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

Optimal. Leaf size=549 \[ \frac{2 a d (c+d x) \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac{2 a d (c+d x) \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )}-\frac{2 a d^2 \text{PolyLog}\left (3,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac{2 a d^2 \text{PolyLog}\left (3,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac{2 d (c+d x) \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac{2 d (c+d x) \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac{a (c+d x)^2 \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac{a (c+d x)^2 \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac{b (c+d x)^2 \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}-\frac{(c+d x)^2}{f \left (a^2+b^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.03665, antiderivative size = 549, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3324, 3322, 2264, 2190, 2531, 2282, 6589, 5561, 2279, 2391} \[ \frac{2 a d (c+d x) \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac{2 a d (c+d x) \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )}-\frac{2 a d^2 \text{PolyLog}\left (3,-\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac{2 a d^2 \text{PolyLog}\left (3,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac{2 d (c+d x) \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac{2 d (c+d x) \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac{a (c+d x)^2 \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac{a (c+d x)^2 \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac{b (c+d x)^2 \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}-\frac{(c+d x)^2}{f \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

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 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]

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

Mathematica [A]  time = 1.93532, size = 428, normalized size = 0.78 \[ \frac{-\frac{a \left (-2 d f (c+d x) \text{PolyLog}\left (2,\frac{b e^{e+f x}}{\sqrt{a^2+b^2}-a}\right )+2 d f (c+d x) \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )+2 d^2 \text{PolyLog}\left (3,\frac{b e^{e+f x}}{\sqrt{a^2+b^2}-a}\right )-2 d^2 \text{PolyLog}\left (3,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )-f^2 (c+d x)^2 \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )+f^2 (c+d x)^2 \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )\right )}{\sqrt{a^2+b^2}}+2 d^2 \text{PolyLog}\left (2,\frac{b e^{e+f x}}{\sqrt{a^2+b^2}-a}\right )+2 d^2 \text{PolyLog}\left (2,-\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}\right )+2 d f (c+d x) \log \left (\frac{b e^{e+f x}}{a-\sqrt{a^2+b^2}}+1\right )+2 d f (c+d x) \log \left (\frac{b e^{e+f x}}{\sqrt{a^2+b^2}+a}+1\right )-\frac{b f^2 (c+d x)^2 \cosh (e+f x)}{a+b \sinh (e+f x)}-f^2 (c+d x)^2}{f^3 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.29, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{2}}{ \left ( a+b\sinh \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.52775, size = 8910, normalized size = 16.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a^2*b + b^3)*d^2*e^2 - 4*(a^2*b + b^3)*c*d*e*f + 2*(a^2*b + b^3)*c^2*f^2 + 2*((a^2*b + b^3)*d^2*f^2*x^2 +
2*(a^2*b + b^3)*c*d*f^2*x - (a^2*b + b^3)*d^2*e^2 + 2*(a^2*b + b^3)*c*d*e*f)*cosh(f*x + e)^2 + 2*((a^2*b + b^
3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*f^2*x - (a^2*b + b^3)*d^2*e^2 + 2*(a^2*b + b^3)*c*d*e*f)*sinh(f*x + e)^2
+ 2*(a*b^2*d^2*cosh(f*x + e)^2 + a*b^2*d^2*sinh(f*x + e)^2 + 2*a^2*b*d^2*cosh(f*x + e) - a*b^2*d^2 + 2*(a*b^2*
d^2*cosh(f*x + e) + a^2*b*d^2)*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2)*polylog(3, (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) - 2*(a*b^2*d^2*cosh(f*x + e)^2 + a*b^2*d^2
*sinh(f*x + e)^2 + 2*a^2*b*d^2*cosh(f*x + e) - a*b^2*d^2 + 2*(a*b^2*d^2*cosh(f*x + e) + a^2*b*d^2)*sinh(f*x +
e))*sqrt((a^2 + b^2)/b^2)*polylog(3, (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) + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*c*d*f^2*x - 2*(a^3 + a*b^2)*d^2*e^2
+ 4*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(f*x + e) - 2*((a^2*b + b^3)*d^2*cosh(f*x + e)^2 + (a^
2*b + b^3)*d^2*sinh(f*x + e)^2 + 2*(a^3 + a*b^2)*d^2*cosh(f*x + e) - (a^2*b + b^3)*d^2 + 2*((a^2*b + b^3)*d^2*
cosh(f*x + e) + (a^3 + a*b^2)*d^2)*sinh(f*x + e) - (a*b^2*d^2*f*x + a*b^2*c*d*f - (a*b^2*d^2*f*x + a*b^2*c*d*f
)*cosh(f*x + e)^2 - (a*b^2*d^2*f*x + a*b^2*c*d*f)*sinh(f*x + e)^2 - 2*(a^2*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x +
e) - 2*(a^2*b*d^2*f*x + a^2*b*c*d*f + (a*b^2*d^2*f*x + a*b^2*c*d*f)*cosh(f*x + e))*sinh(f*x + e))*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) - 2*((a^2*b + b^3)*d^2*cosh(f*x + e)^2 + (a^2*b + b^3)*d^2*sinh(f*x + e)^2 + 2*(a^3 + a*b^2)*d^
