3.2.74 \(\int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx\) [174]

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

[Out]

-(d*x+c)^2/(a^2+b^2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)/f^2+a*(d*x+c)^2*ln(1+b*exp
(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)/f^
2-a*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f+2*d^2*polylog(2,-b*exp(f*x+e)/(a-(a^2+b
^2)^(1/2)))/(a^2+b^2)/f^3+2*a*d*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^2+2*d^2
*polylog(2,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)/f^3-2*a*d*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a+(a^2+b^2)
^(1/2)))/(a^2+b^2)^(3/2)/f^2-2*a*d^2*polylog(3,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^3+2*a*d^2*
polylog(3,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^3-b*(d*x+c)^2*cosh(f*x+e)/(a^2+b^2)/f/(a+b*sinh
(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3405, 3403, 2296, 2221, 2611, 2320, 6724, 5680, 2317, 2438} \begin {gather*} \frac {2 a d (c+d x) \text {Li}_2\left (-\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 {Li}_2\left (-\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 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 )}+\frac {2 d^2 \text {Li}_2\left (-\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 {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}-\frac {2 a d^2 \text {Li}_3\left (-\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 {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}} \end {gather*}

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 2221

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]
- 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 2296

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 2317

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 2320

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 2438

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]

Rule 2611

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 3403

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 3405

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 5680

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 6724

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 {(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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.12, size = 428, normalized size = 0.78 \begin {gather*} \frac {-f^2 (c+d x)^2+2 d f (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )+2 d f (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-\frac {a \left (-f^2 (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )+f^2 (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \text {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d f (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-2 d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-\frac {b f^2 (c+d x)^2 \cosh (e+f x)}{a+b \sinh (e+f x)}}{\left (a^2+b^2\right ) f^3} \end {gather*}

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 = 1.84, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{2}}{\left (a +b \sinh \left (f x +e \right )\right )^{2}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

2*a*d^2*f*integrate(x^2*e^(f*x + e)/(a^2*b*f*e^(2*f*x + 2*e) + b^3*f*e^(2*f*x + 2*e) + 2*a^3*f*e^(f*x + e) + 2
*a*b^2*f*e^(f*x + e) - a^2*b*f - b^3*f), x) + 4*a*c*d*f*integrate(x*e^(f*x + e)/(a^2*b*f*e^(2*f*x + 2*e) + b^3
*f*e^(2*f*x + 2*e) + 2*a^3*f*e^(f*x + e) + 2*a*b^2*f*e^(f*x + e) - a^2*b*f - b^3*f), x) + 2*b*c*d*(a*log((b*e^
(f*x + e) + a - sqrt(a^2 + b^2))/(b*e^(f*x + e) + a + sqrt(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*f^2) -
2*(f*x + e)/((a^2*b + b^3)*f^2) + log(b*e^(2*f*x + 2*e) + 2*a*e^(f*x + e) - b)/((a^2*b + b^3)*f^2)) - 4*a*d^2*
integrate(x*e^(f*x + e)/(a^2*b*f*e^(2*f*x + 2*e) + b^3*f*e^(2*f*x + 2*e) + 2*a^3*f*e^(f*x + e) + 2*a*b^2*f*e^(
f*x + e) - a^2*b*f - b^3*f), x) + 4*b*d^2*integrate(x/(a^2*b*f*e^(2*f*x + 2*e) + b^3*f*e^(2*f*x + 2*e) + 2*a^3
*f*e^(f*x + e) + 2*a*b^2*f*e^(f*x + e) - a^2*b*f - b^3*f), x) + c^2*(a*log((b*e^(-f*x - e) - a - sqrt(a^2 + b^
2))/(b*e^(-f*x - e) - a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*f) - 2*(a*e^(-f*x - e) + b)/((a^2*b + b^3 + 2*(
a^3 + a*b^2)*e^(-f*x - e) - (a^2*b + b^3)*e^(-2*f*x - 2*e))*f)) - 2*a*c*d*log((b*e^(f*x + e) + a - sqrt(a^2 +
b^2))/(b*e^(f*x + e) + a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*f^2) + 2*(b*d^2*x^2 + 2*b*c*d*x - (a*d^2*x^2*e
^e + 2*a*c*d*x*e^e)*e^(f*x))/(a^2*b*f + b^3*f - (a^2*b*f + b^3*f)*e^(2*f*x + 2*e) - 2*(a^3*f + a*b^2*f)*e^(f*x
+ e))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5925 vs. \(2 (515) = 1030\).
time = 0.44, size = 5925, normalized size = 10.79 \begin {gather*} \text {Too large to display} \end {gather*}

