Integrand size = 26, antiderivative size = 330 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b (e+f x)^3}{3 a^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{a d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{a d} \]
1/3*b*(f*x+e)^3/a^2/f-2*f*(f*x+e)*cosh(d*x+c)/a/d^2-b*(f*x+e)^2*ln(1+a*exp (d*x+c)/(b-(a^2+b^2)^(1/2)))/a^2/d-b*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b+(a^2+b ^2)^(1/2)))/a^2/d-2*b*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2) ))/a^2/d^2-2*b*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^2/ d^2+2*b*f^2*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^2/d^3+2*b*f^2*p olylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^2/d^3+2*f^2*sinh(d*x+c)/a/d^ 3+(f*x+e)^2*sinh(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(937\) vs. \(2(330)=660\).
Time = 11.57 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.84 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {f^2 \text {csch}(c+d x) \left (\frac {4 b x^3}{-1+e^{2 c}}-2 b x^3 \coth (c)-\frac {6 a^2 b \left (d^2 x^2 \log \left (1+\frac {\left (b-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 d x \operatorname {PolyLog}\left (2,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 \operatorname {PolyLog}\left (3,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (-b+\sqrt {a^2+b^2}\right ) d^3}-\frac {6 a^2 b \left (d^2 x^2 \log \left (1+\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 \operatorname {PolyLog}\left (3,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right ) d^3}+\frac {6 b^2 \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}-\frac {6 b^2 \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}+\frac {6 a \cosh (d x) \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right )}{d^3}+\frac {6 a \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right ) \sinh (d x)}{d^3}\right ) (b+a \sinh (c+d x))}{6 a^2 (a+b \text {csch}(c+d x))}-\frac {e^2 \text {csch}(c+d x) \left (\frac {b \log (b+a \sinh (c+d x))}{a^2}-\frac {\sinh (c+d x)}{a}\right ) (b+a \sinh (c+d x))}{d (a+b \text {csch}(c+d x))}+\frac {e f \text {csch}(c+d x) (b+a \sinh (c+d x)) \left (-2 a \cosh (c+d x)-b \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 c \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )+2 a d x \sinh (c+d x)\right )}{a^2 d^2 (a+b \text {csch}(c+d x))} \]
(f^2*Csch[c + d*x]*((4*b*x^3)/(-1 + E^(2*c)) - 2*b*x^3*Coth[c] - (6*a^2*b* (d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x*PolyLog[2 , ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a]))/(Sqrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) - (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x* PolyLog[2, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)] - 2*PolyLog[3, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b ^2])*d^3) + (6*b^2*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*b^2* (d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*E^(c + d*x) )/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d*x *Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2*d *x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(6*a^2*(a + b*Csch[c + d*x])) - (e^2*Csch[c + d*x]*((b*Log[b + a*Sinh[c + d*x]])/a^2 - Sinh[c + d *x]/a)*(b + a*Sinh[c + d*x]))/(d*(a + b*Csch[c + d*x])) + (e*f*Csch[c + d* x]*(b + a*Sinh[c + d*x])*(-2*a*Cosh[c + d*x] - b*(2*c*(c + d*x) - (c + d*x )^2 + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] - 2*c*Log[a - 2*b*E...
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {6128, 6113, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 6095, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle -\frac {b \left (\int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 a f}\right )}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \left (-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^3}{3 a f}\right )}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \left (-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^3}{3 a f}\right )}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \left (-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^3}{3 a f}\right )}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {b \left (-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^3}{3 a f}\right )}{a}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{a}\) |
-((b*(-1/3*(e + f*x)^3/(a*f) + ((e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - S qrt[a^2 + b^2])])/(a*d) + ((e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a ^2 + b^2])])/(a*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b ^2]))])/d^2))/(a*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/d^2))/(a*d)))/a) + (((e + f*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I* (e + f*x)*Cosh[c + d*x])/d - (I*f*Sinh[c + d*x])/d^2))/d)/a
3.1.18.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (308) = 616\).
Time = 0.29 (sec) , antiderivative size = 1265, normalized size of antiderivative = 3.83 \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
-1/6*(3*a*d^2*f^2*x^2 + 3*a*d^2*e^2 + 6*a*d*e*f + 6*a*f^2 - 3*(a*d^2*f^2*x ^2 + a*d^2*e^2 - 2*a*d*e*f + 2*a*f^2 + 2*(a*d^2*e*f - a*d*f^2)*x)*cosh(d*x + c)^2 - 3*(a*d^2*f^2*x^2 + a*d^2*e^2 - 2*a*d*e*f + 2*a*f^2 + 2*(a*d^2*e* f - a*d*f^2)*x)*sinh(d*x + c)^2 + 6*(a*d^2*e*f + a*d*f^2)*x - 2*(b*d^3*f^2 *x^3 + 3*b*d^3*e*f*x^2 + 3*b*d^3*e^2*x + 6*b*c*d^2*e^2 - 6*b*c^2*d*e*f + 2 *b*c^3*f^2)*cosh(d*x + c) + 12*((b*d*f^2*x + b*d*e*f)*cosh(d*x + c) + (b*d *f^2*x + b*d*e*f)*sinh(d*x + c))*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 12*((b*d*f^2*x + b*d*e*f)*cosh(d*x + c) + (b*d*f^2*x + b*d*e*f)*sinh(d*x + c))*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh( d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 6*((b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d*x + c) + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sinh(d* x + c))*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a ^2) + 2*b) + 6*((b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d*x + c) + (b*d ^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sinh(d*x + c))*log(2*a*cosh(d*x + c) + 2 *a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 6*((b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*cosh(d*x + c) + (b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sinh(d*x + c))*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^ 2)/a^2) - a)/a) + 6*((b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c...
\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh {\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
-1/2*e^2*(2*(d*x + c)*b/(a^2*d) - e^(d*x + c)/(a*d) + e^(-d*x - c)/(a*d) + 2*b*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^2*d)) - 1/6*(2*b*d ^3*f^2*x^3*e^c + 6*b*d^3*e*f*x^2*e^c - 3*(a*d^2*f^2*x^2*e^(2*c) + 2*(d^2*e *f - d*f^2)*a*x*e^(2*c) - 2*(d*e*f - f^2)*a*e^(2*c))*e^(d*x) + 3*(a*d^2*f^ 2*x^2 + 2*(d^2*e*f + d*f^2)*a*x + 2*(d*e*f + f^2)*a)*e^(-d*x))*e^(-c)/(a^2 *d^3) + integrate(-2*(a*b*f^2*x^2 + 2*a*b*e*f*x - (b^2*f^2*x^2*e^c + 2*b^2 *e*f*x*e^c)*e^(d*x))/(a^3*e^(2*d*x + 2*c) + 2*a^2*b*e^(d*x + c) - a^3), x)
\[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]