3.5.20 \(\int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [420]

Optimal. Leaf size=451 \[ -\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {6 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^4}-\frac {6 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^4}+\frac {3 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4} \]

[Out]

(f*x+e)^3*ln(1-exp(2*d*x+2*c))/a/d-(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/d-(f*x+e)^3*ln(1+b*exp(d
*x+c)/(a+(a^2+b^2)^(1/2)))/a/d+3/2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2-3*f*(f*x+e)^2*polylog(2,-b*exp(
d*x+c)/(a-(a^2+b^2)^(1/2)))/a/d^2-3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/d^2-3/2*f^2*(f*
x+e)*polylog(3,exp(2*d*x+2*c))/a/d^3+6*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/d^3+6*f^2*(f
*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/d^3+3/4*f^3*polylog(4,exp(2*d*x+2*c))/a/d^4-6*f^3*polylog
(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/d^4-6*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/d^4

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Rubi [A]
time = 0.54, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5688, 3797, 2221, 2611, 6744, 2320, 6724, 5680} \begin {gather*} -\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^4}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^4}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d}+\frac {3 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a d^4}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Coth[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-(((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*d)) - ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(
a + Sqrt[a^2 + b^2])])/(a*d) + ((e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a*d) - (3*f*(e + f*x)^2*PolyLog[2, -((b
*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -((b
*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))])/(a*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) - (6*f^3*PolyLog[4, -((b*E^(c + d*
x))/(a - Sqrt[a^2 + b^2]))])/(a*d^4) - (6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^4) +
(3*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a*d^4)

