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

Optimal. Leaf size=400 \[ -\frac{a f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{a^2 f \cosh (c+d x)}{b^3 d^2}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}+\frac{a f \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac{a f x}{4 b^2 d}-\frac{f \cosh ^3(c+d x)}{9 b d^2}-\frac{2 f \cosh (c+d x)}{3 b d^2}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]

[Out]

-(a*f*x)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^2)/(2*b^4*f) - (a^2*f*Cosh[c + d*x])/(b^3*d^2) - (2*f*Cosh[c + d
*x])/(3*b*d^2) - (f*Cosh[c + d*x]^3)/(9*b*d^2) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^
2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a
^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a*(a^2 + b^2)*f*PolyLog[2, -((b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (a^2*(e + f*x)*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)*Sinh[c
+ d*x])/(3*b*d) + (a*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(
3*b*d) - (a*(e + f*x)*Sinh[c + d*x]^2)/(2*b^2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.484925, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5579, 3310, 3296, 2638, 5565, 5446, 2635, 8, 5561, 2190, 2279, 2391} \[ -\frac{a f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{a^2 f \cosh (c+d x)}{b^3 d^2}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}+\frac{a f \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac{a f x}{4 b^2 d}-\frac{f \cosh ^3(c+d x)}{9 b d^2}-\frac{2 f \cosh (c+d x)}{3 b d^2}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*f*x)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^2)/(2*b^4*f) - (a^2*f*Cosh[c + d*x])/(b^3*d^2) - (2*f*Cosh[c + d
*x])/(3*b*d^2) - (f*Cosh[c + d*x]^3)/(9*b*d^2) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^
2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a
^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a*(a^2 + b^2)*f*PolyLog[2, -((b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (a^2*(e + f*x)*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)*Sinh[c
+ d*x])/(3*b*d) + (a*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(
3*b*d) - (a*(e + f*x)*Sinh[c + d*x]^2)/(2*b^2*d)

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 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 2638

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

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 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \cosh ^3(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{f \cosh ^3(c+d x)}{9 b d^2}+\frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}+\frac{a^2 \int (e+f x) \cosh (c+d x) \, dx}{b^3}-\frac{a \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{2 \int (e+f x) \cosh (c+d x) \, dx}{3 b}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}\\ &=\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac{f \cosh ^3(c+d x)}{9 b d^2}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac{\left (a^2 f\right ) \int \sinh (c+d x) \, dx}{b^3 d}+\frac{(a f) \int \sinh ^2(c+d x) \, dx}{2 b^2 d}-\frac{(2 f) \int \sinh (c+d x) \, dx}{3 b d}\\ &=\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac{a^2 f \cosh (c+d x)}{b^3 d^2}-\frac{2 f \cosh (c+d x)}{3 b d^2}-\frac{f \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac{(a f) \int 1 \, dx}{4 b^2 d}+\frac{\left (a \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}+\frac{\left (a \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}\\ &=-\frac{a f x}{4 b^2 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac{a^2 f \cosh (c+d x)}{b^3 d^2}-\frac{2 f \cosh (c+d x)}{3 b d^2}-\frac{f \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}+\frac{\left (a \left (a^2+b^2\right ) f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a-\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}+\frac{\left (a \left (a^2+b^2\right ) f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}\\ &=-\frac{a f x}{4 b^2 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac{a^2 f \cosh (c+d x)}{b^3 d^2}-\frac{2 f \cosh (c+d x)}{3 b d^2}-\frac{f \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{a \left (a^2+b^2\right ) f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}+\frac{a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x) \sinh (c+d x)}{3 b d}+\frac{a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}\\ \end{align*}

