3.250 \(\int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=420 \[ -\frac {b^2 f \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {b^2 f \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}+\frac {b (e+f x) \coth (c+d x)}{a^2 d}-\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}-\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {f \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {f \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac {f \text {csch}(c+d x)}{2 a d^2}+\frac {(e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.73, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5575, 4185, 4182, 2279, 2391, 4184, 3475, 3322, 2264, 2190} \[ -\frac {b^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}+\frac {b^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^3 d^2 \sqrt {a^2+b^2}}-\frac {b^2 f \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {b^2 f \text {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}+\frac {f \text {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {f \text {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \sqrt {a^2+b^2}}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}+\frac {b (e+f x) \coth (c+d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}+\frac {(e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)*Coth[c
+ d*x])/(a^2*d) - (f*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*
x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (b*f*Log[Sinh[c + d*x]])/(a^2*d^2) + (f*PolyLog[2, -E^
(c + d*x)])/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (
b^2*f*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*S
qrt[a^2 + b^2]*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2)

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2264

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 2279

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 2391

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 3322

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4182

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 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 5575

Int[(Csch[(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*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[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]

Rubi steps

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

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Mathematica [C]  time = 7.91, size = 736, normalized size = 1.75 \[ -\frac {i b^2 f \left (i \left (\text {Li}_2\left (-e^{-c-d x}\right )-\text {Li}_2\left (e^{-c-d x}\right )\right )+i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )}{a^3 d^2}-\frac {b^2 c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d^2}+\frac {b^2 e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d}+\frac {\text {csch}\left (\frac {1}{2} (c+d x)\right ) \left (-a f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (a f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}+\frac {b^3 \left (2 d e \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )-2 c f \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )\right )}{a^3 d^2 \sqrt {a^2+b^2}}+\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right ) (-f (c+d x)+c f-d e)}{8 a d^2}+\frac {\text {sech}^2\left (\frac {1}{2} (c+d x)\right ) (-f (c+d x)+c f-d e)}{8 a d^2}+\frac {i f \left (i \left (\text {Li}_2\left (-e^{-c-d x}\right )-\text {Li}_2\left (e^{-c-d x}\right )\right )+i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )}{2 a d^2}+\frac {c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}-\frac {e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d*x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*
x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) - (b*f*Lo
g[Sinh[c + d*x]])/(a^2*d^2) - (e*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (b^2*e*Log[Tanh[(c + d*x)/2]])/(a^3*d) + (c
*f*Log[Tanh[(c + d*x)/2]])/(2*a*d^2) - (b^2*c*f*Log[Tanh[(c + d*x)/2]])/(a^3*d^2) + ((I/2)*f*(I*(c + d*x)*(Log
[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - PolyLog[2, E^(-c - d*x)])))/(a*d^
2) - (I*b^2*f*(I*(c + d*x)*(Log[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - Po
lyLog[2, E^(-c - d*x)])))/(a^3*d^2) + (b^3*(2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*c*f*ArcTanh
[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*
x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^3*Sqrt[a^2 + b^2]*d^2) + ((-(d*e) + c*f - f*(c + d*x)
)*Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(2*b*d*e*Sinh[(c + d*x)/2] + a*f*Sinh[(c + d*x)/2] - 2*b
*c*f*Sinh[(c + d*x)/2] + 2*b*f*(c + d*x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)

