3.363 \(\int \frac{(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=449 \[ \frac{2 a^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 a^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2}-\frac{2 a^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 a^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3}+\frac{a^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.716799, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5579, 5446, 3310, 3296, 2637, 5561, 2190, 2531, 2282, 6589} \[ \frac{2 a^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 a^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2}-\frac{2 a^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 a^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3}+\frac{a^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) - (a^2*(e + f*x)^3)/(3*b^3*f) + (2*a*f*(e + f*x)*Cosh[c + d*x])/(b^2*d^2)
+ (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
))])/(b^3*d^2) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (2*a^2*f
^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(
a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a*f^2*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x])/(b^2*d)
- (f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d^2) + (f^2*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^2*Sinh[c
+ d*x]^2)/(2*b*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 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 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 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 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 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)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{a \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac{f \int (e+f x) \sinh ^2(c+d x) \, dx}{b d}\\ &=-\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac{a^2 \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac{a^2 \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac{(2 a f) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d}+\frac{f \int (e+f x) \, dx}{2 b d}\\ &=\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^3 d}-\frac{\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^3 d}-\frac{\left (2 a f^2\right ) \int \cosh (c+d x) \, dx}{b^2 d^2}\\ &=\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{\left (2 a^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac{\left (2 a^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^3 d^2}\\ &=\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d^2}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac{\left (2 a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}-\frac{\left (2 a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}\\ &=\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}-\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{a^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2}+\frac{2 a^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d^2}-\frac{2 a^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 a^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 d^3}-\frac{2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac{f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh ^2(c+d x)}{2 b d}\\ \end{align*}

Mathematica [B]  time = 11.766, size = 1496, normalized size = 3.33 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

-(e^2*Log[a + b*Sinh[c + d*x]])/(4*b*d) - (e*f*(-x^2/(2*b) + (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
)/(b*d) + (x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2
+ b^2])]/(b*d^2) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]/(b*d^2)))/2 - (f^2*(-x^3/(3*b) + (x^2
*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/
(b*d) + (2*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (2*x*PolyLog[2, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/(b*d^2) - (2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b*d^3) - (2*PolyLo
g[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3)))/4 + (f^2*(2*(4*a^2 + b^2)*x^3*Coth[c] - (24*a*b*Cosh
[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (3*b^2*Cosh[2*d*x]*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Si
nh[2*c]))/d^3 - (4*a^2 + b^2)*(-1 + Coth[c])*(2*x^3 + (6*a*(d^2*x^2*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x])
)/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])] - 2*Po
lyLog[3, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])])*Sinh[c]*(Cosh[c] + Sinh[c]))/(Sqrt[a^2 +
b^2]*d^3) - (3*b^2*(d^2*x^2*Log[1 + ((a - Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*d*x*PolyLo
g[2, ((-a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 + b^2])*(Cosh[
c + d*x] - Sinh[c + d*x]))/b])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^3) - (3
*b^2*(d^2*x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*d*x*PolyLog[2, ((a + Sqrt
[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b] - 2*PolyLog[3, ((a + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh
[c + d*x]))/b])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3) - (3*a*(d^2*x^2*Log[
1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c
+ d*x]))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))]
)*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*d^3)) - (24*a*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[
d*x])/d^3 + (3*b^2*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^3))/(24*b^3) + (e^2*((4*a^2 +
b^2)*Log[a + b*Sinh[c + d*x]] - 4*a*b*Sinh[c + d*x] + 2*b^2*Sinh[c + d*x]^2))/(4*b^3*d) + (e*f*(8*a*b*Cosh[c
+ d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + 2*(4*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]))]) - 8
*a*b*d*x*Sinh[c + d*x] - b^2*Sinh[2*(c + d*x)]))/(4*b^3*d^2)

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Maple [F]  time = 0.302, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}\cosh \left ( dx+c \right ) \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, e^{2}{\left (\frac{8 \,{\left (d x + c\right )} a^{2}}{b^{3} d} - \frac{{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} + \frac{8 \, a^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d} + \frac{4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d}\right )} + \frac{{\left (16 \, a^{2} d^{3} f^{2} x^{3} e^{\left (2 \, c\right )} + 48 \, a^{2} d^{3} e f x^{2} e^{\left (2 \, c\right )} + 3 \,{\left (2 \, b^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \,{\left (2 \, d^{2} e f - d f^{2}\right )} b^{2} x e^{\left (4 \, c\right )} -{\left (2 \, d e f - f^{2}\right )} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 24 \,{\left (a b d^{2} f^{2} x^{2} e^{\left (3 \, c\right )} + 2 \,{\left (d^{2} e f - d f^{2}\right )} a b x e^{\left (3 \, c\right )} - 2 \,{\left (d e f - f^{2}\right )} a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} + 24 \,{\left (a b d^{2} f^{2} x^{2} e^{c} + 2 \,{\left (d^{2} e f + d f^{2}\right )} a b x e^{c} + 2 \,{\left (d e f + f^{2}\right )} a b e^{c}\right )} e^{\left (-d x\right )} + 3 \,{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 2 \,{\left (2 \, d^{2} e f + d f^{2}\right )} b^{2} x +{\left (2 \, d e f + f^{2}\right )} b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{48 \, b^{3} d^{3}} - \int -\frac{2 \,{\left (a^{2} b f^{2} x^{2} + 2 \, a^{2} b e f x -{\left (a^{3} f^{2} x^{2} e^{c} + 2 \, a^{3} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} - b^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

