3.22 \(\int \frac{(e+f x)^3 \cosh ^2(c+d x)}{a+b \text{csch}(c+d x)} \, dx\)
Optimal. Leaf size=696 \[ \frac{6 b f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^3}-\frac{3 b f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^2}-\frac{6 b f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^4}+\frac{6 b f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^4}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^3 d}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}+\frac{3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}+\frac{(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f} \]
[Out]
(3*e*f^2*x)/(4*a*d^2) + (3*f^3*x^2)/(8*a*d^2) + (e + f*x)^4/(8*a*f) + (b^2*(e + f*x)^4)/(4*a^3*f) - (6*b*f^2*(
e + f*x)*Cosh[c + d*x])/(a^2*d^3) - (b*(e + f*x)^3*Cosh[c + d*x])/(a^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*a*d^4)
- (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3
*d) - (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*
b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (6*b*Sqrt[a^
2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*
f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyL
og[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((a*E^(c + d
*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*f^3*Sinh[c + d*x])/(a^2*d^4) + (3*b*f*(e + f*x)^2*Sinh[c + d*x]
)/(a^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*
x])/(2*a*d)
________________________________________________________________________________________
Rubi [A] time = 1.28301, antiderivative size = 696, normalized size of antiderivative = 1.,
number of steps used = 24, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used
= {5594, 5579, 3311, 32, 3310, 5565, 3296, 2637, 3322, 2264, 2190, 2531, 6609, 2282, 6589}
\[ \frac{6 b f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^3}-\frac{3 b f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^2}-\frac{6 b f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^4}+\frac{6 b f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^4}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^3 d}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}+\frac{3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}+\frac{(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f} \]
Antiderivative was successfully verified.
[In]
Int[((e + f*x)^3*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
[Out]
(3*e*f^2*x)/(4*a*d^2) + (3*f^3*x^2)/(8*a*d^2) + (e + f*x)^4/(8*a*f) + (b^2*(e + f*x)^4)/(4*a^3*f) - (6*b*f^2*(
e + f*x)*Cosh[c + d*x])/(a^2*d^3) - (b*(e + f*x)^3*Cosh[c + d*x])/(a^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*a*d^4)
- (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3
*d) - (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*
b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (6*b*Sqrt[a^
2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*
f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyL
og[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((a*E^(c + d
*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*f^3*Sinh[c + d*x])/(a^2*d^4) + (3*b*f*(e + f*x)^2*Sinh[c + d*x]
)/(a^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*
x])/(2*a*d)
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 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 3311
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 3310
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Rule 5565
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
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 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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 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 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 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 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]
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 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]
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \cosh ^2(c+d x)}{a+b \text{csch}(c+d x)} \, dx &=\int \frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac{\int (e+f x)^3 \cosh ^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\int (e+f x)^3 \, dx}{2 a}-\frac{b \int (e+f x)^3 \sinh (c+d x) \, dx}{a^2}+\frac{b^2 \int (e+f x)^3 \, dx}{a^3}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^3}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac{\left (3 f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 a d^2}\\ &=\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (2 b \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx}{a^3}+\frac{(3 b f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a^2 d}+\frac{\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (2 b \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 b-2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}+\frac{\left (2 b \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 b+2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}-\frac{\left (6 b f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a^2 d^2}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\left (3 b \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}-\frac{\left (3 b \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}+\frac{\left (6 b f^3\right ) \int \cosh (c+d x) \, dx}{a^2 d^3}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\left (6 b \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d^2}-\frac{\left (6 b \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d^2}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^3}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (6 b \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d^3}+\frac{\left (6 b \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d^3}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^3}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (6 b \sqrt{a^2+b^2} 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^3 d^4}+\frac{\left (6 b \sqrt{a^2+b^2} 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^3 d^4}\\ &=\frac{3 e f^2 x}{4 a d^2}+\frac{3 f^3 x^2}{8 a d^2}+\frac{(e+f x)^4}{8 a f}+\frac{b^2 (e+f x)^4}{4 a^3 f}-\frac{6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac{b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac{3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac{3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{3 b \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{6 b \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^4}+\frac{6 b \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^4}+\frac{6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac{3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac{3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac{(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}\\ \end{align*}
Mathematica [C] time = 14.4774, size = 3187, normalized size = 4.58 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((e + f*x)^3*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
[Out]
(e^3*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d))*Csch[c + d*x]*(
b + a*Sinh[c + d*x]))/(4*a*(a + b*Csch[c + d*x])) + (3*e^2*f*Csch[c + d*x]*(x^2 + (2*b*((I*Pi*ArcTanh[(-a + b*
Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (2*(c + I*ArcCos[((-I)*b)/a])*ArcTan[((a - I*b)*Cot[((2
*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + ((-2*I)*c + Pi - (2*I)*d*x)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c
+ Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)
*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((a + I*b)*(a - I*b + Sqrt[-a^2 - b^2])*(1 + I*Cot[((2*I)*c + Pi + (2*I)*d*x)
/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - (ArcCos[((-I)*b)/a] + 2*ArcTan[(
(a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[(I*(a + I*b)*(-a + I*b + Sqrt[-a^2 - b^2])
*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))
] + (ArcCos[((-I)*b)/a] + 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*Arc
Tanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^
(-c/2 - (d*x)/2))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x]]))] + (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*
b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I
)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((-1)^(1/4)*Sqrt[-a^2 - b^2]*E^((c + d*x)/2))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b +
a*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*b + Sqrt[-a^2 - b^2])*(a + I*b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi
+ (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((b + I*S
qrt[-a^2 - b^2])*(I*a - b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b
^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))]))/Sqrt[-a^2 - b^2]))/d^2)*(b + a*Sinh[c + d*x]))/(8*a*(a + b*Csch[c +
d*x])) + (e*f^2*Csch[c + d*x]*(x^3 - (3*b*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^2*x^2*L
og[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*d
*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]
+ 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3))*(b + a*Sinh[c + d*x]))/(4*a
*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*(x^4 - (4*b*(d^3*x^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]
)] - d^3*x^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[
a^2 + b^2])] - 3*d^2*x^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (a*E^(c + d*x
))/(-b + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (a*E^(c
+ d*x))/(-b + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d
^4))*(b + a*Sinh[c + d*x]))/(16*a*(a + b*Csch[c + d*x])) + (e*f^2*Csch[c + d*x]*(2*(a^2 + 4*b^2)*x^3 - (6*b*(3
*a^2 + 4*b^2)*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + S
qrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((a*E^(c + d*x
))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((a*E^(c + d
*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c
]))/d^3 + (3*a^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c]))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (
2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*a^2*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3)*
(b + a*Sinh[c + d*x]))/(8*a^3*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*((a^2 + 4*b^2)*x^4 - (4*b*(3*a^2 + 4
*b^2)*(d^3*x^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^3*x^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2
+ b^2])] + 3*d^2*x^2*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 3*d^2*x^2*PolyLog[2, -((a*E^(c + d*x
))/(b + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((a*
E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((
a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4) - (16*a*b*Cosh[d*x]*(d*x*(6 + d^2*x^2)*Cosh[c]
- 3*(2 + d^2*x^2)*Sinh[c]))/d^4 + (a^2*Cosh[2*d*x]*(-3*(1 + 2*d^2*x^2)*Cosh[2*c] + 2*d*x*(3 + 2*d^2*x^2)*Sinh[
2*c]))/d^4 - (16*a*b*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^4 + (a^2*(2*d*x*(3 +
2*d^2*x^2)*Cosh[2*c] - 3*(1 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^4)*(b + a*Sinh[c + d*x]))/(16*a^3*(a + b*Cs
ch[c + d*x])) + (e^3*Csch[c + d*x]*(b + a*Sinh[c + d*x])*((a^2 + 4*b^2)*(c + d*x) - (2*b*(3*a^2 + 4*b^2)*ArcTa
n[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(c + d*x)])
)/(4*a^3*d*(a + b*Csch[c + d*x])) + (3*e^2*f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*((a^2 + 4*b^2)*(-c + d*x)*(c
+ d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] - (2*b*(3*a^2 + 4*b^2)*(2*c*ArcTanh[(b + a*Cosh[c + d
*x] + a*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b - Sqrt[a^2
