Integrand size = 20, antiderivative size = 200 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
1/3*(d*x+c)^2/a^2/f-4/3*d*(d*x+c)*ln(1+exp(f*x+e))/a^2/f^2-4/3*d^2*polylog (2,-exp(f*x+e))/a^2/f^3+1/3*d*(d*x+c)*sech(1/2*f*x+1/2*e)^2/a^2/f^2-2/3*d^ 2*tanh(1/2*f*x+1/2*e)/a^2/f^3+1/3*(d*x+c)^2*tanh(1/2*f*x+1/2*e)/a^2/f+1/6* (d*x+c)^2*sech(1/2*f*x+1/2*e)^2*tanh(1/2*f*x+1/2*e)/a^2/f
Time = 1.05 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {8 \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (f (c+d x)+2 d \left (1+e^e\right ) \log \left (1+e^{-e-f x}\right )\right )-2 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )\right )}{1+e^e}+\text {sech}\left (\frac {e}{2}\right ) \left (2 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+2 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )-4 d^2 \sinh \left (\frac {f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )+2 d^2 \sinh \left (e+\frac {f x}{2}\right )-2 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )\right )\right )}{3 a^2 f^3 (1+\cosh (e+f x))^2} \]
(Cosh[(e + f*x)/2]*((-8*Cosh[(e + f*x)/2]^3*(f*(c + d*x)*(f*(c + d*x) + 2* d*(1 + E^e)*Log[1 + E^(-e - f*x)]) - 2*d^2*(1 + E^e)*PolyLog[2, -E^(-e - f *x)]))/(1 + E^e) + Sech[e/2]*(2*d*f*(c + d*x)*Cosh[(f*x)/2] + 2*d*f*(c + d *x)*Cosh[e + (f*x)/2] - 4*d^2*Sinh[(f*x)/2] + 3*c^2*f^2*Sinh[(f*x)/2] + 6* c*d*f^2*x*Sinh[(f*x)/2] + 3*d^2*f^2*x^2*Sinh[(f*x)/2] + 2*d^2*Sinh[e + (f* x)/2] - 2*d^2*Sinh[e + (3*f*x)/2] + c^2*f^2*Sinh[e + (3*f*x)/2] + 2*c*d*f^ 2*x*Sinh[e + (3*f*x)/2] + d^2*f^2*x^2*Sinh[e + (3*f*x)/2])))/(3*a^2*f^3*(1 + Cosh[e + f*x])^2)
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3799, 3042, 4674, 3042, 4254, 24, 4672, 26, 3042, 26, 4201, 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.
\(\displaystyle \int \frac {(c+d x)^2}{(a \cosh (e+f x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d x)^2}{\left (a+a \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 3799 |
\(\displaystyle \frac {\int (c+d x)^2 \text {sech}^4\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{4 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^4dx}{4 a^2}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx-\frac {4 d^2 \int \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{3 f^2}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^2dx-\frac {4 d^2 \int \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^2dx}{3 f^2}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^2dx-\frac {8 i d^2 \int 1d\left (-i \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f^3}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^2dx+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 i d \int -i (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \int (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \int -i (c+d x) \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 i d \int (c+d x) \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 i d \left (2 i \int \frac {e^{e+f x} (c+d x)}{1+e^{e+f x}}dx-\frac {i (c+d x)^2}{2 d}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 i d \left (2 i \left (\frac {(c+d x) \log \left (e^{e+f x}+1\right )}{f}-\frac {d \int \log \left (1+e^{e+f x}\right )dx}{f}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 i d \left (2 i \left (\frac {(c+d x) \log \left (e^{e+f x}+1\right )}{f}-\frac {d \int e^{-e-f x} \log \left (1+e^{e+f x}\right )de^{e+f x}}{f^2}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 i d \left (2 i \left (\frac {(c+d x) \log \left (e^{e+f x}+1\right )}{f}+\frac {d \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{f^2}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
((4*d*(c + d*x)*Sech[e/2 + (f*x)/2]^2)/(3*f^2) - (8*d^2*Tanh[e/2 + (f*x)/2 ])/(3*f^3) + (2*(c + d*x)^2*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(3* f) + (2*(((4*I)*d*(((-1/2*I)*(c + d*x)^2)/d + (2*I)*(((c + d*x)*Log[1 + E^ (e + f*x)])/f + (d*PolyLog[2, -E^(e + f*x)])/f^2)))/f + (2*(c + d*x)^2*Tan h[e/2 + (f*x)/2])/f))/3)/(4*a^2)
3.2.17.3.1 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[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[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Time = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.56
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{f x +e} d^{2} f^{2} x^{2}+6 \,{\mathrm e}^{f x +e} c d \,f^{2} x +d^{2} x^{2} f^{2}-2 d^{2} f x \,{\mathrm e}^{2 f x +2 e}+3 \,{\mathrm e}^{f x +e} c^{2} f^{2}+2 c d \,f^{2} x -2 c d f \,{\mathrm e}^{2 f x +2 e}-2 d^{2} f x \,{\mathrm e}^{f x +e}+c^{2} f^{2}-2 c d f \,{\mathrm e}^{f x +e}-2 \,{\mathrm e}^{2 f x +2 e} d^{2}-4 \,{\mathrm e}^{f x +e} d^{2}-2 d^{2}\right )}{3 f^{3} a^{2} \left (1+{\mathrm e}^{f x +e}\right )^{3}}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}-\frac {4 d c \ln \left (1+{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}\) | \(313\) |
-2/3*(3*exp(f*x+e)*d^2*f^2*x^2+6*exp(f*x+e)*c*d*f^2*x+d^2*x^2*f^2-2*d^2*f* x*exp(2*f*x+2*e)+3*exp(f*x+e)*c^2*f^2+2*c*d*f^2*x-2*c*d*f*exp(2*f*x+2*e)-2 *d^2*f*x*exp(f*x+e)+c^2*f^2-2*c*d*f*exp(f*x+e)-2*exp(2*f*x+2*e)*d^2-4*exp( f*x+e)*d^2-2*d^2)/f^3/a^2/(1+exp(f*x+e))^3+4/3/a^2*d/f^2*c*ln(exp(f*x+e))- 4/3/a^2*d/f^2*c*ln(1+exp(f*x+e))+2/3/a^2*d^2/f*x^2+4/3/a^2*d^2/f^2*e*x+2/3 /a^2*d^2/f^3*e^2-4/3/a^2*d^2/f^2*ln(1+exp(f*x+e))*x-4/3*d^2*polylog(2,-exp (f*x+e))/a^2/f^3-4/3/a^2*d^2/f^3*e*ln(exp(f*x+e))
Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (163) = 326\).