1/8*e^2*(8*(d*x + c)*a^2/(b^3*d) - (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*a^2*log(-2*a*e^(-d*x - c
) + b*e^(-2*d*x - 2*c) - b)/(b^3*d) + (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/48*(16*a^2*d^3*f^2*
x^3*e^(2*c) + 48*a^2*d^3*e*f*x^2*e^(2*c) + 3*(2*b^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*b^2*x*e^(4*c)
- (2*d*e*f - f^2)*b^2*e^(4*c))*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^(3*c) + 2*(d^2*e*f - d*f^2)*a*b*x*e^(3*c) - 2
*(d*e*f - f^2)*a*b*e^(3*c))*e^(d*x) + 24*(a*b*d^2*f^2*x^2*e^c + 2*(d^2*e*f + d*f^2)*a*b*x*e^c + 2*(d*e*f + f^2
)*a*b*e^c)*e^(-d*x) + 3*(2*b^2*d^2*f^2*x^2 + 2*(2*d^2*e*f + d*f^2)*b^2*x + (2*d*e*f + f^2)*b^2)*e^(-2*d*x))*e^
(-2*c)/(b^3*d^3) - integrate(-2*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x - (a^3*f^2*x^2*e^c + 2*a^3*e*f*x*e^c)*e^(d*x))/
(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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Fricas [C]  time = 2.6464, size = 5513, normalized size = 12.28 \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)^2*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(6*b^2*d^2*f^2*x^2 + 6*b^2*d^2*e^2 + 6*b^2*d*e*f + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b
^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^4 + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f
+ b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*b^2*f^2 - 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 - 12*(2*a*b*d^2*f^2*x^2 + 2*a*b*d^2
*e^2 - 4*a*b*d*e*f + 4*a*b*f^2 + 4*(a*b*d^2*e*f - a*b*d*f^2)*x - (2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*
e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*(a^2*d^3*f^2*x^3 + 3*a^2*
d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2)*cosh(d*x + c)^2 - 2*(8*a^2*
d^3*f^2*x^3 + 24*a^2*d^3*e*f*x^2 + 24*a^2*d^3*e^2*x + 48*a^2*c*d^2*e^2 - 48*a^2*c^2*d*e*f + 16*a^2*c^3*f^2 - 9
*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^2
+ 36*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c))
*sinh(d*x + c)^2 + 6*(2*b^2*d^2*e*f + b^2*d*f^2)*x + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 + 2*a*b*d*e*f + 2*a*b*f
^2 + 2*(a*b*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x + c) + 96*((a^2*d*f^2*x + a^2*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*d*f
^2*x + a^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d*f^2*x + a^2*d*e*f)*sinh(d*x + c)^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) + 96*((a^2*d*f^2
*x + a^2*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*d*f^2*x + a^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d*f^2*x + a^2
*d*e*f)*sinh(d*x + c)^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) + 48*((a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*e^2
- 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*sinh
(d*x + c)^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) + 48*((a^2*d^2*e^2 -
2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c) + (a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*sinh(d*x + c)^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) + 48*((a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f
^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)*sinh(d
*x + c) + (a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*sinh(d*x + c)^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) + 48*((a^2*d^2*f^2*
x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x +
2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f
- a^2*c^2*f^2)*sinh(d*x + c)^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) - 96*(a^2*f^2*cosh(d*x + c)^2 + 2*a^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*f^2*
sinh(d*x + c)^2)*polylog(3, (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) - 96*(a^2*f^2*cosh(d*x + c)^2 + 2*a^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*f^2*sinh(d*x + c)
^2)*polylog(3, (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) + 4*(6*a*b*d^2*f^2*x^2 + 6*a*b*d^2*e^2 + 12*a*b*d*e*f + 12*a*b*f^2 + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2
- 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^3 - 18*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 + 12*(a*b*d^2*e*f + a*b*d*f^2)*x -
8*(a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2)
*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^3*cosh(d*x + c)^2 + 2*b^3*d^3*cosh(d*x + c)*sinh(d*x + c) + b^3*d^3*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)**2*cosh(d*x+c)*sinh(d*x+c)**2/(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 )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2}}{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)^2*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*cosh(d*x + c)*sinh(d*x + c)^2/(b*sinh(d*x + c) + a), x)