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 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 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 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 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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)^3}{1-e^{2 (c+d x)}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a}\\ &=-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d}\\ &=-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}-\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a d^2}\\ &=-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{2 (c+d x)}\right ) \, dx}{2 a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a d^3}\\ &=-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a d^4}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^4}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^4}+\frac {3 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1002\) vs. \(2(451)=902\).
time = 5.22, size = 1002, normalized size = 2.22 \begin {gather*} -\frac {-4 d^3 e^3 \log \left (1-e^{2 (c+d x)}\right )-12 d^3 e^2 f x \log \left (1-e^{2 (c+d x)}\right )-12 d^3 e f^2 x^2 \log \left (1-e^{2 (c+d x)}\right )-4 d^3 f^3 x^3 \log \left (1-e^{2 (c+d x)}\right )+4 d^3 e^3 \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+12 d^3 e^2 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 )+12 d^3 e 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 d^3 f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+12 d^3 e^2 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 )+12 d^3 e 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 d^3 f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 d^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )+12 d^2 f (e+f x)^2 \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 )+12 d^2 e^2 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 )+24 d^2 e 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 )+12 d^2 f^3 x^2 \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 )+6 d e f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )+6 d f^3 x \text {PolyLog}\left (3,e^{2 (c+d x)}\right )-24 d e 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 )-24 d f^3 x \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 )-24 d e 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 )-24 d f^3 x \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 )-3 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )+24 f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+24 f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{4 a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^3*Coth[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-1/4*(-4*d^3*e^3*Log[1 - E^(2*(c + d*x))] - 12*d^3*e^2*f*x*Log[1 - E^(2*(c + d*x))] - 12*d^3*e*f^2*x^2*Log[1 -
E^(2*(c + d*x))] - 4*d^3*f^3*x^3*Log[1 - E^(2*(c + d*x))] + 4*d^3*e^3*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d
*x))] + 12*d^3*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 12*d^3*e*f^2*x^2*Log[1
+ (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 4*d^3*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 + b^2)*E^(2*c)])] + 12*d^3*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 1
2*d^3*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 4*d^3*f^3*x^3*Log[1 + (b*E^(2
*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d^2*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))] + 12*d^2*f*(
e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 12*d^2*e^2*f*PolyLog[2, -((b
*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 24*d^2*e*f^2*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)]))] + 12*d^2*f^3*x^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c
)]))] + 6*d*e*f^2*PolyLog[3, E^(2*(c + d*x))] + 6*d*f^3*x*PolyLog[3, E^(2*(c + d*x))] - 24*d*e*f^2*PolyLog[3,
-((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 24*d*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c -
Sqrt[(a^2 + b^2)*E^(2*c)]))] - 24*d*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))
] - 24*d*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 3*f^3*PolyLog[4, E^(2*(c
+ d*x))] + 24*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 24*f^3*PolyLog[4, -(
(b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/(a*d^4)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*coth(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*coth(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)^3*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-(log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/
(a*d))*e^3 + 3*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*f*e^2/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) +
dilog(e^(d*x + c)))*f*e^2/(a*d^2) + 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)))*f^2*e/(a*d^3) + 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)))*f^2*e/(a*d^3) + (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -
e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d
*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) - 1/2*(d^4*f^3*x^4 + 4*d^4*f
^2*x^3*e + 6*d^4*f*x^2*e^2)/(a*d^4) + integrate(-2*(b*f^3*x^3 + 3*b*f^2*x^2*e + 3*b*f*x*e^2 - (a*f^3*x^3*e^c +
3*a*f^2*x^2*e^(c + 1) + 3*a*f*x*e^(c + 2))*e^(d*x))/(a*b*e^(2*d*x + 2*c) + 2*a^2*e^(d*x + c) - a*b), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1816 vs. \(2 (427) = 854\).
time = 0.42, size = 1816, normalized size = 4.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(6*f^3*polylog(4, (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) + 6*f^3*polylog(4, (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) - 6*f^3*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 6*f^3*polylog(4, -cosh(d*x + c) - sinh(d*x
+ c)) + 3*(d^2*f^3*x^2 + 2*d^2*f^2*x*cosh(1) + d^2*f*cosh(1)^2 + d^2*f*sinh(1)^2 + 2*(d^2*f^2*x + d^2*f*cosh(
1))*sinh(1))*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) + 3*(d^2*f^3*x^2 + 2*d^2*f^2*x*cosh(1) + d^2*f*cosh(1)^2 + d^2*f*sinh(1)^2 + 2*(d^2*f^2*x + d^
2*f*cosh(1))*sinh(1))*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) - 3*(d^2*f^3*x^2 + 2*d^2*f^2*x*cosh(1) + d^2*f*cosh(1)^2 + d^2*f*sinh(1)^2 + 2*(d^2*f
^2*x + d^2*f*cosh(1))*sinh(1))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 3*(d^2*f^3*x^2 + 2*d^2*f^2*x*cosh(1) + d
^2*f*cosh(1)^2 + d^2*f*sinh(1)^2 + 2*(d^2*f^2*x + d^2*f*cosh(1))*sinh(1))*dilog(-cosh(d*x + c) - sinh(d*x + c)
) - (c^3*f^3 - 3*c^2*d*f^2*cosh(1) + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 - d^3*sinh(1)^3 + 3*(c*d^2*f - d^3*co
sh(1))*sinh(1)^2 - 3*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2)*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) - (c^3*f^3 - 3*c^2*d*f^2*cosh(1) + 3*c*d^2*f*cosh(1)^2 - d^3*cosh
(1)^3 - d^3*sinh(1)^3 + 3*(c*d^2*f - d^3*cosh(1))*sinh(1)^2 - 3*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2
)*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) + (d^3*f^3*x^3 + c^3*f
^3 + 3*(d^3*f*x + c*d^2*f)*cosh(1)^2 + 3*(d^3*f*x + c*d^2*f)*sinh(1)^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*cosh(1) +
3*(d^3*f^2*x^2 - c^2*d*f^2 + 2*(d^3*f*x + c*d^2*f)*cosh(1))*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) + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*
d^2*f)*cosh(1)^2 + 3*(d^3*f*x + c*d^2*f)*sinh(1)^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*cosh(1) + 3*(d^3*f^2*x^2 - c^
2*d*f^2 + 2*(d^3*f*x + c*d^2*f)*cosh(1))*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) - (d^3*f^3*x^3 + 3*d^3*f^2*x^2*cosh(1) + 3*d^3*f*x*cosh(1)^2 +
d^3*cosh(1)^3 + d^3*sinh(1)^3 + 3*(d^3*f*x + d^3*cosh(1))*sinh(1)^2 + 3*(d^3*f^2*x^2 + 2*d^3*f*x*cosh(1) + d^
3*cosh(1)^2)*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (c^3*f^3 - 3*c^2*d*f^2*cosh(1) + 3*c*d^2*f*cosh
(1)^2 - d^3*cosh(1)^3 - d^3*sinh(1)^3 + 3*(c*d^2*f - d^3*cosh(1))*sinh(1)^2 - 3*(c^2*d*f^2 - 2*c*d^2*f*cosh(1)
+ d^3*cosh(1)^2)*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) - 1) - (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*d^
2*f)*cosh(1)^2 + 3*(d^3*f*x + c*d^2*f)*sinh(1)^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*cosh(1) + 3*(d^3*f^2*x^2 - c^2*
d*f^2 + 2*(d^3*f*x + c*d^2*f)*cosh(1))*sinh(1))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 6*(d*f^3*x + d*f^2*c
osh(1) + d*f^2*sinh(1))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sq
rt((a^2 + b^2)/b^2))/b) - 6*(d*f^3*x + d*f^2*cosh(1) + d*f^2*sinh(1))*polylog(3, (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) + 6*(d*f^3*x + d*f^2*cosh(1) + d*f^2*sin
h(1))*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + 6*(d*f^3*x + d*f^2*cosh(1) + d*f^2*sinh(1))*polylog(3, -cosh
(d*x + c) - sinh(d*x + c)))/(a*d^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{3} \coth {\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)**3*coth(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*coth(c + d*x)/(a + b*sinh(c + d*x)), 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((f*x+e)^3*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*coth(d*x + c)/(b*sinh(d*x + c) + a), x)

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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 )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)

[Out]

int((coth(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)

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