2*cosh(f*x + e) - (a^2*b + b^3)*d^2 + 2*((a^2*b + b^3)*d^2*cosh(f*x + e) + (a^3 + a*b^2)*d^2)*sinh(f*x + e) +
(a*b^2*d^2*f*x + a*b^2*c*d*f - (a*b^2*d^2*f*x + a*b^2*c*d*f)*cosh(f*x + e)^2 - (a*b^2*d^2*f*x + a*b^2*c*d*f)*s
inh(f*x + e)^2 - 2*(a^2*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x + e) - 2*(a^2*b*d^2*f*x + a^2*b*c*d*f + (a*b^2*d^2*f
*x + a*b^2*c*d*f)*cosh(f*x + e))*sinh(f*x + e))*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) - (2*(a^2*b + b^3)*d^2*e - 2*(a^2*b
+ b^3)*c*d*f - 2*((a^2*b + b^3)*d^2*e - (a^2*b + b^3)*c*d*f)*cosh(f*x + e)^2 - 2*((a^2*b + b^3)*d^2*e - (a^2*b
+ b^3)*c*d*f)*sinh(f*x + e)^2 - 4*((a^3 + a*b^2)*d^2*e - (a^3 + a*b^2)*c*d*f)*cosh(f*x + e) - 4*((a^3 + a*b^2
)*d^2*e - (a^3 + a*b^2)*c*d*f + ((a^2*b + b^3)*d^2*e - (a^2*b + b^3)*c*d*f)*cosh(f*x + e))*sinh(f*x + e) + (a*
b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 - (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(f*x + e
)^2 - (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sinh(f*x + e)^2 - 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f +
a^2*b*c^2*f^2)*cosh(f*x + e) - 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*
c*d*e*f + a*b^2*c^2*f^2)*cosh(f*x + e))*sinh(f*x + e))*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) - (2*(a^2*b + b^3)*d^2*e - 2*(a^2*b + b^3)*c*d*f - 2*((a^2*b + b^
3)*d^2*e - (a^2*b + b^3)*c*d*f)*cosh(f*x + e)^2 - 2*((a^2*b + b^3)*d^2*e - (a^2*b + b^3)*c*d*f)*sinh(f*x + e)^
2 - 4*((a^3 + a*b^2)*d^2*e - (a^3 + a*b^2)*c*d*f)*cosh(f*x + e) - 4*((a^3 + a*b^2)*d^2*e - (a^3 + a*b^2)*c*d*f
+ ((a^2*b + b^3)*d^2*e - (a^2*b + b^3)*c*d*f)*cosh(f*x + e))*sinh(f*x + e) - (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f
+ a*b^2*c^2*f^2 - (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(f*x + e)^2 - (a*b^2*d^2*e^2 - 2*a*b^
2*c*d*e*f + a*b^2*c^2*f^2)*sinh(f*x + e)^2 - 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(f*x + e)
- 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh
(f*x + e))*sinh(f*x + e))*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) + (2*(a^2*b + b^3)*d^2*f*x + 2*(a^2*b + b^3)*d^2*e - 2*((a^2*b + b^3)*d^2*f*x + (a^2*b + b^3)*
d^2*e)*cosh(f*x + e)^2 - 2*((a^2*b + b^3)*d^2*f*x + (a^2*b + b^3)*d^2*e)*sinh(f*x + e)^2 - 4*((a^3 + a*b^2)*d^
2*f*x + (a^3 + a*b^2)*d^2*e)*cosh(f*x + e) - 4*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*d^2*e + ((a^2*b + b^3)*d
^2*f*x + (a^2*b + b^3)*d^2*e)*cosh(f*x + e))*sinh(f*x + e) + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^
2*e^2 + 2*a*b^2*c*d*e*f - (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*cosh(f*x +
e)^2 - (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*sinh(f*x + e)^2 - 2*(a^2*b*d
^2*f^2*x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f)*cosh(f*x + e) - 2*(a^2*b*d^2*f^2*x^2 + 2*a^2
*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*
b^2*c*d*e*f)*cosh(f*x + e))*sinh(f*x + e))*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) + (2*(a^2*b + b^3)*d^2*f*x + 2*(a^2*b + b^3)*d
^2*e - 2*((a^2*b + b^3)*d^2*f*x + (a^2*b + b^3)*d^2*e)*cosh(f*x + e)^2 - 2*((a^2*b + b^3)*d^2*f*x + (a^2*b + b
^3)*d^2*e)*sinh(f*x + e)^2 - 4*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*d^2*e)*cosh(f*x + e) - 4*((a^3 + a*b^2)*
d^2*f*x + (a^3 + a*b^2)*d^2*e + ((a^2*b + b^3)*d^2*f*x + (a^2*b + b^3)*d^2*e)*cosh(f*x + e))*sinh(f*x + e) - (
a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f - (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2
*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*cosh(f*x + e)^2 - (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2
+ 2*a*b^2*c*d*e*f)*sinh(f*x + e)^2 - 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e
*f)*cosh(f*x + e) - 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^
2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*cosh(f*x + e))*sinh(f*x + e))*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
) + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*c*d*f^2*x - 2*(a^3 + a*b^2)*d^2*e^2 + 4*(a^3 + a*b^2)*c*d*e
*f - (a^3 + a*b^2)*c^2*f^2 + 2*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*f^2*x - (a^2*b + b^3)*d^2*e^2
+ 2*(a^2*b + b^3)*c*d*e*f)*cosh(f*x + e))*sinh(f*x + e))/((a^4*b + 2*a^2*b^3 + b^5)*f^3*cosh(f*x + e)^2 + (a^4
*b + 2*a^2*b^3 + b^5)*f^3*sinh(f*x + e)^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*f^3*cosh(f*x + e) - (a^4*b + 2*a^2*b^3
+ b^5)*f^3 + 2*((a^4*b + 2*a^2*b^3 + b^5)*f^3*cosh(f*x + e) + (a^5 + 2*a^3*b^2 + a*b^4)*f^3)*sinh(f*x + e))

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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