3.3.59 \(\int \frac {x}{a+b \sinh ^2(x)} \, dx\) [259]

Optimal. Leaf size=215 \[ \frac {x \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {\text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5748, 3401, 2296, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sinh[x]^2),x]

[Out]

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

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

Rule 3401

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(
(-I)*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5748

Int[(x_)^(m_.)*((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]^2)^(n_), x_Symbol] :> Dist[1/2^n, Int[x^m*(2*a - b + b*C
osh[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n,
-1] || (EqQ[m, 1] && EqQ[n, -2]))

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.64, size = 576, normalized size = 2.68 \begin {gather*} -\frac {4 x \text {ArcTan}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )-2 i \text {ArcCos}\left (1-\frac {2 a}{b}\right ) \text {ArcTan}\left (\frac {\sqrt {-a^2+a b} \tanh (x)}{a}\right )+\left (\text {ArcCos}\left (1-\frac {2 a}{b}\right )+2 \left (\text {ArcTan}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )+\text {ArcTan}\left (\frac {\sqrt {-a^2+a b} \tanh (x)}{a}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a (-a+b)} e^{-x}}{\sqrt {b} \sqrt {2 a-b+b \cosh (2 x)}}\right )+\left (\text {ArcCos}\left (1-\frac {2 a}{b}\right )-2 \left (\text {ArcTan}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )+\text {ArcTan}\left (\frac {\sqrt {-a^2+a b} \tanh (x)}{a}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a (-a+b)} e^x}{\sqrt {b} \sqrt {2 a-b+b \cosh (2 x)}}\right )-\left (\text {ArcCos}\left (1-\frac {2 a}{b}\right )+2 \text {ArcTan}\left (\frac {\sqrt {-a^2+a b} \tanh (x)}{a}\right )\right ) \log \left (\frac {2 a \left (-i a+i b+\sqrt {a (-a+b)}\right ) (-1+\tanh (x))}{-i a b+b \sqrt {a (-a+b)} \tanh (x)}\right )-\left (\text {ArcCos}\left (1-\frac {2 a}{b}\right )-2 \text {ArcTan}\left (\frac {\sqrt {-a^2+a b} \tanh (x)}{a}\right )\right ) \log \left (\frac {2 a \left (i a-i b+\sqrt {a (-a+b)}\right ) (1+\tanh (x))}{-i a b+b \sqrt {a (-a+b)} \tanh (x)}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (-2 a+b-2 i \sqrt {a (-a+b)}\right ) \left (i a+\sqrt {a (-a+b)} \tanh (x)\right )}{-i a b+b \sqrt {a (-a+b)} \tanh (x)}\right )+\text {PolyLog}\left (2,\frac {\left (-2 a+b+2 i \sqrt {a (-a+b)}\right ) \left (i a+\sqrt {a (-a+b)} \tanh (x)\right )}{-i a b+b \sqrt {a (-a+b)} \tanh (x)}\right )\right )}{4 \sqrt {a (-a+b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sinh[x]^2),x]

[Out]

-1/4*(4*x*ArcTan[(a*Coth[x])/Sqrt[-(a*(a - b))]] - (2*I)*ArcCos[1 - (2*a)/b]*ArcTan[(Sqrt[-a^2 + a*b]*Tanh[x])
/a] + (ArcCos[1 - (2*a)/b] + 2*(ArcTan[(a*Coth[x])/Sqrt[-(a*(a - b))]] + ArcTan[(Sqrt[-a^2 + a*b]*Tanh[x])/a])
)*Log[(Sqrt[2]*Sqrt[a*(-a + b)])/(Sqrt[b]*E^x*Sqrt[2*a - b + b*Cosh[2*x]])] + (ArcCos[1 - (2*a)/b] - 2*(ArcTan
[(a*Coth[x])/Sqrt[-(a*(a - b))]] + ArcTan[(Sqrt[-a^2 + a*b]*Tanh[x])/a]))*Log[(Sqrt[2]*Sqrt[a*(-a + b)]*E^x)/(
Sqrt[b]*Sqrt[2*a - b + b*Cosh[2*x]])] - (ArcCos[1 - (2*a)/b] + 2*ArcTan[(Sqrt[-a^2 + a*b]*Tanh[x])/a])*Log[(2*
a*((-I)*a + I*b + Sqrt[a*(-a + b)])*(-1 + Tanh[x]))/((-I)*a*b + b*Sqrt[a*(-a + b)]*Tanh[x])] - (ArcCos[1 - (2*
a)/b] - 2*ArcTan[(Sqrt[-a^2 + a*b]*Tanh[x])/a])*Log[(2*a*(I*a - I*b + Sqrt[a*(-a + b)])*(1 + Tanh[x]))/((-I)*a
*b + b*Sqrt[a*(-a + b)]*Tanh[x])] + I*(-PolyLog[2, ((-2*a + b - (2*I)*Sqrt[a*(-a + b)])*(I*a + Sqrt[a*(-a + b)
]*Tanh[x]))/((-I)*a*b + b*Sqrt[a*(-a + b)]*Tanh[x])] + PolyLog[2, ((-2*a + b + (2*I)*Sqrt[a*(-a + b)])*(I*a +
Sqrt[a*(-a + b)]*Tanh[x]))/((-I)*a*b + b*Sqrt[a*(-a + b)]*Tanh[x])]))/Sqrt[a*(-a + b)]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(171)=342\).
time = 0.86, size = 505, normalized size = 2.35

method	result	size
risch	\(\frac {x \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \sqrt {a \left (a -b \right )}}-\frac {x^{2}}{2 \sqrt {a \left (a -b \right )}}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \sqrt {a \left (a -b \right )}}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) x}{-2 \sqrt {a \left (a -b \right )}-2 a +b}-\frac {x^{2}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a x}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b x}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {a \,x^{2}}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {x^{2} b}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{-4 \sqrt {a \left (a -b \right )}-4 a +2 b}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}\)	\(505\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(x/(b*sinh(x)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (165) = 330\).
time = 0.50, size = 837, normalized size = 3.89 \begin {gather*} -\frac {b x \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} + b}{b}\right ) + b x \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} - b}{b}\right ) - b x \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} + b}{b}\right ) - b x \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} - b}{b}\right ) + b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} + b}{b} + 1\right ) + b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} - b}{b} + 1\right ) - b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} + b}{b} + 1\right ) - b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} - b}{b} + 1\right )}{2 \, {\left (a^{2} - a b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a + b \sinh ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a + b*sinh(x)**2), x)

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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/(a+b*sinh(x)^2),x, algorithm="giac")

[Out]

integrate(x/(b*sinh(x)^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*sinh(x)^2),x)

[Out]

int(x/(a + b*sinh(x)^2), x)

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