3.117 \(\int \frac{(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{4 d^2 \text{PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}-\frac{4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{(c+d x)^2}{3 a^2 f}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3} \]

[Out]

(c + d*x)^2/(3*a^2*f) - (4*d*(c + d*x)*Log[1 + E^(e + f*x)])/(3*a^2*f^2) - (4*d^2*PolyLog[2, -E^(e + f*x)])/(3
*a^2*f^3) + (d*(c + d*x)*Sech[e/2 + (f*x)/2]^2)/(3*a^2*f^2) - (2*d^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f^3) + ((c +
d*x)^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

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Rubi [A]  time = 0.252144, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3318, 4186, 3767, 8, 4184, 3718, 2190, 2279, 2391} \[ -\frac{4 d^2 \text{PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}-\frac{4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{(c+d x)^2}{3 a^2 f}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^2/(3*a^2*f) - (4*d*(c + d*x)*Log[1 + E^(e + f*x)])/(3*a^2*f^2) - (4*d^2*PolyLog[2, -E^(e + f*x)])/(3
*a^2*f^3) + (d*(c + d*x)*Sech[e/2 + (f*x)/2]^2)/(3*a^2*f^2) - (2*d^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f^3) + ((c +
d*x)^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

Rule 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (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])

Rule 4186

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4184

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 3718

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] + Dist[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]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx &=\frac{\int (c+d x)^2 \csc ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\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}+\frac{\int (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{6 a^2}-\frac{d^2 \int \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{3 a^2 f^2}\\ &=\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\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}-\frac{\left (2 i d^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{3 a^2 f^3}-\frac{(2 d) \int (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac{(c+d x)^2}{3 a^2 f}+\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}-\frac{(4 d) \int \frac{e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1+e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=\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{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}+\frac{\left (4 d^2\right ) \int \log \left (1+e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=\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{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}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=\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 \text{Li}_2\left (-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}\\ \end{align*}

Mathematica [C]  time = 6.45686, size = 637, normalized size = 3.18 \[ \frac{16 d^2 \text{csch}\left (\frac{e}{2}\right ) \text{sech}\left (\frac{e}{2}\right ) \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right ) \left (-\frac{1}{4} f^2 x^2 e^{-\tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )}+\frac{i \coth \left (\frac{e}{2}\right ) \left (i \text{PolyLog}\left (2,e^{2 i \left (i \tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )+\frac{i f x}{2}\right )}\right )-\frac{1}{2} f x \left (-\pi +2 i \tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )\right )-2 \left (i \tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )+\frac{i f x}{2}\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )+\frac{i f x}{2}\right )}\right )+2 i \tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\coth \left (\frac{e}{2}\right )\right )+\frac{f x}{2}\right )\right )-\pi \log \left (e^{f x}+1\right )+\pi \log \left (\cosh \left (\frac{f x}{2}\right )\right )\right )}{\sqrt{1-\coth ^2\left (\frac{e}{2}\right )}}\right )}{3 f^3 \sqrt{\text{csch}^2\left (\frac{e}{2}\right ) \left (\sinh ^2\left (\frac{e}{2}\right )-\cosh ^2\left (\frac{e}{2}\right )\right )} (a \cosh (e+f x)+a)^2}+\frac{\text{sech}\left (\frac{e}{2}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}\right ) \left (c^2 f^2 \sinh \left (e+\frac{3 f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac{f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac{3 f x}{2}\right )+2 c d f \cosh \left (e+\frac{f x}{2}\right )+6 c d f^2 x \sinh \left (\frac{f x}{2}\right )+2 c d f \cosh \left (\frac{f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac{3 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 )+2 d^2 f x \cosh \left (e+\frac{f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac{f x}{2}\right )-4 d^2 \sinh \left (\frac{f x}{2}\right )+2 d^2 f x \cosh \left (\frac{f x}{2}\right )\right )}{3 f^3 (a \cosh (e+f x)+a)^2}-\frac{16 c d \text{sech}\left (\frac{e}{2}\right ) \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right ) \left (\cosh \left (\frac{e}{2}\right ) \log \left (\sinh \left (\frac{e}{2}\right ) \sinh \left (\frac{f x}{2}\right )+\cosh \left (\frac{e}{2}\right ) \cosh \left (\frac{f x}{2}\right )\right )-\frac{1}{2} f x \sinh \left (\frac{e}{2}\right )\right )}{3 f^2 \left (\cosh ^2\left (\frac{e}{2}\right )-\sinh ^2\left (\frac{e}{2}\right )\right ) (a \cosh (e+f x)+a)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-16*c*d*Cosh[e/2 + (f*x)/2]^4*Sech[e/2]*(Cosh[e/2]*Log[Cosh[e/2]*Cosh[(f*x)/2] + Sinh[e/2]*Sinh[(f*x)/2]] - (
f*x*Sinh[e/2])/2))/(3*f^2*(a + a*Cosh[e + f*x])^2*(Cosh[e/2]^2 - Sinh[e/2]^2)) + (16*d^2*Cosh[e/2 + (f*x)/2]^4
*Csch[e/2]*(-(f^2*x^2)/(4*E^ArcTanh[Coth[e/2]]) + (I*Coth[e/2]*(-(f*x*(-Pi + (2*I)*ArcTanh[Coth[e/2]]))/2 - Pi
*Log[1 + E^(f*x)] - 2*((I/2)*f*x + I*ArcTanh[Coth[e/2]])*Log[1 - E^((2*I)*((I/2)*f*x + I*ArcTanh[Coth[e/2]]))]
+ Pi*Log[Cosh[(f*x)/2]] + (2*I)*ArcTanh[Coth[e/2]]*Log[I*Sinh[(f*x)/2 + ArcTanh[Coth[e/2]]]] + I*PolyLog[2, E
^((2*I)*((I/2)*f*x + I*ArcTanh[Coth[e/2]]))]))/Sqrt[1 - Coth[e/2]^2])*Sech[e/2])/(3*f^3*(a + a*Cosh[e + f*x])^
2*Sqrt[Csch[e/2]^2*(-Cosh[e/2]^2 + Sinh[e/2]^2)]) + (Cosh[e/2 + (f*x)/2]*Sech[e/2]*(2*c*d*f*Cosh[(f*x)/2] + 2*
d^2*f*x*Cosh[(f*x)/2] + 2*c*d*f*Cosh[e + (f*x)/2] + 2*d^2*f*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*f^3*(a + a*Cosh[e + f*x])^2)

