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

Optimal. Leaf size=335 \[ -\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2 \sqrt{a^2+b^2}}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^2 e x}{b^3}+\frac{a^2 f x^2}{2 b^3}+\frac{a f \sinh (c+d x)}{b^2 d^2}-\frac{a (e+f x) \cosh (c+d x)}{b^2 d}-\frac{f \sinh ^2(c+d x)}{4 b d^2}+\frac{(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac{e x}{2 b}-\frac{f x^2}{4 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.593089, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5557, 3310, 3296, 2637, 3322, 2264, 2190, 2279, 2391} \[ -\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2 \sqrt{a^2+b^2}}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^2 e x}{b^3}+\frac{a^2 f x^2}{2 b^3}+\frac{a f \sinh (c+d x)}{b^2 d^2}-\frac{a (e+f x) \cosh (c+d x)}{b^2 d}-\frac{f \sinh ^2(c+d x)}{4 b d^2}+\frac{(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac{e x}{2 b}-\frac{f x^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5557

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*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]

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 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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 2.68747, size = 307, normalized size = 0.92 \[ \frac{\frac{8 a^3 \left (-f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+2 d e \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-2 c f \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )\right )}{\sqrt{a^2+b^2}}-2 \left (2 a^2-b^2\right ) (c+d x) (c f-d (2 e+f x))-8 a b d (e+f x) \cosh (c+d x)+8 a b f \sinh (c+d x)+2 b^2 d (e+f x) \sinh (2 (c+d x))-b^2 f \cosh (2 (c+d x))}{8 b^3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*a^2 - b^2)*(c + d*x)*(c*f - d*(2*e + f*x)) - 8*a*b*d*(e + f*x)*Cosh[c + d*x] - b^2*f*Cosh[2*(c + d*x)]
+ (8*a^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 + Sq
rt[a^2 + b^2])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + S
qrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*f*Sinh[c + d*x] + 2*b^2*d*(e + f*x)*Sinh[2*(c + d*x)])/(8*b^3*d^2)

________________________________________________________________________________________

Maple [A]  time = 0.067, size = 589, normalized size = 1.8 \begin{align*}{\frac{{a}^{2}f{x}^{2}}{2\,{b}^{3}}}-{\frac{f{x}^{2}}{4\,b}}+{\frac{{a}^{2}ex}{{b}^{3}}}-{\frac{ex}{2\,b}}+{\frac{ \left ( 2\,dfx+2\,de-f \right ){{\rm e}^{2\,dx+2\,c}}}{16\,{d}^{2}b}}-{\frac{a \left ( dfx+de-f \right ){{\rm e}^{dx+c}}}{2\,{b}^{2}{d}^{2}}}-{\frac{a \left ( dfx+de+f \right ){{\rm e}^{-dx-c}}}{2\,{b}^{2}{d}^{2}}}-{\frac{ \left ( 2\,dfx+2\,de+f \right ){{\rm e}^{-2\,dx-2\,c}}}{16\,{d}^{2}b}}+2\,{\frac{{a}^{3}e}{{b}^{3}d\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,b{{\rm e}^{dx+c}}+2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{{a}^{3}fx}{{b}^{3}d}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}-{\frac{{a}^{3}fc}{{b}^{3}{d}^{2}}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}+{\frac{{a}^{3}fx}{{b}^{3}d}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}+{\frac{{a}^{3}fc}{{b}^{3}{d}^{2}}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}-{\frac{{a}^{3}f}{{b}^{3}{d}^{2}}{\it dilog} \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}+{\frac{{a}^{3}f}{{b}^{3}{d}^{2}}{\it dilog} \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}-2\,{\frac{{a}^{3}fc}{{b}^{3}{d}^{2}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,b{{\rm e}^{dx+c}}+2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*f*x^2/b^3-1/4*f*x^2/b+a^2*e*x/b^3-1/2*e*x/b+1/16*(2*d*f*x+2*d*e-f)/d^2/b*exp(2*d*x+2*c)-1/2*a*(d*f*x+d
*e-f)/b^2/d^2*exp(d*x+c)-1/2*a*(d*f*x+d*e+f)/b^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/d^2/b*exp(-2*d*x-2*c)+
2*a^3/b^3/d*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-a^3/b^3/d*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-a^3/b^3/d^2*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+a^3/b^3/d*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+a^3/b^3/d^2*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-a^3/
b^3/d^2*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)))+a^3/b^3/d^2*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)))-2*a^3/b^3/d^2*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))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.90955, size = 3992, normalized size = 11.92 \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)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/16*((2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*e - (a^2*b^2 + b^4)*f)*cosh(d*x + c)^4 + (2*(a^2*b^2 + b^
4)*d*f*x + 2*(a^2*b^2 + b^4)*d*e - (a^2*b^2 + b^4)*f)*sinh(d*x + c)^4 - 2*(a^2*b^2 + b^4)*d*f*x - 8*((a^3*b +
a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e - (a^3*b + a*b^3)*f)*cosh(d*x + c)^3 - 4*(2*(a^3*b + a*b^3)*d*f*x + 2*(a^3*
b + a*b^3)*d*e - 2*(a^3*b + a*b^3)*f - (2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*e - (a^2*b^2 + b^4)*f)*c
osh(d*x + c))*sinh(d*x + c)^3 - 2*(a^2*b^2 + b^4)*d*e + 4*((2*a^4 + a^2*b^2 - b^4)*d^2*f*x^2 + 2*(2*a^4 + a^2*
b^2 - b^4)*d^2*e*x)*cosh(d*x + c)^2 + 2*(2*(2*a^4 + a^2*b^2 - b^4)*d^2*f*x^2 + 4*(2*a^4 + a^2*b^2 - b^4)*d^2*e
*x + 3*(2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*e - (a^2*b^2 + b^4)*f)*cosh(d*x + c)^2 - 12*((a^3*b + a*
b^3)*d*f*x + (a^3*b + a*b^3)*d*e - (a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 16*(a^3*b*f*cosh(d*x +
c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2)*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) + 16*(a^3*b*f
*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilo
g((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) +
16*((a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*
d*e - a^3*b*c*f)*sinh(d*x + c)^2)*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) - 16*((a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)
*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*si
nh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 16*((a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*
x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^2)*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) +
16*((a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3
*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^2)*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) - (a^2*b^2 + b^4)*f - 8*((a^3*b + a*b^3)*d*f*x + (a
^3*b + a*b^3)*d*e + (a^3*b + a*b^3)*f)*cosh(d*x + c) - 4*(2*(a^3*b + a*b^3)*d*f*x - (2*(a^2*b^2 + b^4)*d*f*x +
2*(a^2*b^2 + b^4)*d*e - (a^2*b^2 + b^4)*f)*cosh(d*x + c)^3 + 2*(a^3*b + a*b^3)*d*e + 6*((a^3*b + a*b^3)*d*f*x
+ (a^3*b + a*b^3)*d*e - (a^3*b + a*b^3)*f)*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*f - 2*((2*a^4 + a^2*b^2 - b^4)
*d^2*f*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^2*e*x)*cosh(d*x + c))*sinh(d*x + c))/((a^2*b^3 + b^5)*d^2*cosh(d*x +
c)^2 + 2*(a^2*b^3 + b^5)*d^2*cosh(d*x + c)*sinh(d*x + c) + (a^2*b^3 + b^5)*d^2*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)*sinh(d*x+c)**3/(a+b*sinh(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 )} \sinh \left (d x + c\right )^{3}}{b \sinh \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)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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