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

Optimal. Leaf size=548 \[ -\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {2 i b f^2 \text {Li}_2\left (i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{d^3 \left (a^2+b^2\right )}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d^2 \left (a^2+b^2\right )}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a (e+f x)^2 \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^2}{d \left (a^2+b^2\right )} \]

[Out]

a*(f*x+e)^2/(a^2+b^2)/d-4*b*f*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)/d^2-2*a*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2
+b^2)/d^2+b^2*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d-b^2*(f*x+e)^2*ln(1+b*exp(d*x+
c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d+2*I*b*f^2*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^3-2*I*b*f^2*polylog(2
,I*exp(d*x+c))/(a^2+b^2)/d^3-a*f^2*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)/d^3+2*b^2*f*(f*x+e)*polylog(2,-b*exp(d
*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-2*b^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a
^2+b^2)^(3/2)/d^2-2*b^2*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^3+2*b^2*f^2*polylog
(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^3+b*(f*x+e)^2*sech(d*x+c)/(a^2+b^2)/d+a*(f*x+e)^2*tanh
(d*x+c)/(a^2+b^2)/d

________________________________________________________________________________________

Rubi [A]  time = 1.30, antiderivative size = 548, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5573, 3322, 2264, 2190, 2531, 2282, 6589, 6742, 4184, 3718, 2279, 2391, 5451, 4180} \[ \frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 i b f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {2 i b f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {a f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{d^3 \left (a^2+b^2\right )}-\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d^2 \left (a^2+b^2\right )}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a (e+f x)^2 \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^2}{d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e + f*x)^2)/((a^2 + b^2)*d) - (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d^2) + (b^2*(e + f*x)^2*L
og[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x)
)/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (2*a*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2)
+ ((2*I)*b*f^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - ((2*I)*b*f^2*PolyLog[2, I*E^(c + d*x)])/((a^
2 + b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^
2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (a*f^2
*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) - (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
]))])/((a^2 + b^2)^(3/2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^
(3/2)*d^3) + (b*(e + f*x)^2*Sech[c + d*x])/((a^2 + b^2)*d) + (a*(e + f*x)^2*Tanh[c + d*x])/((a^2 + b^2)*d)

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 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 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 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 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]

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 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 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {\int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^2+b^2}+\frac {\left (2 b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}\\ &=\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {a \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{a^2+b^2}-\frac {b \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}\\ &=\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {(2 a f) \int (e+f x) \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}-\frac {(2 b f) \int (e+f x) \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d}\\ &=\frac {a (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(4 a f) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 i b f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (2 i b f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^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 )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 b^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 )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\left (2 a f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (a f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [A]  time = 8.13, size = 905, normalized size = 1.65 \[ \frac {\left (-2 e^2 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^2+f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2-f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2+2 f (e+f x) \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d-2 f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) b^2}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \left (-\frac {2 \tan ^{-1}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right ) \tanh ^{-1}(\coth (c))}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}-\frac {i \text {csch}(c) \left (i \left (d x+\tanh ^{-1}(\coth (c))\right ) \left (\log \left (1-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\log \left (1+e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )+i \left (\text {Li}_2\left (-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text {Li}_2\left (e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt {1-\coth ^2(c)}}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {4 e f \tan ^{-1}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right ) b}{\left (a^2+b^2\right ) d^2 \sqrt {\cosh ^2(c)-\sinh ^2(c)}}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (b \cosh (c) e^2+a \sinh (d x) e^2+2 b f x \cosh (c) e+2 a f x \sinh (d x) e+b f^2 x^2 \cosh (c)+a f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \text {csch}(c) \left (d^2 e^{-\tanh ^{-1}(\coth (c))} x^2-\frac {i \coth (c) \left (-d x \left (2 i \tanh ^{-1}(\coth (c))-\pi \right )-\pi \log \left (1+e^{2 d x}\right )-2 \left (i d x+i \tanh ^{-1}(\coth (c))\right ) \log \left (1-e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )+\pi \log (\cosh (d x))+2 i \tanh ^{-1}(\coth (c)) \log \left (i \sinh \left (d x+\tanh ^{-1}(\coth (c))\right )\right )+i \text {Li}_2\left (e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )\right )}{\sqrt {1-\coth ^2(c)}}\right ) \text {sech}(c)}{\left (a^2+b^2\right ) d^3 \sqrt {\text {csch}^2(c) \left (\sinh ^2(c)-\cosh ^2(c)\right )}}-\frac {2 a e f \text {sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c))}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b^2*(-2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x)
)/(a + Sqrt[a^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLo
g[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^
2]))] - 2*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))]))/((a^2 + b^2)^(3/2)*d^3) - (2*a*e*f*Sech[c]*(Cosh[c]*Log[Cosh[c]*Cosh[d*x] + Sinh[c]*Sinh[d*x
]] - d*x*Sinh[c]))/((a^2 + b^2)*d^2*(Cosh[c]^2 - Sinh[c]^2)) - (4*b*e*f*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2
])/Sqrt[Cosh[c]^2 - Sinh[c]^2]])/((a^2 + b^2)*d^2*Sqrt[Cosh[c]^2 - Sinh[c]^2]) - (a*f^2*Csch[c]*((d^2*x^2)/E^A
rcTanh[Coth[c]] - (I*Coth[c]*(-(d*x*(-Pi + (2*I)*ArcTanh[Coth[c]])) - Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*Arc
Tanh[Coth[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))] + Pi*Log[Cosh[d*x]] + (2*I)*ArcTanh[Coth[c]]*Lo
g[I*Sinh[d*x + ArcTanh[Coth[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))]))/Sqrt[1 - Coth[c]^2]
)*Sech[c])/((a^2 + b^2)*d^3*Sqrt[Csch[c]^2*(-Cosh[c]^2 + Sinh[c]^2)]) - (2*b*f^2*(((-I)*Csch[c]*(I*(d*x + ArcT
anh[Coth[c]])*(Log[1 - E^(-(d*x) - ArcTanh[Coth[c]])] - Log[1 + E^(-(d*x) - ArcTanh[Coth[c]])]) + I*(PolyLog[2
, -E^(-(d*x) - ArcTanh[Coth[c]])] - PolyLog[2, E^(-(d*x) - ArcTanh[Coth[c]])])))/Sqrt[1 - Coth[c]^2] - (2*ArcT
an[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]]*ArcTanh[Coth[c]])/Sqrt[Cosh[c]^2 - Sinh[c]^2
]))/((a^2 + b^2)*d^3) + (Sech[c]*Sech[c + d*x]*(b*e^2*Cosh[c] + 2*b*e*f*x*Cosh[c] + b*f^2*x^2*Cosh[c] + a*e^2*
Sinh[d*x] + 2*a*e*f*x*Sinh[d*x] + a*f^2*x^2*Sinh[d*x]))/((a^2 + b^2)*d)