+ b^2])] - (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2])] + PolyLog[2, (a*(Cosh[
c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqr
t[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*a^2*d*x*Sinh[2*(c + d*x)]))/(8*a^3*d^2*(a + b*Csch
[c + d*x]))
________________________________________________________________________________________
Maple [F] time = 0.405, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{a+b{\rm csch} \left (dx+c\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x)
[Out]
int((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 2.33321, size = 8699, normalized size = 12.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="fricas")
[Out]
-1/32*(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 + 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*a^2*f^3 - (4*a^2*d^3*f^3*x^3 +
4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2
*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^4 - (4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*
e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f
^2 + a^2*d*f^3)*x)*sinh(d*x + c)^4 + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a
*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x +
c)^3 + 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 - 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 - 24*a*b*f^3 + 12*(a*b*d^3*e*
f^2 - a*b*d^2*f^3)*x^2 + 12*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - (4*a^2*d^3*f^3*x^3 + 4*a^2*d^3
*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*
f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(2*a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 - 4
*((a^2 + 2*b^2)*d^4*f^3*x^4 + 4*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 6*(a^2 + 2*b^2)*d^4*e^2*f*x^2 + 4*(a^2 + 2*b^2)*
d^4*e^3*x)*cosh(d*x + c)^2 - 2*(2*(a^2 + 2*b^2)*d^4*f^3*x^4 + 8*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 12*(a^2 + 2*b^2)
*d^4*e^2*f*x^2 + 8*(a^2 + 2*b^2)*d^4*e^3*x + 3*(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*
e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*
x)*cosh(d*x + c)^2 - 24*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*
d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x +
c)^2 + 96*((a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b*
d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*
f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) - 96*((a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(
d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^
3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x + c) + b*s
inh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 32*((a*b*d^3*e^3 - 3*a*
b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*
c^2*d*e*f^2 - a*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2
- a*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^
2 + b^2)/a^2) + 2*b) + 32*((a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*cosh(d*x + c)^2
+ 2*(a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^
3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*co
sh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 32*((a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^
2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a*b*d^3*f^3*
x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x
+ c)*sinh(d*x + c) + (a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^
2*d*e*f^2 + a*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*c
osh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) - 32*((a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3
*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a*b*d^3*f^3*x^3 +
3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)*
sinh(d*x + c) + (a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e
*f^2 + a*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) + 192*(a*b*f^3*cosh(d*x + c)^2 + 2*a*b*f^3*cosh(d*x +
c)*sinh(d*x + c) + a*b*f^3*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) - 192*(a*b*f^3*cosh(d*x + c)^2 + 2*a*b*f^3*
cosh(d*x + c)*sinh(d*x + c) + a*b*f^3*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*polylog(4, (b*cosh(d*x + c) + b*s
inh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 192*((a*b*d*f^3*x + a*b*d*e*f^2
)*cosh(d*x + c)^2 + 2*(a*b*d*f^3*x + a*b*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^3*x + a*b*d*e*f^2)*si
nh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) + 192*((a*b*d*f^3*x + a*b*d*e*f^2)*cosh(d*x + c)^2 + 2*(a*b*d*f^3*x + a*b
*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^3*x + a*b*d*e*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*pol
ylog(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) + 6
*(2*a^2*d^3*e^2*f + 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 + 3*a*b*d^2*e^2*f + 6*a
*b*d*e*f^2 + 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^
3)*x)*cosh(d*x + c) + 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 + 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 + 24*a*b*f^3 -
(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^
2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^3 + 12*(a*b*d^3*e*f^2 + a*b*d^
2*f^3)*x^2 + 12*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^
2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 + 12*(a*b*d^3*e^2*
f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - 2*((a^2 + 2*b^2)*d^4*f^3*x^4 + 4*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 6*(a^2 +
2*b^2)*d^4*e^2*f*x^2 + 4*(a^2 + 2*b^2)*d^4*e^3*x)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^4*cosh(d*x + c)^2 + 2*
a^3*d^4*cosh(d*x + c)*sinh(d*x + c) + a^3*d^4*sinh(d*x + c)^2)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**3*cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")
[Out]
integrate((f*x + e)^3*cosh(d*x + c)^2/(b*csch(d*x + c) + a), x)