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fricas [B]  time = 0.58, size = 4720, normalized size = 11.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*c*f)*cosh(d*x + c)^4 + 4*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b
^3)*c*f)*sinh(d*x + c)^4 - 2*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*e + (a^4 + a^2*b^2)*f)*cosh(d*x + c)^3
- 2*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*e + (a^4 + a^2*b^2)*f - 8*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*
b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a^3*b + a*b^3)*d*e + 4*(a^3*b + a*b^3)*c*f - 4*((a^3*b + a*b^3)*
d*f*x - (a^3*b + a*b^3)*d*e + 2*(a^3*b + a*b^3)*c*f)*cosh(d*x + c)^2 - 2*(2*(a^3*b + a*b^3)*d*f*x - 2*(a^3*b +
a*b^3)*d*e + 4*(a^3*b + a*b^3)*c*f - 12*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*c*f)*cosh(d*x + c)^2 + 3*((a
^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*e + (a^4 + a^2*b^2)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b^4*f*cosh(
d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c)^4 - 2*b^4*f*cosh(d*x + c)^2 + b^4*f +
2*(3*b^4*f*cosh(d*x + c)^2 - b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*x + c)^3 - b^4*f*cosh(d*x + c))*sinh(d*
x + c))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(b^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*si
nh(d*x + c)^4 - 2*b^4*f*cosh(d*x + c)^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 - b^4*f)*sinh(d*x + c)^2 + 4*(b^4
*f*cosh(d*x + c)^3 - b^4*f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^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^4*d*e - b^4*c*f + (b^
4*d*e - b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*e - b^4*c*f)*s
inh(d*x + c)^4 - 2*(b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 - 2*(b^4*d*e - b^4*c*f - 3*(b^4*d*e - b^4*c*f)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*e - b^4*c*f)*cosh(d*x + c)^3 - (b^4*d*e - b^4*c*f)*cosh(d*x + c))*sinh(d*
x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2
*(b^4*d*e - b^4*c*f + (b^4*d*e - b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^
3 + (b^4*d*e - b^4*c*f)*sinh(d*x + c)^4 - 2*(b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 - 2*(b^4*d*e - b^4*c*f - 3*(b^
4*d*e - b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*e - b^4*c*f)*cosh(d*x + c)^3 - (b^4*d*e - b^4*c*
f)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a
^2 + b^2)/b^2) + 2*a) - 2*(b^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*f*x + b^4*c*
f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^4 - 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x +
c)^2 - 2*(b^4*d*f*x + b^4*c*f - 3*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f*x + b^4
*c*f)*cosh(d*x + c)^3 - (b^4*d*f*x + b^4*c*f)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*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) + 2*(b^4*d*f*x
+ b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (
b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^4 - 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2 - 2*(b^4*d*f*x + b^4*c*f - 3*(b
^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^3 - (b^4*d*f*x +
b^4*c*f)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cos
h(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*e -
(a^4 + a^2*b^2)*f)*cosh(d*x + c) - ((a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^4 + 4*(a^4 - a^2*b^2 - 2*b^4)*f*c
osh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^2*b^2 - 2*b^4)*f*sinh(d*x + c)^4 - 2*(a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*
x + c)^2 + 2*(3*(a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^2 - (a^4 - a^2*b^2 - 2*b^4)*f)*sinh(d*x + c)^2 + (a^4
- a^2*b^2 - 2*b^4)*f + 4*((a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^3 - (a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c))
*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + ((a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^4 + 4*(a^4 - a
^2*b^2 - 2*b^4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^2*b^2 - 2*b^4)*f*sinh(d*x + c)^4 - 2*(a^4 - a^2*b^2
- 2*b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^2 - (a^4 - a^2*b^2 - 2*b^4)*f)*sinh
(d*x + c)^2 + (a^4 - a^2*b^2 - 2*b^4)*f + 4*((a^4 - a^2*b^2 - 2*b^4)*f*cosh(d*x + c)^3 - (a^4 - a^2*b^2 - 2*b^
4)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (
a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*cosh(d*x + c)^4 + 4*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 -
a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (
a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*sinh(d*x + c)^4 + (a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2
*b^2 - 2*b^4)*d*e - 2*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*cosh
(d*x + c)^2 - 2*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*d*e - 3*((a^4 - a^2*b^2 - 2*b^4)*d*f*
x + (a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*cosh(d*x + c)^2 - 2*(a^3*b + a*b^3)*f)*sinh(d*x + c)^2
- 2*(a^3*b + a*b^3)*f + 4*(((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)
*cosh(d*x + c)^3 - ((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*d*e - 2*(a^3*b + a*b^3)*f)*cosh(d*
x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (((a^4 - a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a
*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c)^4 + 4*((a^4 - a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a*b^3 - (
a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 - a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a*b^3
- (a^4 - a^2*b^2 - 2*b^4)*c)*f)*sinh(d*x + c)^4 + (a^4 - a^2*b^2 - 2*b^4)*d*e - 2*((a^4 - a^2*b^2 - 2*b^4)*d*
e + (2*a^3*b + 2*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c)^2 - 2*((a^4 - a^2*b^2 - 2*b^4)*d*e - 3*((
