\(\int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\) [331]

Optimal result

Integrand size = 26, antiderivative size = 306 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {a f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {a f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}+\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac {f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5572, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac {f^2 \log (a+b \sinh (c+d x))}{b d^3 \left (a^2+b^2\right )}+\frac {a f (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {a f (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {f (e+f x) \cosh (c+d x)}{d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2} \]

[In]

Int[((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 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 5572

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(623\) vs. \(2(306)=612\).

Time = 6.68 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.04 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {f^2 x \coth (c)}{b \left (a^2+b^2\right ) d^2}+\frac {2 e^c f \left (-e^c f x+e^{-c} \left (-1+e^{2 c}\right ) f x-\frac {a e e^{-c} \left (-1+e^{2 c}\right ) \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {a e^{-c} \left (-1+e^{2 c}\right ) f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d}+\frac {1}{2} e^{-c} \left (-1+e^{2 c}\right ) f \left (-2 x+\frac {2 a \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}+\frac {\log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}\right )+\frac {a \left (-1+e^{2 c}\right ) f \left (d x \left (\log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{2 d \sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{b \left (a^2+b^2\right ) d^2 \left (-1+e^{2 c}\right )}-\frac {f^2 x \cosh (c) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right ) \left (a e f \cosh (c)+a f^2 x \cosh (c)+b e f \sinh (d x)+b f^2 x \sinh (d x)\right )}{2 b \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]

[In]

Integrate[((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3,x]

[Out]

(f^2*x*Coth[c])/(b*(a^2 + b^2)*d^2) + (2*E^c*f*(-(E^c*f*x) + ((-1 + E^(2*c))*f*x)/E^c - (a*e*(-1 + E^(2*c))*Ar
cTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*E^c) + (a*(-1 + E^(2*c))*f*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d*E^c) + ((-1 + E^(2*c))*f*(-2*x + (2*a*ArcTan[(a + b*E^(c + d*x))/Sq
rt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d) + Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))]/d))/(2*E^c) + (a*(-1 +
E^(2*c))*f*(d*x*(Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - Log[1 + (b*E^(2*c + d*x))/(
a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])]) + PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] -
PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(2*d*Sqrt[(a^2 + b^2)*E^(2*c)])))/(b*(a
^2 + b^2)*d^2*(-1 + E^(2*c))) - (f^2*x*Cosh[c]*Csch[c/2]*Sech[c/2])/(2*b*(a^2 + b^2)*d^2) - (e + f*x)^2/(2*b*d
*(a + b*Sinh[c + d*x])^2) + (Csch[c/2]*Sech[c/2]*(a*e*f*Cosh[c] + a*f^2*x*Cosh[c] + b*e*f*Sinh[d*x] + b*f^2*x*
Sinh[d*x]))/(2*b*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(804\) vs. \(2(284)=568\).

Time = 15.88 (sec) , antiderivative size = 805, normalized size of antiderivative = 2.63

method	result	size
risch	\(-\frac {2 \left (a^{2} d \,f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+b^{2} d \,f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+2 a^{2} d e f x \,{\mathrm e}^{2 d x +2 c}-a b \,f^{2} x \,{\mathrm e}^{3 d x +3 c}+2 b^{2} d e f x \,{\mathrm e}^{2 d x +2 c}+a^{2} d \,e^{2} {\mathrm e}^{2 d x +2 c}-2 a^{2} f^{2} x \,{\mathrm e}^{2 d x +2 c}-a b e f \,{\mathrm e}^{3 d x +3 c}+b^{2} d \,e^{2} {\mathrm e}^{2 d x +2 c}+b^{2} f^{2} x \,{\mathrm e}^{2 d x +2 c}-2 a^{2} e f \,{\mathrm e}^{2 d x +2 c}+3 a b \,f^{2} x \,{\mathrm e}^{d x +c}+b^{2} e f \,{\mathrm e}^{2 d x +2 c}+3 a b e f \,{\mathrm e}^{d x +c}-b^{2} f^{2} x -b^{2} e f \right )}{b \,d^{2} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )^{2} \left (a^{2}+b^{2}\right )}-\frac {2 f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{\left (a^{2}+b^{2}\right ) d^{3} b}+\frac {f^{2} \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{\left (a^{2}+b^{2}\right ) d^{3} b}-\frac {2 f a e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}+\frac {f^{2} a \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}-\frac {f^{2} a \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}+\frac {f^{2} a \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} b}-\frac {f^{2} a \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} b}+\frac {f^{2} a \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} b}-\frac {f^{2} a \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} b}+\frac {2 f^{2} a c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} b}\)	\(805\)

