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

Optimal. Leaf size=448 \[ -\frac {6 b f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^4}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d}+\frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.73, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5594, 5579, 3296, 2638, 5561, 2190, 2531, 6609, 2282, 6589} \[ \frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}\right )}{a^2 d^3}-\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}\right )}{a^2 d^2}-\frac {6 b f^3 \text {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \text {PolyLog}\left (4,-\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}\right )}{a^2 d^4}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d}+\frac {b (e+f x)^4}{4 a^2 f}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2282

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 2531

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 2638

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

Rule 3296

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

Rule 5561

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 5579

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

Rule 5594

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rule 6589

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 6609

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

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Mathematica [C]  time = 27.51, size = 2122, normalized size = 4.74 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e*f^2*Csch[c + d*x]*(2*b*x^3*(-1 + Coth[c]) - 2*b*x^3*Coth[c] - (6*a^2*b*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^
2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a] - 2*d*x*PolyLog[2, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d
*x]))/a] - 2*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a]))/(Sqrt[a^2 + b^2]*(-b + S
qrt[a^2 + b^2])*d^3) - (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a] -
2*d*x*PolyLog[2, ((b + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a] - 2*PolyLog[3, ((b + Sqrt[a^2 + b
^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d^3) + (6*b^2*(d^2*x^2*Log[
1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*(Cosh[c + d*x] + Sinh[c +
d*x]))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])]))/(
Sqrt[a^2 + b^2]*d^3) - (6*b^2*(d^2*x^2*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2])] + 2*
d*x*PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*(Cosh[c + d*x
] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^
2*x^2)*Sinh[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(2*
a^2*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*(-1/4*(b*x^4*Cosh[c]*Csch[c/2]*Sech[c/2])/a^2 + (2*Cosh[d*x]*(
-6*Cosh[c] - 3*d^2*x^2*Cosh[c] + 6*d*x*Sinh[c] + d^3*x^3*Sinh[c]))/(a*d^4) + (b*(x^4 - (2*a^2*(d^3*x^3*Log[1 +
((b - Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a] - 3*d^2*x^2*PolyLog[2, ((-b + Sqrt[a^2 + b^2])*(Co
sh[c + d*x] - Sinh[c + d*x]))/a] - 6*d*x*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a
] - 6*PolyLog[4, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a])*(-1 + Cosh[2*c] + Sinh[2*c]))/(S
qrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^4) - (2*a^2*(d^3*x^3*Log[1 + ((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - S
inh[c + d*x]))/a] - 3*d^2*x^2*PolyLog[2, -(((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a)] - 6*d*x
*PolyLog[3, -(((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a)] - 6*PolyLog[4, -(((b + Sqrt[a^2 + b^
2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a)])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])
*d^4) + (2*b*(d^3*x^3*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b - Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2
, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])] - 6*d*x*PolyLog[3, (a*(Cosh[c + d*x] + Sinh[c +
d*x]))/(-b + Sqrt[a^2 + b^2])] + 6*PolyLog[4, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])])*(-1
+ Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*d^4) - (2*b*(d^3*x^3*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(
b + Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))] - 6
*d*x*PolyLog[3, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, -((a*(Cosh[c + d*
x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*d^4)))/(a^2*(-1 +
Cosh[2*c] + Sinh[2*c])) + (2*(6*d*x*Cosh[c] + d^3*x^3*Cosh[c] - 6*Sinh[c] - 3*d^2*x^2*Sinh[c])*Sinh[d*x])/(a*
d^4))*(b + a*Sinh[c + d*x]))/(2*(a + b*Csch[c + d*x])) - (e^3*Csch[c + d*x]*((b*Log[b + a*Sinh[c + d*x]])/a^2
- Sinh[c + d*x]/a)*(b + a*Sinh[c + d*x]))/(d*(a + b*Csch[c + d*x])) + (3*e^2*f*Csch[c + d*x]*(b + a*Sinh[c + d
*x])*(-(a*Cosh[c + d*x]) - b*(c + d*x)*Log[b + a*Sinh[c + d*x]] + b*c*Log[1 + (a*Sinh[c + d*x])/b] + I*b*((-1/
8*I)*(2*c + I*Pi + 2*d*x)^2 - (4*I)*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*ArcTan[((I*a + b)*Cot[((2*I)*c + Pi + (2
*I)*d*x)/4])/Sqrt[a^2 + b^2]] - (((-2*I)*c + Pi - (2*I)*d*x + 4*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]])*Log[1 + ((-
b + Sqrt[a^2 + b^2])*E^(c + d*x))/a])/2 - (((-2*I)*c + Pi - (2*I)*d*x - 4*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]])*L
og[1 - ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a])/2 + (Pi/2 - I*(c + d*x))*Log[b + a*Sinh[c + d*x]] + I*(PolyLog[
2, ((b - Sqrt[a^2 + b^2])*E^(c + d*x))/a] + PolyLog[2, ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a])) + a*d*x*Sinh[c
+ d*x]))/(a^2*d^2*(a + b*Csch[c + d*x]))