a^4 - a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c)^2 + (2*a^3*b + 2
*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*sinh(d*x + c)^2 + (2*a^3*b + 2*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f + 4
*(((a^4 - a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c)^3 - ((a^4 -
a^2*b^2 - 2*b^4)*d*e + (2*a^3*b + 2*a*b^3 - (a^4 - a^2*b^2 - 2*b^4)*c)*f)*cosh(d*x + c))*sinh(d*x + c))*log(co
sh(d*x + c) + sinh(d*x + c) - 1) - (((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f)*cosh(d*x + c
)^4 + 4*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 -
a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f)*sinh(d*x + c)^4 + (a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 -
a^2*b^2 - 2*b^4)*c*f - 2*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f)*cosh(d*x + c)^2 - 2*((
a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f - 3*((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2
- 2*b^4)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*
c*f)*cosh(d*x + c)^3 - ((a^4 - a^2*b^2 - 2*b^4)*d*f*x + (a^4 - a^2*b^2 - 2*b^4)*c*f)*cosh(d*x + c))*sinh(d*x +
c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*((a^4 + a^2*b^2)*d*f*x - 8*((a^3*b + a*b^3)*d*f*x + (a^3*b +
a*b^3)*c*f)*cosh(d*x + c)^3 + (a^4 + a^2*b^2)*d*e + 3*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*e + (a^4 + a^
2*b^2)*f)*cosh(d*x + c)^2 - (a^4 + a^2*b^2)*f + 4*((a^3*b + a*b^3)*d*f*x - (a^3*b + a*b^3)*d*e + 2*(a^3*b + a*
b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + a^3*b^2)*d^2*cosh(d*x + c)^4 + 4*(a^5 + a^3*b^2)*d^2*cosh(d*x
+ c)*sinh(d*x + c)^3 + (a^5 + a^3*b^2)*d^2*sinh(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d^2*cosh(d*x + c)^2 + (a^5 + a^
3*b^2)*d^2 + 2*(3*(a^5 + a^3*b^2)*d^2*cosh(d*x + c)^2 - (a^5 + a^3*b^2)*d^2)*sinh(d*x + c)^2 + 4*((a^5 + a^3*b
^2)*d^2*cosh(d*x + c)^3 - (a^5 + a^3*b^2)*d^2*cosh(d*x + c))*sinh(d*x + c))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.26, size = 861, normalized size = 2.05 \[ -\frac {a d f x \,{\mathrm e}^{3 d x +3 c}+a d e \,{\mathrm e}^{3 d x +3 c}-2 b d f x \,{\mathrm e}^{2 d x +2 c}+a d f x \,{\mathrm e}^{d x +c}+a f \,{\mathrm e}^{3 d x +3 c}-2 b d e \,{\mathrm e}^{2 d x +2 c}+a d e \,{\mathrm e}^{d x +c}+2 b d f x -a f \,{\mathrm e}^{d x +c}+2 b d e}{d^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {b^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d \,a^{3}}-\frac {2 b^{3} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {2 b^{3} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{3}}-\frac {b^{2} f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{3}}-\frac {b^{3} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) f x}{2 a d}-\frac {b^{2} f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{3}}-\frac {b^{2} f \dilog \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}+\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}-\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{2}}+\frac {2 b f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{2}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{2}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{2}}+\frac {f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{2 d^{2} a}+\frac {f \dilog \left ({\mathrm e}^{d x +c}\right )}{2 d^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (8 \, b^{3} \int \frac {x e^{\left (d x + c\right )}}{4 \, {\left (a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} e^{\left (d x + c\right )} - a^{3} b\right )}}\,{d x} + 8 \, a^{2} d \int \frac {x}{16 \, {\left (a^{3} d e^{\left (d x + c\right )} + a^{3} d\right )}}\,{d x} - 16 \, b^{2} d \int \frac {x}{16 \, {\left (a^{3} d e^{\left (d x + c\right )} + a^{3} d\right )}}\,{d x} + 8 \, a^{2} d \int \frac {x}{16 \, {\left (a^{3} d e^{\left (d x + c\right )} - a^{3} d\right )}}\,{d x} - 16 \, b^{2} d \int \frac {x}{16 \, {\left (a^{3} d e^{\left (d x + c\right )} - a^{3} d\right )}}\,{d x} - a b {\left (\frac {d x + c}{a^{3} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3} d^{2}}\right )} - a b {\left (\frac {d x + c}{a^{3} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} - 1\right )}{a^{3} d^{2}}\right )} - \frac {2 \, b d x e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b d x - {\left (a d x e^{\left (3 \, c\right )} + a e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (a d x e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{a^{2} d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d^{2}}\right )} f - \frac {1}{2} \, e {\left (\frac {2 \, b^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d} - \frac {2 \, {\left (a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-3 \, d x - 3 \, c\right )} - 2 \, b\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*b^3*integrate(1/4*x*e^(d*x + c)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b), x) + 8*a^2*d*integrat
e(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 8*a^2*d
*integrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d), x) - 16*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d), x)
- a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1)/(a^3*d^2)) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)
/(a^3*d^2)) - (2*b*d*x*e^(2*d*x + 2*c) - 2*b*d*x - (a*d*x*e^(3*c) + a*e^(3*c))*e^(3*d*x) - (a*d*x*e^c - a*e^c)
*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2))*f - 1/2*e*(2*b^3*log((b*e^(-d*x - c
) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d) - 2*(a*e^(-d*x - c) +
2*b*e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) -
(a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x)*csch(c + d*x)**3/(a + b*sinh(c + d*x)), x)

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