Integrand size = 26, antiderivative size = 306 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {a f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {a f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}+\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {a f^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac {f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]
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))
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 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {f^2 x \coth (c)}{b \left (a^2+b^2\right ) d^2}+\frac {2 e^c f \left (-e^c f x+e^{-c} \left (-1+e^{2 c}\right ) f x-\frac {a e e^{-c} \left (-1+e^{2 c}\right ) \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {a e^{-c} \left (-1+e^{2 c}\right ) f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d}+\frac {1}{2} e^{-c} \left (-1+e^{2 c}\right ) f \left (-2 x+\frac {2 a \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}+\frac {\log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}\right )+\frac {a \left (-1+e^{2 c}\right ) f \left (d x \left (\log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{2 d \sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{b \left (a^2+b^2\right ) d^2 \left (-1+e^{2 c}\right )}-\frac {f^2 x \cosh (c) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right ) \left (a e f \cosh (c)+a f^2 x \cosh (c)+b e f \sinh (d x)+b f^2 x \sinh (d x)\right )}{2 b \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]
(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))*ArcTanh[(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))/Sqrt[-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)])]) + Pol yLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - PolyLo g[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]))
Time = 1.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5987, 3042, 3805, 3042, 3147, 16, 3803, 25, 2694, 27, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 5987 |
\(\displaystyle \frac {f \int \frac {e+f x}{(a+b \sinh (c+d x))^2}dx}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \int \frac {e+f x}{(a-i b \sin (i c+i d x))^2}dx}{b d}\) |
\(\Big \downarrow \) 3805 |
\(\displaystyle \frac {f \left (\frac {a \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {b f \int \frac {\cosh (c+d x)}{a+b \sinh (c+d x)}dx}{d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \left (\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {b f \int \frac {\cos (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle -\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \left (\frac {f \int \frac {1}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}+\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \left (\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}\) |
\(\Big \downarrow \) 3803 |
\(\displaystyle \frac {f \left (\frac {2 a \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {f \left (-\frac {2 a \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {f \left (-\frac {2 a \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {f \left (-\frac {2 a \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {f \left (-\frac {2 a \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {f \left (-\frac {2 a \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {f \left (-\frac {2 a \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{b d}-\frac {(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}\) |
-1/2*(e + f*x)^2/(b*d*(a + b*Sinh[c + d*x])^2) + (f*((f*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)*d^2) - (2*a*(-1/2*(b*(((e + f*x)*Log[1 + (b*E^(c + d* x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - S qrt[a^2 + b^2]))])/(b*d^2)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)*Log[1 + (b*E ^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x) )/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(2*Sqrt[a^2 + b^2])))/(a^2 + b^2) - ( b*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d*(a + b*Sinh[c + d*x]))))/(b*d)
3.4.31.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
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]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[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] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[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]
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] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[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]
Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sinh[ (c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Simp[(e + f*x)^m*((a + b*Sinh[c + d*x])^(n + 1)/(b*d*(n + 1))), x] - Simp[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]
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}\) | \(805\) |
-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*exp(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*e xp(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))
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 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
(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...
\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]