[In]

int((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2/b*(a^2*d*f^2*x^2*exp(2*d*x+2*c)+b^2*d*f^2*x^2*exp(2*d*x+2*c)+2*a^2*d*e*f*x*exp(2*d*x+2*c)-a*b*f^2*x*exp(3*d
*x+3*c)+2*b^2*d*e*f*x*exp(2*d*x+2*c)+a^2*d*e^2*exp(2*d*x+2*c)-2*a^2*f^2*x*exp(2*d*x+2*c)-a*b*e*f*exp(3*d*x+3*c
)+b^2*d*e^2*exp(2*d*x+2*c)+b^2*f^2*x*exp(2*d*x+2*c)-2*a^2*e*f*exp(2*d*x+2*c)+3*a*b*f^2*x*exp(d*x+c)+b^2*e*f*ex
p(2*d*x+2*c)+3*a*b*e*f*exp(d*x+c)-b^2*f^2*x-b^2*e*f)/d^2/(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)^2/(a^2+b^2)-2/(a^
2+b^2)/d^3/b*f^2*ln(exp(d*x+c))+1/(a^2+b^2)/d^3/b*f^2*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/(a^2+b^2)^(3/2)/
d^2/b*f*a*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(3/2)/d^2/b*f^2*a*ln((-b*exp(d*x+c)+
(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(3/2)/d^2/b*f^2*a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(
a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(3/2)/d^3/b*f^2*a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))
*c-1/(a^2+b^2)^(3/2)/d^3/b*f^2*a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/(a^2+b^2)^(3/2)/
d^3/b*f^2*a*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)^(3/2)/d^3/b*f^2*a*dilog(
(b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/(a^2+b^2)^(3/2)/d^3/b*f^2*a*c*arctanh(1/2*(2*b*exp(d*x
+c)+2*a)/(a^2+b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5233 vs. \(2 (282) = 564\).

Time = 0.34 (sec) , antiderivative size = 5233, normalized size of antiderivative = 17.10 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**2*cosh(d*x+c)/(a+b*sinh(d*x+c))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

(2*a*d*integrate(x*e^(d*x + c)/(a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + 2*a^3*b*d^2*e^(d*x + c
) + 2*a*b^3*d^2*e^(d*x + c) - a^2*b^2*d^2 - b^4*d^2), x) + b*(a*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 + b^4)*sqrt(a^2 + b^2)*d^3) - 2*(d*x + c)/((a^2*b^2 + b^4)*d^3) +
log(b*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - b)/((a^2*b^2 + b^4)*d^3)) + 2*(a*b*x*e^(3*d*x + 3*c) - 3*a*b*x*e^(d
*x + c) + b^2*x - ((a^2*d*e^(2*c) + b^2*d*e^(2*c))*x^2 - (2*a^2*e^(2*c) - b^2*e^(2*c))*x)*e^(2*d*x))/(a^2*b^3*
d^2 + b^5*d^2 + (a^2*b^3*d^2*e^(4*c) + b^5*d^2*e^(4*c))*e^(4*d*x) + 4*(a^3*b^2*d^2*e^(3*c) + a*b^4*d^2*e^(3*c)
)*e^(3*d*x) + 2*(2*a^4*b*d^2*e^(2*c) + a^2*b^3*d^2*e^(2*c) - b^5*d^2*e^(2*c))*e^(2*d*x) - 4*(a^3*b^2*d^2*e^c +
a*b^4*d^2*e^c)*e^(d*x)) - a*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^3))*f^2 + e*f*(2*(a*b*e^(3*d*x + 3*c) - 3*a*b*e^(d*x + c) + b^2 + (2*a^2*e^(2
*c) - b^2*e^(2*c) - 2*(a^2*d*e^(2*c) + b^2*d*e^(2*c))*x)*e^(2*d*x))/(a^2*b^3*d^2 + b^5*d^2 + (a^2*b^3*d^2*e^(4
*c) + b^5*d^2*e^(4*c))*e^(4*d*x) + 4*(a^3*b^2*d^2*e^(3*c) + a*b^4*d^2*e^(3*c))*e^(3*d*x) + 2*(2*a^4*b*d^2*e^(2
*c) + a^2*b^3*d^2*e^(2*c) - b^5*d^2*e^(2*c))*e^(2*d*x) - 4*(a^3*b^2*d^2*e^c + a*b^4*d^2*e^c)*e^(d*x)) + a*log(
(b*e^(d*x + 2*c) + a*e^c - sqrt(a^2 + b^2)*e^c)/(b*e^(d*x + 2*c) + a*e^c + sqrt(a^2 + b^2)*e^c))/((a^2*b + b^3
)*sqrt(a^2 + b^2)*d^2)) - 2*e^2*e^(-2*d*x - 2*c)/((4*a*b^2*e^(-d*x - c) - 4*a*b^2*e^(-3*d*x - 3*c) + b^3*e^(-4
*d*x - 4*c) + b^3 + 2*(2*a^2*b - b^3)*e^(-2*d*x - 2*c))*d)

Giac [F]

\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*cosh(d*x + c)/(b*sinh(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int((cosh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x))^3,x)

[Out]

int((cosh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x))^3, x)