3.5.50 \(\int \frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [450]

Optimal. Leaf size=419 \[ -\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 b f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 b f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {2 b f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {b f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3} \]

[Out]

-4*f*(f*x+e)*arctanh(exp(d*x+c))/a/d^2-(f*x+e)^2*csch(d*x+c)/a/d-b*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^2/d+b*(f*x
+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d+b*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d-2*
f^2*polylog(2,-exp(d*x+c))/a/d^3+2*f^2*polylog(2,exp(d*x+c))/a/d^3-b*f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^2/d
^2+2*b*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2+2*b*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a
+(a^2+b^2)^(1/2)))/a^2/d^2+1/2*b*f^2*polylog(3,exp(2*d*x+2*c))/a^2/d^3-2*b*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/a^2/d^3-2*b*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^3

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Rubi [A]
time = 0.56, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5706, 5560, 4267, 2317, 2438, 5688, 3797, 2221, 2611, 2320, 6724, 5680} \begin {gather*} -\frac {2 b f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}+\frac {b f^2 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(-4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + (b*(e + f*x)^2*Log[1 + (b*
E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/
(a^2*d) - (b*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2
*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^
2*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*(e + f*x)*Pol
yLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3)
- (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (b*f^2*PolyLog[3, E^(2*(c + d*x)
)])/(2*a^2*d^3)

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 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 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 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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5688

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[c +
d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5706

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

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 {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}-\frac {b \int (e+f x)^2 \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(2 f) \int (e+f x) \text {csch}(c+d x) \, dx}{a d}\\ &=-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {(2 b f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {\left (b f^2\right ) \int \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (2 b f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^2}-\frac {\left (2 b f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^2}\\ &=-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}\\ &=-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 b f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 9.42, size = 783, normalized size = 1.87 \begin {gather*} \frac {-8 a d e f \tanh ^{-1}\left (e^{c+d x}\right )-2 a d^2 e^2 \text {csch}(c+d x)-4 a d^2 e f x \text {csch}(c+d x)-2 a d^2 f^2 x^2 \text {csch}(c+d x)+4 a d f^2 x \log \left (1-e^{c+d x}\right )-4 a d f^2 x \log \left (1+e^{c+d x}\right )-2 b d^2 e^2 \log \left (1-e^{2 (c+d x)}\right )-4 b d^2 e f x \log \left (1-e^{2 (c+d x)}\right )-2 b d^2 f^2 x^2 \log \left (1-e^{2 (c+d x)}\right )+2 b d^2 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+4 b d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 b d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-4 a f^2 \text {PolyLog}\left (2,-e^{c+d x}\right )+4 a f^2 \text {PolyLog}\left (2,e^{c+d x}\right )-2 b d e f \text {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 b d f^2 x \text {PolyLog}\left (2,e^{2 (c+d x)}\right )+4 b d e f \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 b d f^2 x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 b d e f \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 b d f^2 x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+b f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )-4 b f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-4 b f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{2 a^2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(-8*a*d*e*f*ArcTanh[E^(c + d*x)] - 2*a*d^2*e^2*Csch[c + d*x] - 4*a*d^2*e*f*x*Csch[c + d*x] - 2*a*d^2*f^2*x^2*C
sch[c + d*x] + 4*a*d*f^2*x*Log[1 - E^(c + d*x)] - 4*a*d*f^2*x*Log[1 + E^(c + d*x)] - 2*b*d^2*e^2*Log[1 - E^(2*
(c + d*x))] - 4*b*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] - 2*b*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 2*b*d^2*e^2*
Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 4*b*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 +
b^2)*E^(2*c)])] + 2*b*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 4*b*d^2*e*
f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*b*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))
/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 4*a*f^2*PolyLog[2, -E^(c + d*x)] + 4*a*f^2*PolyLog[2, E^(c + d*x)] - 2
*b*d*e*f*PolyLog[2, E^(2*(c + d*x))] - 2*b*d*f^2*x*PolyLog[2, E^(2*(c + d*x))] + 4*b*d*e*f*PolyLog[2, -((b*E^(
2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 4*b*d*f^2*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(
a^2 + b^2)*E^(2*c)]))] + 4*b*d*e*f*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 4*b*
d*f^2*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + b*f^2*PolyLog[3, E^(2*(c + d*x)
)] - 4*b*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 4*b*f^2*PolyLog[3, -((b*E^
(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/(2*a^2*d^3)

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Maple [F]
time = 2.12, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \coth \left (d x +c \right ) \mathrm {csch}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

