Integrand size = 20, antiderivative size = 404 \[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4} \]
(d*x+c)^3*ln(1+b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f/(a^2+b^2)^(1/2)-(d*x+c) ^3*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f/(a^2+b^2)^(1/2)+3*d*(d*x+c)^2* polylog(2,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f^2/(a^2+b^2)^(1/2)-3*d*(d*x+ c)^2*polylog(2,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f^2/(a^2+b^2)^(1/2)-6*d^ 2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f^3/(a^2+b^2)^(1/2) +6*d^2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f^3/(a^2+b^2)^ (1/2)+6*d^3*polylog(4,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f^4/(a^2+b^2)^(1/ 2)-6*d^3*polylog(4,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f^4/(a^2+b^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+\frac {3 d \left (f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \operatorname {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (4,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )\right )}{f^3}-\frac {3 d \left (f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{f^3}}{\sqrt {a^2+b^2} f} \]
((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])] - (c + d*x)^3* Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])] + (3*d*(f^2*(c + d*x)^2*Pol yLog[2, (b*E^(e + f*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*f*(c + d*x)*PolyLog[ 3, (b*E^(e + f*x))/(-a + Sqrt[a^2 + b^2])] + 2*d^2*PolyLog[4, (b*E^(e + f* x))/(-a + Sqrt[a^2 + b^2])]))/f^3 - (3*d*(f^2*(c + d*x)^2*PolyLog[2, -((b* E^(e + f*x))/(a + Sqrt[a^2 + b^2]))] - 2*d*f*(c + d*x)*PolyLog[3, -((b*E^( e + f*x))/(a + Sqrt[a^2 + b^2]))] + 2*d^2*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))]))/f^3)/(Sqrt[a^2 + b^2]*f)
Time = 1.67 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3803, 25, 2694, 27, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d x)^3}{a-i b \sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 3803 |
\(\displaystyle 2 \int -\frac {e^{e+f x} (c+d x)^3}{-2 e^{e+f x} a-b e^{2 (e+f x)}+b}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{e+f x} (c+d x)^3}{-2 e^{e+f x} a-b e^{2 (e+f x)}+b}dx\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle -2 \left (\frac {b \int -\frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 \left (\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2+b^2}}\right )\) |
-2*(-1/2*(b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/ (b*f) - (3*d*(-(((c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/f) + (2*d*(((c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/f - (d*PolyLog[4, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/f^ 2))/f))/(b*f)))/Sqrt[a^2 + b^2] + (b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x)) /(a + Sqrt[a^2 + b^2])])/(b*f) - (3*d*(-(((c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/f) + (2*d*(((c + d*x)*PolyLog[3, -((b*E^ (e + f*x))/(a + Sqrt[a^2 + b^2]))])/f - (d*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/f^2))/f))/(b*f)))/(2*Sqrt[a^2 + b^2]))
3.2.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
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]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
\[\int \frac {\left (d x +c \right )^{3}}{a +b \sinh \left (f x +e \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (362) = 724\).
Time = 0.28 (sec) , antiderivative size = 1004, normalized size of antiderivative = 2.49 \[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\text {Too large to display} \]
(6*b*d^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2))/b) - 6*b*d ^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(f*x + e) + a*sinh(f*x + e) - ( b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2))/b) + 3*(b*d^3*f^ 2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh (f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^ 2 + b^2)/b^2) - b)/b + 1) - 3*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f ^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(f*x + e) + a*sinh(f*x + e) - (b*co sh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (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)*sqrt((a^2 + b^2)/b^2) *log(2*b*cosh(f*x + e) + 2*b*sinh(f*x + e) + 2*b*sqrt((a^2 + b^2)/b^2) + 2 *a) - (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)*sqrt((a^ 2 + b^2)/b^2)*log(2*b*cosh(f*x + e) + 2*b*sinh(f*x + e) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*sqrt((a^2 + b^2)/b^2)*log(- (a*cosh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*s qrt((a^2 + b^2)/b^2) - b)/b) - (b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^ 2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*sqrt((a^2 + b^2 )/b^2)*log(-(a*cosh(f*x + e) + a*sinh(f*x + e) - (b*cosh(f*x + e) + b*sinh (f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b) - 6*(b*d^3*f*x + b*c*d^2*f)*sq...
\[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\int \frac {\left (c + d x\right )^{3}}{a + b \sinh {\left (e + f x \right )}}\, dx \]
\[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \sinh \left (f x + e\right ) + a} \,d x } \]
c^3*log((b*e^(-f*x - e) - a - sqrt(a^2 + b^2))/(b*e^(-f*x - e) - a + sqrt( a^2 + b^2)))/(sqrt(a^2 + b^2)*f) + integrate(2*d^3*x^3/(b*(e^(f*x + e) - e ^(-f*x - e)) + 2*a) + 6*c*d^2*x^2/(b*(e^(f*x + e) - e^(-f*x - e)) + 2*a) + 6*c^2*d*x/(b*(e^(f*x + e) - e^(-f*x - e)) + 2*a), x)
\[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \sinh \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {sinh}\left (e+f\,x\right )} \,d x \]