Mathematica [A]  time = 3.43536, size = 551, normalized size = 1.38 \[ -\frac{36 b^2 f \left (a \left (\text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+(c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+(c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-c \log (a+b \sinh (c+d x))-\frac{1}{2} (c+d x)^2\right )-b d x \sinh (c+d x)+b \cosh (c+d x)\right )+f \left (36 a \left (2 a^2+b^2\right ) \left (\text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+(c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+(c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-c \log (a+b \sinh (c+d x))-\frac{1}{2} (c+d x)^2\right )-18 b d x \left (4 a^2+b^2\right ) \sinh (c+d x)+18 b \left (4 a^2+b^2\right ) \cosh (c+d x)-9 a b^2 \sinh (2 (c+d x))+18 a b^2 d x \cosh (2 (c+d x))-6 b^3 d x \sinh (3 (c+d x))+2 b^3 \cosh (3 (c+d x))\right )+12 d e \left (-3 b \left (2 a^2+b^2\right ) \sinh (c+d x)+3 a \left (2 a^2+b^2\right ) \log (a+b \sinh (c+d x))+3 a b^2 \sinh ^2(c+d x)-2 b^3 \sinh ^3(c+d x)\right )-36 b^2 d e (b \sinh (c+d x)-a \log (a+b \sinh (c+d x)))}{72 b^4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-36*b^2*d*e*(-(a*Log[a + b*Sinh[c + d*x]]) + b*Sinh[c + d*x]) + 36*b^2*f*(b*Cosh[c + d*x] + a*(-(c + d*x)^2/
2 + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2])] - c*Log[a + b*Sinh[c + d*x]] + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - b*d*x*Sinh[c + d*x]) + 12*d*e*(3*a*(2*a^2 + b^2)*Log[a + b*Sinh[c + d*
x]] - 3*b*(2*a^2 + b^2)*Sinh[c + d*x] + 3*a*b^2*Sinh[c + d*x]^2 - 2*b^3*Sinh[c + d*x]^3) + f*(18*b*(4*a^2 + b^
2)*Cosh[c + d*x] + 18*a*b^2*d*x*Cosh[2*(c + d*x)] + 2*b^3*Cosh[3*(c + d*x)] + 36*a*(2*a^2 + b^2)*(-(c + d*x)^2
/2 + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^
2 + b^2])] - c*Log[a + b*Sinh[c + d*x]] + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + PolyLog[2, -((b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - 18*b*(4*a^2 + b^2)*d*x*Sinh[c + d*x] - 9*a*b^2*Sinh[2*(c + d*x)] - 6*
b^3*d*x*Sinh[3*(c + d*x)]))/(72*b^4*d^2)

