3.3.50 \(\int \frac {(e+f x) \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [250]

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

[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 = 0.52, 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 = {5694, 4270, 4267, 2317, 2438, 4269, 3556, 3403, 2296, 2221} \begin {gather*} -\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} \end {gather*}

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 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 3403

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

Rule 3556

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

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 4269

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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5694

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 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac {f \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 7.13, size = 736, normalized size = 1.75 \begin {gather*} \frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \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 ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}-\frac {e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d}+\frac {b^2 e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d}+\frac {c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}-\frac {b^2 c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d^2}+\frac {i f \left (i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{-c-d x}\right )-\text {PolyLog}\left (2,e^{-c-d x}\right )\right )\right )}{2 a d^2}-\frac {i b^2 f \left (i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{-c-d x}\right )-\text {PolyLog}\left (2,e^{-c-d x}\right )\right )\right )}{a^3 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 )-2 c f \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {(-d e+c f-f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )+a f \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} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(386)=772\).
time = 1.74, size = 861, normalized size = 2.05

method	result	size
risch	\(-\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 e d}{d^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{2 d a}-\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 d^{2} a}-\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 ) 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 {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 {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^{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 {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 {2 b f \ln \left ({\mathrm e}^{d x +c}\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}-\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}}}\)	\(861\)

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,method=_RETURNVERBOSE)

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

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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)*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*(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))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5442 vs. \(2 (386) = 772\).
time = 0.42, size = 5442, normalized size = 12.96 \begin {gather*} \text {Too large to display} \end {gather*}

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*cosh(1) + (a^4 + a^2*b^2)*d*sinh(1) +
(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*cosh(1) + (a^4 + a^2*b^2)*d*
sinh(1) + (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)*c*f - 4*(a^3*b + a*b^3)*d*cosh(1) - 4*((a^3*b + a*b^3)*d*f*x + 2*(a^3*b + a*b^3)*c*f - (a^3
*b + a*b^3)*d*cosh(1) - (a^3*b + a*b^3)*d*sinh(1))*cosh(d*x + c)^2 - 4*(a^3*b + a*b^3)*d*sinh(1) - 2*(2*(a^3*b
+ a*b^3)*d*f*x + 4*(a^3*b + a*b^3)*c*f - 2*(a^3*b + a*b^3)*d*cosh(1) - 12*((a^3*b + a*b^3)*d*f*x + (a^3*b + a
*b^3)*c*f)*cosh(d*x + c)^2 - 2*(a^3*b + a*b^3)*d*sinh(1) + 3*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*d*cosh(1
) + (a^4 + a^2*b^2)*d*sinh(1) + (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))*sqrt((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*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))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sin
h(1) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^4 + 4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))
*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*sinh(d*x + c)^4 - 2*(b^4*c*f - b^4*
d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2 - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) - 3*(b^4*c*f - b^4*d*c
osh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d
*x + c)^3 - (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*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*c*f - b^4*d*cosh(1) - b^4*d*
sinh(1) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^4 + 4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(
1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*sinh(d*x + c)^4 - 2*(b^4*c*f - b
^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2 - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) - 3*(b^4*c*f - b^4*
d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cos
h(d*x + c)^3 - (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*l
og(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*cosh(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*cosh(1) + (a^4 + a^2*b^2)*d*sinh(1) - (a^4 + a^2*b^2)*f)*co
sh(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)*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)*...

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