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)*c^2*f^2 - 4*(a^2*b + b^3)*c*d*f*cosh(1) + 2*(a^2*b + b^3)*d^2*cosh(1)^2 + 2*(a^2*b + b^3)*d^
2*sinh(1)^2 + 2*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*f^2*x + 2*(a^2*b + b^3)*c*d*f*cosh(1) - (a^2*
b + b^3)*d^2*cosh(1)^2 - (a^2*b + b^3)*d^2*sinh(1)^2 + 2*((a^2*b + b^3)*c*d*f - (a^2*b + b^3)*d^2*cosh(1))*sin
h(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*f^2*x + 2*(a^2*b +
b^3)*c*d*f*cosh(1) - (a^2*b + b^3)*d^2*cosh(1)^2 - (a^2*b + b^3)*d^2*sinh(1)^2 + 2*((a^2*b + b^3)*c*d*f - (a^2
*b + b^3)*d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a*b^2*d^2*cosh(f*x + cosh(1) + sinh(1))^2
+ a*b^2*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*a^2*b*d^2*cosh(f*x + cosh(1) + sinh(1)) - a*b^2*d^2 + 2*(a*b^
2*d^2*cosh(f*x + cosh(1) + sinh(1)) + a^2*b*d^2)*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2)*polylog(
3, (a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) + (b*cosh(f*x + cosh(1) + sinh(1)) + b*s
inh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2))/b) - 2*(a*b^2*d^2*cosh(f*x + cosh(1) + sinh(1))^2 + a*b^2
*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*a^2*b*d^2*cosh(f*x + cosh(1) + sinh(1)) - a*b^2*d^2 + 2*(a*b^2*d^2*co
sh(f*x + cosh(1) + sinh(1)) + a^2*b*d^2)*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*co
sh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) - (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x
+ cosh(1) + sinh(1)))*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 - (
a^3 + a*b^2)*c^2*f^2 + 4*(a^3 + a*b^2)*c*d*f*cosh(1) - 2*(a^3 + a*b^2)*d^2*cosh(1)^2 - 2*(a^3 + a*b^2)*d^2*sin
h(1)^2 + 4*((a^3 + a*b^2)*c*d*f - (a^3 + a*b^2)*d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) - 2*((a^2*
b + b^3)*d^2*cosh(f*x + cosh(1) + sinh(1))^2 + (a^2*b + b^3)*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^3 + a*
b^2)*d^2*cosh(f*x + cosh(1) + sinh(1)) - (a^2*b + b^3)*d^2 + 2*((a^2*b + b^3)*d^2*cosh(f*x + cosh(1) + sinh(1)
) + (a^3 + a*b^2)*d^2)*sinh(f*x + cosh(1) + sinh(1)) - (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 + cosh(1) + sinh(1))^2 - (a*b^2*d^2*f*x + a*b^2*c*d*f)*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(a^2
*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x + cosh(1) + sinh(1)) - 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 + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2))*dilog((a*cosh(
f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) + (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + c
osh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*((a^2*b + b^3)*d^2*cosh(f*x + cosh(1) + sinh(1))^2 +
(a^2*b + b^3)*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^3 + a*b^2)*d^2*cosh(f*x + cosh(1) + sinh(1)) - (a^2*b
+ b^3)*d^2 + 2*((a^2*b + b^3)*d^2*cosh(f*x + cosh(1) + sinh(1)) + (a^3 + a*b^2)*d^2)*sinh(f*x + cosh(1) + sin
h(1)) + (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 + cosh(1) + sinh(1))^2 - (a*b^2*
d^2*f*x + a*b^2*c*d*f)*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(a^2*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x + cosh(1) +
sinh(1)) - 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 + cosh(1) + sinh(1)))*sinh(
f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2))*dilog((a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1)
+ sinh(1)) - (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2) - b)/b
+ 1) + (2*(a^2*b + b^3)*c*d*f - 2*(a^2*b + b^3)*d^2*cosh(1) - 2*(a^2*b + b^3)*d^2*sinh(1) - 2*((a^2*b + b^3)*
c*d*f - (a^2*b + b^3)*d^2*cosh(1) - (a^2*b + b^3)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 2*((a^2*b + b
^3)*c*d*f - (a^2*b + b^3)*d^2*cosh(1) - (a^2*b + b^3)*d^2*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 4*((a^3 +
a*b^2)*c*d*f - (a^3 + a*b^2)*d^2*cosh(1) - (a^3 + a*b^2)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) - 4*((a^3
+ a*b^2)*c*d*f - (a^3 + a*b^2)*d^2*cosh(1) - (a^3 + a*b^2)*d^2*sinh(1) + ((a^2*b + b^3)*c*d*f - (a^2*b + b^3)
*d^2*cosh(1) - (a^2*b + b^3)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)) - (a*b^
2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - (a*b^2*c^2*f^2 - 2*a*b^2*c*d*f
*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*cosh(f*x +
cosh(1) + sinh(1))^2 - (a*b^2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - 2
*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(a^2*b*c^2*f^2 - 2*a^2*b*c*d*f
*cosh(1) + a^2*b*d^2*cosh(1)^2 + a^2*b*d^2*sinh(1)^2 - 2*(a^2*b*c*d*f - a^2*b*d^2*cosh(1))*sinh(1))*cosh(f*x +
cosh(1) + sinh(1)) - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1) - 2*(a^2*b*c^2*f^2 - 2*a^2*b*c*d*f*cosh(1) +
a^2*b*d^2*cosh(1)^2 + a^2*b*d^2*sinh(1)^2 + (a*b^2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*
b^2*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) - 2*(a^2*b*c*d*
f - a^2*b*d^2*cosh(1))*sinh(1))*sinh(f*x + cosh...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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