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fricas [C]  time = 1.06, size = 3600, normalized size = 6.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*(a^3 + a*b^2)*d^2*e^2 - 8*(a^3 + a*b^2)*c*d*e*f + 4*(a^3 + a*b^2)*c^2*f^2 - 4*((a^3 + a*b^2)*d^2*f^2*x
^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2 - 4*((a^3 +
a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*sinh(d*x + c
)^2 - 4*(b^3*d*f^2*x + b^3*d*e*f + (b^3*d*f^2*x + b^3*d*e*f)*cosh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*e*f)*cos
h(d*x + c)*sinh(d*x + c) + (b^3*d*f^2*x + b^3*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x
+ c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 4*(b^3*d*f^2*
x + b^3*d*e*f + (b^3*d*f^2*x + b^3*d*e*f)*cosh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*e*f)*cosh(d*x + c)*sinh(d*x
+ c) + (b^3*d*f^2*x + b^3*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x +
c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(b^3*d^2*e^2 - 2*b^3*c*d*e*f +
b^3*c^2*f^2 + (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)^2 + 2*(b^3*d^2*e^2 - 2*b^3*c*d*e*f +
b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*sinh(d*x + c)^2)*sqrt((
a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(b^3*d^2*e^2
- 2*b^3*c*d*e*f + b^3*c^2*f^2 + (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)^2 + 2*(b^3*d^2*e^2 -
2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*sinh(d
*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a)
- 2*(b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2 + (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*
b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)^2 + 2*(b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^
2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2)*sinh(d*x +
c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt
((a^2 + b^2)/b^2) - b)/b) + 2*(b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2 + (b^3*d^2*f^2*
x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)^2 + 2*(b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x +
2*b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f
- b^3*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 4*(b^3*f^2*cosh(d*x + c)^2 + 2*b^3*f^2*cosh(d*x + c)*sin
h(d*x + c) + b^3*f^2*sinh(d*x + c)^2 + b^3*f^2)*sqrt((a^2 + b^2)/b^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*(b^3*f^2*cosh(d*x + c)^2 + 2*b^3*f^2
*cosh(d*x + c)*sinh(d*x + c) + b^3*f^2*sinh(d*x + c)^2 + b^3*f^2)*sqrt((a^2 + b^2)/b^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*((a^2*b + b^3)*d^2
*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + (a^2*b + b^3)*d^2*e^2)*cosh(d*x + c) + 4*((a^3 + a*b^2)*f^2 + I*(a^2*b
+ b^3)*f^2 + ((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3
)*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d
*x + c) + I*sinh(d*x + c)) + 4*((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2 + ((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)
*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^
2)*f^2 - I*(a^2*b + b^3)*f^2)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + (4*(a^3 + a*b^2)*d*
e*f + 4*I*(a^2*b + b^3)*d*e*f - 4*(a^3 + a*b^2)*c*f^2 - 4*I*(a^2*b + b^3)*c*f^2 + (4*(a^3 + a*b^2)*d*e*f + 4*I
*(a^2*b + b^3)*d*e*f - 4*(a^3 + a*b^2)*c*f^2 - 4*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (8*(a^3 + a*b^2)*d*e
*f + 8*I*(a^2*b + b^3)*d*e*f - 8*(a^3 + a*b^2)*c*f^2 - 8*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) +
(4*(a^3 + a*b^2)*d*e*f + 4*I*(a^2*b + b^3)*d*e*f - 4*(a^3 + a*b^2)*c*f^2 - 4*I*(a^2*b + b^3)*c*f^2)*sinh(d*x +
c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) + (4*(a^3 + a*b^2)*d*e*f - 4*I*(a^2*b + b^3)*d*e*f - 4*(a^3 + a*
b^2)*c*f^2 + 4*I*(a^2*b + b^3)*c*f^2 + (4*(a^3 + a*b^2)*d*e*f - 4*I*(a^2*b + b^3)*d*e*f - 4*(a^3 + a*b^2)*c*f^
