3.28 \(\int \cosh (x) \log ^2(1+\cosh ^2(x)) \, dx\)

Optimal. Leaf size=159 \[ 4 i \sqrt{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 \sqrt{2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+8 \sqrt{2} \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \]

[Out]

-8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]] + (4*I)*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]^2 + 8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2
]]*Log[(2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 4*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[2 + Sinh[x]^2] + (4*I)*Sqrt[
2]*PolyLog[2, 1 - (2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 8*Sinh[x] - 4*Log[2 + Sinh[x]^2]*Sinh[x] + Log[2 + Sinh
[x]^2]^2*Sinh[x]

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Rubi [A]  time = 0.202866, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4358, 2450, 2476, 2448, 321, 203, 2470, 12, 4920, 4854, 2402, 2315} \[ 4 i \sqrt{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 \sqrt{2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+8 \sqrt{2} \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Log[1 + Cosh[x]^2]^2,x]

[Out]

-8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]] + (4*I)*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]^2 + 8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2
]]*Log[(2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 4*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[2 + Sinh[x]^2] + (4*I)*Sqrt[
2]*PolyLog[2, 1 - (2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 8*Sinh[x] - 4*Log[2 + Sinh[x]^2]*Sinh[x] + Log[2 + Sinh
[x]^2]^2*Sinh[x]

Rule 4358

Int[Cosh[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(
b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*
(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]

Rule 2450

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

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx &=\operatorname{Subst}\left (\int \log ^2\left (2+x^2\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname{Subst}\left (\int \frac{x^2 \log \left (2+x^2\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname{Subst}\left (\int \left (\log \left (2+x^2\right )-\frac{2 \log \left (2+x^2\right )}{2+x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname{Subst}\left (\int \log \left (2+x^2\right ) \, dx,x,\sinh (x)\right )+8 \operatorname{Subst}\left (\int \frac{\log \left (2+x^2\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+8 \operatorname{Subst}\left (\int \frac{x^2}{2+x^2} \, dx,x,\sinh (x)\right )-16 \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2} \left (2+x^2\right )} \, dx,x,\sinh (x)\right )\\ &=4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-16 \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\sinh (x)\right )-\left (8 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2+4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+8 \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{i-\frac{x}{\sqrt{2}}} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2+8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-8 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i x}{\sqrt{2}}}\right )}{1+\frac{x^2}{2}} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2+8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+\left (8 i \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sinh (x)}{\sqrt{2}}}\right )\\ &=-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2+8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \log \left (2+\sinh ^2(x)\right )+4 i \sqrt{2} \text{Li}_2\left (1-\frac{4}{2+i \sqrt{2} \sinh (x)}\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0862541, size = 122, normalized size = 0.77 \[ 4 i \sqrt{2} \text{PolyLog}\left (2,\frac{\sqrt{2} \sinh (x)+2 i}{\sqrt{2} \sinh (x)-2 i}\right )+\sinh (x) \left (\log ^2\left (\sinh ^2(x)+2\right )-4 \log \left (\sinh ^2(x)+2\right )+8\right )+4 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \left (\log \left (\sinh ^2(x)+2\right )+2 \log \left (\frac{4 i}{-\sqrt{2} \sinh (x)+2 i}\right )+i \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )-2\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cosh[x]*Log[1 + Cosh[x]^2]^2,x]

[Out]

4*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*(-2 + I*ArcTan[Sinh[x]/Sqrt[2]] + 2*Log[(4*I)/(2*I - Sqrt[2]*Sinh[x])] + Log
[2 + Sinh[x]^2]) + (4*I)*Sqrt[2]*PolyLog[2, (2*I + Sqrt[2]*Sinh[x])/(-2*I + Sqrt[2]*Sinh[x])] + (8 - 4*Log[2 +
Sinh[x]^2] + Log[2 + Sinh[x]^2]^2)*Sinh[x]

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Maple [F]  time = 2.18, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ( x \right ) \left ( \ln \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*ln(1+cosh(x)^2)^2,x)

[Out]

int(cosh(x)*ln(1+cosh(x)^2)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*log(1+cosh(x)^2)^2,x, algorithm="maxima")

[Out]

1/2*(e^(2*x) - 1)*e^(-x)*log(e^(4*x) + 6*e^(2*x) + 1)^2 - 2*(e^(-x) + integrate((e^(2*x) + 6)*e^x/(e^(4*x) + 6
*e^(2*x) + 1), x))*log(2)^2 + 2*(e^x - integrate((6*e^(2*x) + 1)*e^x/(e^(4*x) + 6*e^(2*x) + 1), x))*log(2)^2 +
14*integrate(e^(3*x)/(e^(4*x) + 6*e^(2*x) + 1), x)*log(2)^2 + 14*integrate(e^x/(e^(4*x) + 6*e^(2*x) + 1), x)*
log(2)^2 + 4*integrate(x*e^(6*x)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 28*integrate(x*e^(4*x)/(e^(5*x) + 6*
e^(3*x) + e^x), x)*log(2) + 28*integrate(x*e^(2*x)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 2*integrate(e^(6*x
)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*integrate(e^(4*x)*log(e^(4*x) + 6*e
^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*integrate(e^(2*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x
) + 6*e^(3*x) + e^x), x)*log(2) + 4*integrate(x/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 2*integrate(log(e^(4*
x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 2*integrate(x^2*e^(6*x)/(e^(5*x) + 6*e^(3*x) + e^
x), x) + 14*integrate(x^2*e^(4*x)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 14*integrate(x^2*e^(2*x)/(e^(5*x) + 6*e^(3
*x) + e^x), x) - 2*integrate(x*e^(6*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 14*integ
rate(x*e^(4*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 14*integrate(x*e^(2*x)*log(e^(4*
x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 2*integrate(x^2/(e^(5*x) + 6*e^(3*x) + e^x), x) - 2*inte
grate(x*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 4*integrate(e^(6*x)*log(e^(4*x) + 6*e^(
2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 8*integrate(e^(4*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3
*x) + e^x), x) + 12*integrate(e^(2*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*log(1+cosh(x)^2)^2,x, algorithm="fricas")

[Out]

integral(cosh(x)*log(cosh(x)^2 + 1)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\cosh ^{2}{\left (x \right )} + 1 \right )}^{2} \cosh{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*ln(1+cosh(x)**2)**2,x)

[Out]

Integral(log(cosh(x)**2 + 1)**2*cosh(x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*log(1+cosh(x)^2)^2,x, algorithm="giac")

[Out]

integrate(cosh(x)*log(cosh(x)^2 + 1)^2, x)