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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((f*x+e)^2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) + b*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*d) - b*l
og(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e^2 - 2*(f^2*x^2*e^c + 2*f*x*e^(c + 1))*e^(d*x
)/(a*d*e^(2*d*x + 2*c) - a*d) - 2*f*e*log(e^(d*x + c) + 1)/(a*d^2) + 2*f*e*log(e^(d*x + c) - 1)/(a*d^2) - (d^2
*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*b*f^2/(a^2*d^3) - (d^2*x^2
*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*b*f^2/(a^2*d^3) - 2*(b*d*f*e +
a*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d*f*e - a*f^2)*(d*x*log(-e^(d*x + c)
+ 1) + dilog(e^(d*x + c)))/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*f*e + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^
3*f^2*x^3 + 3*(b*d*f*e - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(-2*(b^2*f^2*x^2 + 2*b^2*f*x*e - (a*b*f^2*x^2*e^
c + 2*a*b*f*x*e^(c + 1))*e^(d*x))/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3408 vs. \(2 (399) = 798\).
time = 0.46, size = 3408, normalized size = 8.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a*d^2*f^2*x^2 + 2*a*d^2*f*x*cosh(1) + a*d^2*cosh(1)^2 + a*d^2*sinh(1)^2 + 2*(a*d^2*f*x + a*d^2*cosh(1))*s
inh(1))*cosh(d*x + c) + 2*(b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1) - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh
(1))*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x
+ b*d*f*cosh(1) + b*d*f*sinh(1))*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1) - (b*d*f^2
*x + b*d*f*cosh(1) + b*d*f*sinh(1))*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*cosh(d*x +
c)*sinh(d*x + c) - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*si
nh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(b*d*f^2*x + b*d*f*cos
h(1) + b*d*f*sinh(1) - a*f^2 - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1) - a*f^2)*cosh(d*x + c)^2 - 2*(b*d*f^
2*x + b*d*f*cosh(1) + b*d*f*sinh(1) - a*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*
sinh(1) - a*f^2)*sinh(d*x + c)^2)*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(b*d*f^2*x + b*d*f*cosh(1) + b*d*f*
sinh(1) + a*f^2 - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1) + a*f^2)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*f*c
osh(1) + b*d*f*sinh(1) + a*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1) + a*f
^2)*sinh(d*x + c)^2)*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2
+ b*d^2*sinh(1)^2 - (b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*co
sh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*
c*d*f - b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2
+ b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1
))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*c^2*f^2 - 2*b*c*d*f*cosh(
1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - (b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 -
2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*
d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*c^2*f^2 - 2*b*c*d*f*cosh
(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^2 - 2*(b*c*d*f -
b*d^2*cosh(1))*sinh(1))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d^2*
f^2*x^2 - b*c^2*f^2 - (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b*c*d*f)*s
inh(1))*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b*c*
d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(
b*d^2*f*x + b*c*d*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b*c*d*f)*sinh
(1))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/
b) + (b*d^2*f^2*x^2 - b*c^2*f^2 - (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x
+ b*c*d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^
2*f*x + b*c*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*c
osh(1) + 2*(b*d^2*f*x + b*c*d*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b
*c*d*f)*sinh(1))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2
)/b^2) - b)/b) - (b*d^2*f^2*x^2 + 2*a*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - (b*d^2*f^2*x^2 + 2*a*d*f^2
*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*(b*d^2*f*x + a*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh(1) + a*d*f)
*sinh(1))*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 + 2*a*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*(b*d^2*f*x
+ a*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh(1) + a*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^2*x^2
+ 2*a*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*(b*d^2*f*x + a*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh
(1) + a*d*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b*d^2*f*x + a*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh(1) + a*d*f)*
sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c + a)*d*f*cosh(1)
+ (b*c^2 + 2*a*c)*f^2 - (b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c + a)*d*f*cosh(1) + (b*c^2 + 2*a*c)*f^2 +
2*(b*d^2*cosh(1) - (b*c + a)*d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c + a
)*d*f*cosh(1) + (b*c^2 + 2*a*c)*f^2 + 2*(b*d^2*cosh(1) - (b*c + a)*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) -
(b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c + a)*d*f*cosh(1) + (b*c^2 + 2*a*c)*f^2 + 2*(b*d^2*cosh(1) - (b*c
+ a)*d*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b*d^2*c...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**2*coth(c + d*x)*csch(c + d*x)/(a + b*sinh(c + d*x)), x)

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Giac [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((f*x+e)^2*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(c + d*x)*(e + f*x)^2)/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

int((coth(c + d*x)*(e + f*x)^2)/(sinh(c + d*x)*(a + b*sinh(c + d*x))), x)

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