/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) - 4*l
og(tanh(x/2) + 1)*tanh(x/2)/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) - 4*log(tanh(x/2) + 1)/(4*tanh
(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) + 2*tanh(x/2)**2/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) -
4) - 6*tanh(x/2)/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4*tanh(x/2) - 4) - 4/(4*tanh(x/2)**3 + 4*tanh(x/2)**2 - 4
*tanh(x/2) - 4)

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