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Maple [A]  time = 0.065, size = 313, normalized size = 1.6 \begin{align*} -{\frac{6\,{f}^{2}{d}^{2}{x}^{2}{{\rm e}^{fx+e}}+12\,{f}^{2}cdx{{\rm e}^{fx+e}}+2\,{d}^{2}{f}^{2}{x}^{2}-4\,{d}^{2}fx{{\rm e}^{2\,fx+2\,e}}+6\,{f}^{2}{c}^{2}{{\rm e}^{fx+e}}+4\,cd{f}^{2}x-4\,cdf{{\rm e}^{2\,fx+2\,e}}-4\,f{d}^{2}x{{\rm e}^{fx+e}}+2\,{c}^{2}{f}^{2}-4\,fcd{{\rm e}^{fx+e}}-4\,{d}^{2}{{\rm e}^{2\,fx+2\,e}}-8\,{d}^{2}{{\rm e}^{fx+e}}-4\,{d}^{2}}{3\,{f}^{3}{a}^{2} \left ({{\rm e}^{fx+e}}+1 \right ) ^{3}}}-{\frac{4\,cd\ln \left ({{\rm e}^{fx+e}}+1 \right ) }{3\,{a}^{2}{f}^{2}}}+{\frac{4\,d\ln \left ({{\rm e}^{fx+e}} \right ) c}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{x}^{2}}{3\,f{a}^{2}}}+{\frac{4\,{d}^{2}ex}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{e}^{2}}{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}\ln \left ({{\rm e}^{fx+e}}+1 \right ) x}{3\,{a}^{2}{f}^{2}}}-{\frac{4\,{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*f^2*d^2*x^2*exp(f*x+e)+6*f^2*c*d*x*exp(f*x+e)+d^2*f^2*x^2-2*d^2*f*x*exp(2*f*x+2*e)+3*f^2*c^2*exp(f*x+e
)+2*c*d*f^2*x-2*c*d*f*exp(2*f*x+2*e)-2*f*d^2*x*exp(f*x+e)+c^2*f^2-2*f*c*d*exp(f*x+e)-2*d^2*exp(2*f*x+2*e)-4*d^
2*exp(f*x+e)-2*d^2)/f^3/a^2/(exp(f*x+e)+1)^3-4/3/f^2*d/a^2*c*ln(exp(f*x+e)+1)+4/3*d/f^2/a^2*ln(exp(f*x+e))*c+2
/3*d^2/f/a^2*x^2+4/3*d^2/f^2/a^2*e*x+2/3*d^2/f^3/a^2*e^2-4/3/f^2*d^2/a^2*ln(exp(f*x+e)+1)*x-4/3*d^2*polylog(2,
-exp(f*x+e))/a^2/f^3-4/3*d^2/f^3/a^2*e*ln(exp(f*x+e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{3} \, d^{2}{\left (\frac{f^{2} x^{2} - 2 \,{\left (f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} +{\left (3 \, f^{2} x^{2} e^{e} - 2 \, f x e^{e} - 4 \, e^{e}\right )} e^{\left (f x\right )} - 2}{a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + a^{2} f^{3}} - 6 \, \int \frac{x}{3 \,{\left (a^{2} f e^{\left (f x + e\right )} + a^{2} f\right )}}\,{d x}\right )} + \frac{4}{3} \, c d{\left (\frac{f x e^{\left (3 \, f x + 3 \, e\right )} +{\left (3 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}} - \frac{\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac{2}{3} \, c^{2}{\left (\frac{3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f} + \frac{1}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Fricas [B]  time = 2.19362, size = 2267, normalized size = 11.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*cosh(f*x + e)^2 + 3*d^2*cosh(f*x + e) + 3*(d^2*cosh(f*x + e) + d^2)*sin
h(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)^3 + 3*a^2*f^3*cosh(f*x + e)^2 + 3*a^2*f^3*cosh(f*x + e) + a^2*f
^3 + 3*(a^2*f^3*cosh(f*x + e) + a^2*f^3)*sinh(f*x + e)^2 + 3*(a^2*f^3*cosh(f*x + e)^2 + 2*a^2*f^3*cosh(f*x + e
) + a^2*f^3)*sinh(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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/(cosh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x))/a**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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