\(\int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [413]

Optimal result

Integrand size = 32, antiderivative size = 454 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {5700, 3801, 3556, 5686, 5559, 3855, 5702, 4269, 5692, 3403, 2296, 2221, 2317, 2438, 6874} \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 d^2 \left (a^2+b^2\right )}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 d \left (a^2+b^2\right )}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 d^2 \left (a^2+b^2\right )}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}+\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 d \left (a^2+b^2\right )}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}+\frac {f \log (\cosh (c+d x))}{b d^2}-\frac {(e+f x) \tanh (c+d x)}{b d}+\frac {e x}{b}+\frac {f x^2}{2 b} \]

[In]

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

[Out]

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

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 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 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 3556

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

Rule 3801

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

Rule 3855

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

Rule 4269

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 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(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 5686

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*Sech
[c + d*x]*(Tanh[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]

Rule 5692

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 5700

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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 5702

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1),
x], x] - Dist[a/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(Tanh[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 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \tanh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a \int (e+f x) \text {sech}(c+d x) \tanh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int (e+f x) \, dx}{b}+\frac {f \int \tanh (c+d x) \, dx}{b d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {(e+f x) \tanh (c+d x)}{b d}+\frac {a^2 \int (e+f x) \text {sech}^2(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac {(a f) \int \text {sech}(c+d x) \, dx}{b^2 d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^3 \int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^2 f\right ) \int \tanh (c+d x) \, dx}{b^3 d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^3 \int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{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 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{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^4 \int (e+f x) \text {sech}^2(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {a^3 \int (e+f x) \text {sech}(c+d x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}-\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}+\frac {\left (a^4 f\right ) \int \tanh (c+d x) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (a^3 f\right ) \int \text {sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (a^3 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (a^3 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^2} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.67 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.79 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {(c+d x) (c f-d (2 e+f x))}{b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a+i b}+\frac {f \log (\cosh (c+d x))}{i a-b}-\frac {f \log (\cosh (c+d x))}{i a+b}+\frac {2 a^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac {2 d (e+f x) \text {sech}(c+d x) (-a+b \sinh (c+d x))}{a^2+b^2}}{2 d^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1896\) vs. \(2(432)=864\).

Time = 2.36 (sec) , antiderivative size = 1897, normalized size of antiderivative = 4.18

method	result	size
risch	\(\text {Expression too large to display}\)	\(1897\)

[In]

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

[Out]

2/d^2/(a^2+b^2)*b*a^2*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-2/d^2/(a^2+b^2)^(5/2)*b*f*arctanh(1/2*(2*b*exp(d*x+
c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/d^2/(a^2+b^2)^(5/2)*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a-4
/d^2/(a^2+b^2)*b^2*f/(2*a^2+2*b^2)*a*arctan(exp(d*x+c))+2*(f*x+e)*(a*exp(d*x+c)+b)/d/(a^2+b^2)/(1+exp(2*d*x+2*
c))-2/d^2/(a^2+b^2)*b*a^2*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/d/(a^2+b^2)^(3/2)*b*a^3*e/(2
*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/2*f*x^2/b+2/d^2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2
+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/d^2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2
)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+2/d^2/(a^2+b^2)^(3/2)*b^3*a*f/(2*a^2+2*b^2)*ar
ctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*e
xp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2
)+a)/(a+(a^2+b^2)^(1/2)))+2/d^2/(a^2+b^2)^(1/2)*b*a*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)
^(1/2))+2/d/(a^2+b^2)^(3/2)/b*a^5*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d^2/(a^2
+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/d/(a^2+b^2)^
(3/2)/b*a^5*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+2/d/(a^2+b^2)^(3/2)/b
*a^5*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-2/d^2/(a^2+b^2)^(3/2)/b*c*a^5*
f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d^2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2)*
ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/d^2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2)*ln((b
*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+e*x/b-2/d^2/(a^2+b^2)*b*f*ln(exp(d*x+c))+1/2/d^2/(a^2+b^
2)^2*b^3*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/b/d*a^3*e/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*ex
p(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/
(-a+(a^2+b^2)^(1/2)))*x+2/d/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+
b^2)^(1/2)))*x-2/d^2/(a^2+b^2)^(3/2)*b*c*a^3*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))
-2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/d^
2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-2/b/d^2*c*a
^3*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/d^2/(a^2+b^2)*a^3*f/(2*
a^2+2*b^2)*arctan(exp(d*x+c))+1/d^2/(a^2+b^2)^2*b*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)*a^2+2/d^2/(a^2+b^2)*
b^3*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-1/d^2/(a^2+b^2)*b^3*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c
)-b)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1571 vs. \(2 (430) = 860\).

Time = 0.31 (sec) , antiderivative size = 1571, normalized size of antiderivative = 3.46 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*x + 4*(a^2*b^2 + b^4)*d*e + ((a^4 + 2
*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c)^2 + ((a
^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*sinh(d*x + c)^2
- 2*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqr
t((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*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x
+ c)^2 + a^3*b*f)*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*(a^3*b*d*e - a^3*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^
2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f)*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*(a^3*b*d*e - a^3
*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^
3*b*d*e - a^3*b*c*f)*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*sq
rt((a^2 + b^2)/b^2) + 2*a) - 2*(a^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b
*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*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*(a^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh
(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*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*((a^3*b + a*b^3)*
f*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^3*b + a*b^3)*f*sinh(d*x + c)^2 + (a^3
*b + a*b^3)*f)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 4*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e)*cosh(d*
x + c) + 2*((a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2*b^2 + b
^4)*f*sinh(d*x + c)^2 + (a^2*b^2 + b^4)*f)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(2*(a^3*b
+ a*b^3)*d*f*x + 2*(a^3*b + a*b^3)*d*e + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e
- 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(d*x + c)^2 + 2*
(a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(d*x + c)*sinh(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh(d*x + c)^2 + (a
^4*b + 2*a^2*b^3 + b^5)*d^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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