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fricas [C]  time = 0.54, size = 1976, normalized size = 4.41 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*d^3*f^3*x^3 + 2*a*d^3*e^3 + 6*a*d^2*e^2*f + 12*a*d*e*f^2 + 12*a*f^3 + 6*(a*d^3*e*f^2 + a*d^2*f^3)*x^
2 - 2*(a*d^3*f^3*x^3 + a*d^3*e^3 - 3*a*d^2*e^2*f + 6*a*d*e*f^2 - 6*a*f^3 + 3*(a*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3
*(a*d^3*e^2*f - 2*a*d^2*e*f^2 + 2*a*d*f^3)*x)*cosh(d*x + c)^2 - 2*(a*d^3*f^3*x^3 + a*d^3*e^3 - 3*a*d^2*e^2*f +
6*a*d*e*f^2 - 6*a*f^3 + 3*(a*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3*(a*d^3*e^2*f - 2*a*d^2*e*f^2 + 2*a*d*f^3)*x)*sinh
(d*x + c)^2 + 6*(a*d^3*e^2*f + 2*a*d^2*e*f^2 + 2*a*d*f^3)*x - (b*d^4*f^3*x^4 + 4*b*d^4*e*f^2*x^3 + 6*b*d^4*e^2
*f*x^2 + 4*b*d^4*e^3*x + 8*b*c*d^3*e^3 - 12*b*c^2*d^2*e^2*f + 8*b*c^3*d*e*f^2 - 2*b*c^4*f^3)*cosh(d*x + c) + 1
2*((b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c) + (b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^
2*f)*sinh(d*x + c))*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 +
b^2)/a^2) - a)/a + 1) + 12*((b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c) + (b*d^2*f^3*x^2 +
2*b*d^2*e*f^2*x + b*d^2*e^2*f)*sinh(d*x + c))*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*
sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 4*((b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f
^3)*cosh(d*x + c) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sinh(d*x + c))*log(2*a*cosh(d*
x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 4*((b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*
f^2 - b*c^3*f^3)*cosh(d*x + c) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sinh(d*x + c))*lo
g(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 4*((b*d^3*f^3*x^3 + 3*b*d^3*e*f^2
*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*cosh(d*x + c) + (b*d^3*f^3*x^3 + 3*b*d
^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*sinh(d*x + c))*log(-(b*cosh(d*
x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 4*((b*d^3*f^3*x
^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*cosh(d*x + c) + (b*d
^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*sinh(d*x + c
))*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a)
+ 24*(b*f^3*cosh(d*x + c) + b*f^3*sinh(d*x + c))*polylog(4, (b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x
+ c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 24*(b*f^3*cosh(d*x + c) + b*f^3*sinh(d*x + c))*polylog(4,
(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 24*((b*d*
f^3*x + b*d*e*f^2)*cosh(d*x + c) + (b*d*f^3*x + b*d*e*f^2)*sinh(d*x + c))*polylog(3, (b*cosh(d*x + c) + b*sinh
(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 24*((b*d*f^3*x + b*d*e*f^2)*cosh(d
*x + c) + (b*d*f^3*x + b*d*e*f^2)*sinh(d*x + c))*polylog(3, (b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x +
c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - (b*d^4*f^3*x^4 + 4*b*d^4*e*f^2*x^3 + 6*b*d^4*e^2*f*x^2 + 4*
b*d^4*e^3*x + 8*b*c*d^3*e^3 - 12*b*c^2*d^2*e^2*f + 8*b*c^3*d*e*f^2 - 2*b*c^4*f^3 + 4*(a*d^3*f^3*x^3 + a*d^3*e^
3 - 3*a*d^2*e^2*f + 6*a*d*e*f^2 - 6*a*f^3 + 3*(a*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3*(a*d^3*e^2*f - 2*a*d^2*e*f^2 +
2*a*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))/(a^2*d^4*cosh(d*x + c) + a^2*d^4*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )}{a +b \,\mathrm {csch}\left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )} b}{a^{2} d} - \frac {e^{\left (d x + c\right )}}{a d} + \frac {e^{\left (-d x - c\right )}}{a d} + \frac {2 \, b \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{2} d}\right )} - \frac {{\left (b d^{4} f^{3} x^{4} e^{c} + 4 \, b d^{4} e f^{2} x^{3} e^{c} + 6 \, b d^{4} e^{2} f x^{2} e^{c} - 2 \, {\left (a d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} a x^{2} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a x e^{\left (2 \, c\right )} - 3 \, {\left (d^{2} e^{2} f - 2 \, d e f^{2} + 2 \, f^{3}\right )} a e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 2 \, {\left (a d^{3} f^{3} x^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} a x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a x + 3 \, {\left (d^{2} e^{2} f + 2 \, d e f^{2} + 2 \, f^{3}\right )} a\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{4 \, a^{2} d^{4}} + \int -\frac {2 \, {\left (a b f^{3} x^{3} + 3 \, a b e f^{2} x^{2} + 3 \, a b e^{2} f x - {\left (b^{2} f^{3} x^{3} e^{c} + 3 \, b^{2} e f^{2} x^{2} e^{c} + 3 \, b^{2} e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} b e^{\left (d x + c\right )} - a^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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