2 + 4*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (8*(a^3 + a*b^2)*d*e*f - 8*I*(a^2*b + b^3)*d*e*f - 8*(a^3 + a*b
^2)*c*f^2 + 8*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + (4*(a^3 + a*b^2)*d*e*f - 4*I*(a^2*b + b^3)*
d*e*f - 4*(a^3 + a*b^2)*c*f^2 + 4*I*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) -
I) + (4*(a^3 + a*b^2)*d*f^2*x - 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f^2 - 4*I*(a^2*b + b^3)*c*f^2 +
(4*(a^3 + a*b^2)*d*f^2*x - 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f^2 - 4*I*(a^2*b + b^3)*c*f^2)*cosh(d
*x + c)^2 + (8*(a^3 + a*b^2)*d*f^2*x - 8*I*(a^2*b + b^3)*d*f^2*x + 8*(a^3 + a*b^2)*c*f^2 - 8*I*(a^2*b + b^3)*c
*f^2)*cosh(d*x + c)*sinh(d*x + c) + (4*(a^3 + a*b^2)*d*f^2*x - 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f
^2 - 4*I*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (4*(a^3 + a*b^2)*d
*f^2*x + 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f^2 + 4*I*(a^2*b + b^3)*c*f^2 + (4*(a^3 + a*b^2)*d*f^2*
x + 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f^2 + 4*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (8*(a^3 + a
*b^2)*d*f^2*x + 8*I*(a^2*b + b^3)*d*f^2*x + 8*(a^3 + a*b^2)*c*f^2 + 8*I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sin
h(d*x + c) + (4*(a^3 + a*b^2)*d*f^2*x + 4*I*(a^2*b + b^3)*d*f^2*x + 4*(a^3 + a*b^2)*c*f^2 + 4*I*(a^2*b + b^3)*
c*f^2)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 4*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b
+ b^3)*d^2*e*f*x + (a^2*b + b^3)*d^2*e^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 +
a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x +
c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*d^3*sinh(d*x + c)^
2 + (a^4 + 2*a^2*b^2 + b^4)*d^3)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 1.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, a e f {\left (\frac {2 \, {\left (d x + c\right )}}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}}\right )} - 4 \, b f^{2} \int \frac {x e^{\left (d x + c\right )}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} + 4 \, a f^{2} \int \frac {x}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} + e^{2} {\left (\frac {b^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} d} + \frac {2 \, {\left (b e^{\left (-d x - c\right )} + a\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d}\right )} - \frac {4 \, b e f \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {2 \, {\left (a f^{2} x^{2} + 2 \, a e f x - {\left (b f^{2} x^{2} e^{c} + 2 \, b e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} d + b^{2} d + {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \int -\frac {2 \, {\left (b^{2} f^{2} x^{2} e^{c} + 2 \, b^{2} e f x e^{c}\right )} e^{\left (d x\right )}}{a^{2} b + b^{3} - {\left (a^{2} b e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} e^{c} + a b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) - 4*b*f^2*integrate(x*e^(
d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + 4*a*f^2*integrate(x/(a^2*d*e^(2
*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + e^2*(b^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))
/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*d) + 2*(b*e^(-d*x - c) + a)/((a^2 + b^2 + (a^2 + b
^2)*e^(-2*d*x - 2*c))*d)) - 4*b*e*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - 2*(a*f^2*x^2 + 2*a*e*f*x - (b*f^2*
x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a^2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) + integrate(-2*(
b^2*f^2*x^2*e^c + 2*b^2*e*f*x*e^c)*e^(d*x)/(a^2*b + b^3 - (a^2*b*e^(2*c) + b^3*e^(2*c))*e^(2*d*x) - 2*(a^3*e^c
+ a*b^2*e^c)*e^(d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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