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Maple [B]  time = 0.099, size = 1102, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*a*(2*d*f*x+2*d*e-f)/b^2/d^2*exp(2*d*x+2*c)-1/8*(4*a^2+3*b^2)*(d*f*x+d*e+f)/b^3/d^2*exp(-d*x-c)-1/16*a*(2
*d*f*x+2*d*e+f)/b^2/d^2*exp(-2*d*x-2*c)+1/2*a^3*f*x^2/b^4+1/2*a*f*x^2/b^2-a^3/b^4/d*f*ln((-b*exp(d*x+c)+(a^2+b
^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a^3/b^4/d^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*
c-a^3/b^4/d*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-a^3/b^4/d^2*f*ln((b*exp(d*x+c)+(a^2+b
^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+a^3/b^4/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2*a^3/b^4/d^2*f*c*ln
(exp(d*x+c))+2*a^3/b^4/d*f*c*x-a^3*e*x/b^4-a*e*x/b^2+a/b^2/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2*a/b
^2/d^2*f*c*ln(exp(d*x+c))-a/b^2/d*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a/b^2/d^2*f*l
n((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-a/b^2/d*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(
a^2+b^2)^(1/2)))*x-a/b^2/d^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2*a/b^2/d*f*c*x-1/72
*(3*d*f*x+3*d*e+f)/d^2/b*exp(-3*d*x-3*c)+1/72*(3*d*f*x+3*d*e-f)/d^2/b*exp(3*d*x+3*c)+1/8*(4*a^2*d*f*x+3*b^2*d*
f*x+4*a^2*d*e+3*b^2*d*e-4*a^2*f-3*b^2*f)/b^3/d^2*exp(d*x+c)+a/b^2/d^2*f*c^2-a/b^2/d*e*ln(b*exp(2*d*x+2*c)+2*a*
exp(d*x+c)-b)+2*a/b^2/d*e*ln(exp(d*x+c))-a/b^2/d^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)
))-a/b^2/d^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+a^3/b^4/d^2*f*c^2-a^3/b^4/d^2*f*d
ilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-a^3/b^4/d^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-
a)/(-a+(a^2+b^2)^(1/2)))-a^3/b^4/d*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2*a^3/b^4/d*e*ln(exp(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{24} \, e{\left (\frac{{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \,{\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{b^{3} d} + \frac{24 \,{\left (a^{3} + a b^{2}\right )}{\left (d x + c\right )}}{b^{4} d} + \frac{3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \,{\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{b^{3} d} + \frac{24 \,{\left (a^{3} + a b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d}\right )} - \frac{1}{144} \, f{\left (\frac{{\left (72 \,{\left (a^{3} d^{2} e^{\left (3 \, c\right )} + a b^{2} d^{2} e^{\left (3 \, c\right )}\right )} x^{2} - 2 \,{\left (3 \, b^{3} d x e^{\left (6 \, c\right )} - b^{3} e^{\left (6 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 9 \,{\left (2 \, a b^{2} d x e^{\left (5 \, c\right )} - a b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 18 \,{\left (4 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, c\right )} -{\left (4 \, a^{2} b d e^{\left (4 \, c\right )} + 3 \, b^{3} d e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (d x\right )} + 18 \,{\left (4 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, b^{3} e^{\left (2 \, c\right )} +{\left (4 \, a^{2} b d e^{\left (2 \, c\right )} + 3 \, b^{3} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (-d x\right )} + 9 \,{\left (2 \, a b^{2} d x e^{c} + a b^{2} e^{c}\right )} e^{\left (-2 \, d x\right )} + 2 \,{\left (3 \, b^{3} d x + b^{3}\right )} e^{\left (-3 \, d x\right )}\right )} e^{\left (-3 \, c\right )}}{b^{4} d^{2}} - 9 \, \int \frac{32 \,{\left ({\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} x e^{\left (d x\right )} -{\left (a^{3} b + a b^{3}\right )} x\right )}}{b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{4} e^{\left (d x + c\right )} - b^{5}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*e*((3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + 24*(a^3 + a
*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + 3*b^2)*e^(-d*x - c))/(b^
3*d) + 24*(a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d)) - 1/144*f*((72*(a^3*d^2*e^(3*
c) + a*b^2*d^2*e^(3*c))*x^2 - 2*(3*b^3*d*x*e^(6*c) - b^3*e^(6*c))*e^(3*d*x) + 9*(2*a*b^2*d*x*e^(5*c) - a*b^2*e
^(5*c))*e^(2*d*x) + 18*(4*a^2*b*e^(4*c) + 3*b^3*e^(4*c) - (4*a^2*b*d*e^(4*c) + 3*b^3*d*e^(4*c))*x)*e^(d*x) + 1
8*(4*a^2*b*e^(2*c) + 3*b^3*e^(2*c) + (4*a^2*b*d*e^(2*c) + 3*b^3*d*e^(2*c))*x)*e^(-d*x) + 9*(2*a*b^2*d*x*e^c +
a*b^2*e^c)*e^(-2*d*x) + 2*(3*b^3*d*x + b^3)*e^(-3*d*x))*e^(-3*c)/(b^4*d^2) - 9*integrate(32*((a^4*e^c + a^2*b^
2*e^c)*x*e^(d*x) - (a^3*b + a*b^3)*x)/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x))