Time = 0.25 (sec) , antiderivative size = 963, normalized size of antiderivative = 4.82 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\text {Too large to display} \]
-2/3*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 - (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)^3 - (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c *d*e*f)*sinh(f*x + e)^3 - (3*d^2*f^2*x^2 - 3*d^2*e^2 + 2*d^2 + 2*(3*c*d*e + c*d)*f + 2*(3*c*d*f^2 + d^2*f)*x)*cosh(f*x + e)^2 - (3*d^2*f^2*x^2 - 3*d ^2*e^2 + 2*d^2 + 2*(3*c*d*e + c*d)*f + 2*(3*c*d*f^2 + d^2*f)*x + 3*(d^2*f^ 2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e))*sinh(f*x + e)^2 - 2*d^2 + (3*d^2*e^2 + 3*c^2*f^2 - 2*d^2*f*x - 4*d^2 - 2*(3*c*d*e + c*d)*f )*cosh(f*x + e) + 2*(d^2*cosh(f*x + e)^3 + d^2*sinh(f*x + e)^3 + 3*d^2*cos h(f*x + e)^2 + 3*d^2*cosh(f*x + e) + 3*(d^2*cosh(f*x + e) + d^2)*sinh(f*x + e)^2 + d^2 + 3*(d^2*cosh(f*x + e)^2 + 2*d^2*cosh(f*x + e) + d^2)*sinh(f* x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 2*(d^2*f*x + (d^2*f*x + c* d*f)*cosh(f*x + e)^3 + (d^2*f*x + c*d*f)*sinh(f*x + e)^3 + c*d*f + 3*(d^2* f*x + c*d*f)*cosh(f*x + e)^2 + 3*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cosh (f*x + e))*sinh(f*x + e)^2 + 3*(d^2*f*x + c*d*f)*cosh(f*x + e) + 3*(d^2*f* x + c*d*f + (d^2*f*x + c*d*f)*cosh(f*x + e)^2 + 2*(d^2*f*x + c*d*f)*cosh(f *x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) + (3*d^2*e^ 2 + 3*c^2*f^2 - 2*d^2*f*x - 3*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d *e*f)*cosh(f*x + e)^2 - 4*d^2 - 2*(3*c*d*e + c*d)*f - 2*(3*d^2*f^2*x^2 - 3 *d^2*e^2 + 2*d^2 + 2*(3*c*d*e + c*d)*f + 2*(3*c*d*f^2 + d^2*f)*x)*cosh(f*x + e))*sinh(f*x + e))/(a^2*f^3*cosh(f*x + e)^3 + a^2*f^3*sinh(f*x + e)^...
\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\int \frac {c^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
(Integral(c**2/(cosh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x) + Integral(d** 2*x**2/(cosh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x) + Integral(2*c*d*x/(co sh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x))/a**2
\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
-2/3*d^2*((f^2*x^2 - 2*(f*x*e^(2*e) + e^(2*e))*e^(2*f*x) + (3*f^2*x^2*e^e - 2*f*x*e^e - 4*e^e)*e^(f*x) - 2)/(a^2*f^3*e^(3*f*x + 3*e) + 3*a^2*f^3*e^( 2*f*x + 2*e) + 3*a^2*f^3*e^(f*x + e) + a^2*f^3) - 6*integrate(1/3*x/(a^2*f *e^(f*x + e) + a^2*f), x)) + 4/3*c*d*((f*x*e^(3*f*x + 3*e) + (3*f*x*e^(2*e ) + e^(2*e))*e^(2*f*x) + e^(f*x + e))/(a^2*f^2*e^(3*f*x + 3*e) + 3*a^2*f^2 *e^(2*f*x + 2*e) + 3*a^2*f^2*e^(f*x + e) + a^2*f^2) - log((e^(f*x + e) + 1 )*e^(-e))/(a^2*f^2)) + 2/3*c^2*(3*e^(-f*x - e)/((3*a^2*e^(-f*x - e) + 3*a^ 2*e^(-2*f*x - 2*e) + a^2*e^(-3*f*x - 3*e) + a^2)*f) + 1/((3*a^2*e^(-f*x - e) + 3*a^2*e^(-2*f*x - 2*e) + a^2*e^(-3*f*x - 3*e) + a^2)*f))
\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \]