3.9.68 \(\int \frac {x^2}{a+b \cosh (x) \sinh (x)} \, dx\) [868]

Optimal. Leaf size=281 \[ \frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}} \]

[Out]

x^2*ln(1+b*exp(2*x)/(2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-x^2*ln(1+b*exp(2*x)/(2*a+(4*a^2+b^2)^(1/2)))/(4
*a^2+b^2)^(1/2)+x*polylog(2,-b*exp(2*x)/(2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-x*polylog(2,-b*exp(2*x)/(2*
a+(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-1/2*polylog(3,-b*exp(2*x)/(2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)+1
/2*polylog(3,-b*exp(2*x)/(2*a+(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5747, 3403, 2296, 2221, 2611, 2320, 6724} \begin {gather*} \frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Cosh[x]*Sinh[x]),x]

[Out]

(x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[
4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (x*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2]
- (x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - PolyLog[3, -((b*E^(2*x))/(2*a
- Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*
a^2 + b^2])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5747

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cosh[(c_.) + (d_.)*(x_)]*(b_.)*Sinh[(c_.) + (d_.)*(x_)])^(n_.), x_Symbo
l] :> Int[(e + f*x)^m*(a + b*(Sinh[2*c + 2*d*x]/2))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {x^2}{a+b \cosh (x) \sinh (x)} \, dx &=\int \frac {x^2}{a+\frac {1}{2} b \sinh (2 x)} \, dx\\ &=2 \int \frac {e^{2 x} x^2}{-\frac {b}{2}+2 a e^{2 x}+\frac {1}{2} b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x^2}{2 a-\sqrt {4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt {4 a^2+b^2}}-\frac {(2 b) \int \frac {e^{2 x} x^2}{2 a+\sqrt {4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {2 \int x \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}+\frac {2 \int x \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\int \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}+\frac {\int \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 210, normalized size = 0.75 \begin {gather*} \frac {2 x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )-2 x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+2 x \text {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-2 x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-\text {PolyLog}\left (3,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )+\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Cosh[x]*Sinh[x]),x]

[Out]

(2*x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])] - 2*x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])] +
2*x*PolyLog[2, (b*E^(2*x))/(-2*a + Sqrt[4*a^2 + b^2])] - 2*x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2
]))] - PolyLog[3, (b*E^(2*x))/(-2*a + Sqrt[4*a^2 + b^2])] + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])
)])/(2*Sqrt[4*a^2 + b^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(529\) vs. \(2(247)=494\).
time = 1.67, size = 530, normalized size = 1.89

method	result	size
risch	\(-\frac {2 x^{3}}{3 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{-2 a -\sqrt {4 a^{2}+b^{2}}}+\frac {x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{-2 a -\sqrt {4 a^{2}+b^{2}}}-\frac {\polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {4 a \,x^{3}}{3 \sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 a \,x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 a x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {a \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {2 x^{3}}{3 \sqrt {4 a^{2}+b^{2}}}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}+\frac {x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}-\frac {\polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}\)	\(530\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*cosh(x)*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

-2/3/(-2*a-(4*a^2+b^2)^(1/2))*x^3+1/(-2*a-(4*a^2+b^2)^(1/2))*x^2*ln(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))+1/(
-2*a-(4*a^2+b^2)^(1/2))*x*polylog(2,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-1/2/(-2*a-(4*a^2+b^2)^(1/2))*polylog(
3,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-4/3/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*a*x^3+2/(4*a^2+b^2)^(1/2
)/(-2*a-(4*a^2+b^2)^(1/2))*a*x^2*ln(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))+2/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^
2)^(1/2))*a*x*polylog(2,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-1/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*a*po
lylog(3,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))-2/3/(4*a^2+b^2)^(1/2)*x^3+1/(4*a^2+b^2)^(1/2)*x^2*ln(1-b*exp(2*x)
/((4*a^2+b^2)^(1/2)-2*a))+1/(4*a^2+b^2)^(1/2)*x*polylog(2,b*exp(2*x)/((4*a^2+b^2)^(1/2)-2*a))-1/2/(4*a^2+b^2)^
(1/2)*polylog(3,b*exp(2*x)/((4*a^2+b^2)^(1/2)-2*a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*cosh(x)*sinh(x)),x, algorithm="maxima")

[Out]

integrate(x^2/(b*cosh(x)*sinh(x) + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1122 vs. \(2 (245) = 490\).
time = 0.37, size = 1122, normalized size = 3.99 \begin {gather*} -\frac {b x^{2} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} + b}{b}\right ) + b x^{2} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} - b}{b}\right ) - b x^{2} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} + b}{b}\right ) - b x^{2} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} - b}{b}\right ) + 2 \, b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} + b}{b} + 1\right ) + 2 \, b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} - b}{b} + 1\right ) - 2 \, b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} + b}{b} + 1\right ) - 2 \, b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} - b}{b} + 1\right ) - 2 \, b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, \frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}}}{b}\right ) - 2 \, b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}}}{b}\right ) + 2 \, b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, \frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}}}{b}\right ) + 2 \, b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}}}{b}\right )}{4 \, a^{2} + b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*cosh(x)*sinh(x)),x, algorithm="fricas")

[Out]

-(b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b
^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b) + b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(-((2*a*cosh(x) + 2
*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) - b)/
b) - b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2
)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b) - b*x^2*sqrt((4*a^2 + b^2)/b^2)*log(-((2*a*cosh(x) +
2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) - b)
/b) + 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 +
b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b + 1) + 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(((2*a
*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2
*a)/b) - b)/b + 1) - 2*b*x*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x)
)*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b + 1) - 2*b*x*sqrt((4*a^2 + b^2)/b^
2)*dilog(((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 +
b^2)/b^2) - 2*a)/b) - b)/b + 1) - 2*b*sqrt((4*a^2 + b^2)/b^2)*polylog(3, (2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(
x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b)/b) - 2*b*sqrt((4*a^2 + b^2
)/b^2)*polylog(3, -(2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt
((4*a^2 + b^2)/b^2) + 2*a)/b)/b) + 2*b*sqrt((4*a^2 + b^2)/b^2)*polylog(3, (2*a*cosh(x) + 2*a*sinh(x) + (b*cosh
(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b)/b) + 2*b*sqrt((4*a^2 + b^2
)/b^2)*polylog(3, -(2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt(
(4*a^2 + b^2)/b^2) - 2*a)/b)/b))/(4*a^2 + b^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*cosh(x)*sinh(x)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*cosh(x)*sinh(x)),x, algorithm="giac")

[Out]

integrate(x^2/(b*cosh(x)*sinh(x) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{a+b\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*cosh(x)*sinh(x)),x)

[Out]

int(x^2/(a + b*cosh(x)*sinh(x)), x)

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