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Fricas [B]  time = 2.56898, size = 5828, normalized size = 14.57 \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)*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/144*(2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*sinh(d*x + c)
^6 - 6*b^3*d*f*x - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*e
- 3*a*b^2*f - 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*b^3*d*e + 18*((4*a^2*b +
3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e - (4*a^2*b + 3*b^3)*f)*cosh(d*x + c)^4 + 3*(6*(4*a^2*b + 3*b^3)*d*f*x + 6
*(4*a^2*b + 3*b^3)*d*e + 10*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^2 - 6*(4*a^2*b + 3*b^3)*f - 15*(2*
a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*b^3*f + 72*((a^3 + a*b^2)*d^2*f*x^2 +
2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^2)*c*d*e - 2*(a^3 + a*b^2)*c^2*f)*cosh(d*x + c)^3 + 2*(36*(a^3 + a*b^2)
*d^2*f*x^2 + 72*(a^3 + a*b^2)*d^2*e*x + 144*(a^3 + a*b^2)*c*d*e - 72*(a^3 + a*b^2)*c^2*f + 20*(3*b^3*d*f*x + 3
*b^3*d*e - b^3*f)*cosh(d*x + c)^3 - 45*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^2 + 36*((4*a^2*b
+ 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e - (4*a^2*b + 3*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 18*((4*a^2*b +
3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e + (4*a^2*b + 3*b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*e -
b^3*f)*cosh(d*x + c)^4 - 3*(4*a^2*b + 3*b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^
3 - 3*(4*a^2*b + 3*b^3)*d*e + 18*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e - (4*a^2*b + 3*b^3)*f)*cosh(
d*x + c)^2 - 3*(4*a^2*b + 3*b^3)*f + 36*((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^2)*c
*d*e - 2*(a^3 + a*b^2)*c^2*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e + a*b^2*f)*cosh(
d*x + c) - 144*((a^3 + a*b^2)*f*cosh(d*x + c)^3 + 3*(a^3 + a*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^3 + a
*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (a^3 + a*b^2)*f*sinh(d*x + c)^3)*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) - 144*((a^3 + a*b^2)*f*cosh(d*x +
c)^3 + 3*(a^3 + a*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^3 + a*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (a^
3 + a*b^2)*f*sinh(d*x + c)^3)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b + 1) - 144*(((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*cosh(d*x + c)^3 + 3*((a^3 + a*
b^2)*d*e - (a^3 + a*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*cosh(d
*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*sinh(d*x + c)^3)*log(2*b*cosh(d*x + c) + 2*b
*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 144*(((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*cosh(d*x + c)
^3 + 3*((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*e - (a^3 + a
*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)*sinh(d*x + c)^3)*log(2*b*co
sh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 144*(((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)
*c*f)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 +
a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*si
nh(d*x + c)^3)*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) - 144*(((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d*f*x + (a^3
+ a*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)*sinh(d
*x + c)^2 + ((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*sinh(d*x + c)^3)*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) - 3*(6*a*b^2*d*f*x - 4*(3*b^3*d*f*x + 3*b
^3*d*e - b^3*f)*cosh(d*x + c)^5 + 6*a*b^2*d*e + 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^4 + 3
*a*b^2*f - 24*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e - (4*a^2*b + 3*b^3)*f)*cosh(d*x + c)^3 - 72*((a
^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^2)*c*d*e - 2*(a^3 + a*b^2)*c^2*f)*cosh(d*x + c)
^2 + 12*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*e + (4*a^2*b + 3*b^3)*f)*cosh(d*x + c))*sinh(d*x + c))/
(b^4*d^2*cosh(d*x + c)^3 + 3*b^4*d^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d^2*cosh(d*x + c)*sinh(d*x + c)^2 +
b^4*d^2*sinh(d*x + c)^3)

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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)*cosh(d*x+c)**3*sinh(d*x+c)/(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 )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )}{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)*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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