\(\int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx\) [117]

Optimal result

Integrand size = 20, antiderivative size = 200 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271, 3852, 8, 4269, 3799, 2221, 2317, 2438} \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=-\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \]

[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 3399

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

Rule 3799

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 3852

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 4269

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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {8 \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (f (c+d x)+2 d \left (1+e^e\right ) \log \left (1+e^{-e-f x}\right )\right )-2 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )\right )}{1+e^e}+\text {sech}\left (\frac {e}{2}\right ) \left (2 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+2 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )-4 d^2 \sinh \left (\frac {f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )+2 d^2 \sinh \left (e+\frac {f x}{2}\right )-2 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )\right )\right )}{3 a^2 f^3 (1+\cosh (e+f x))^2} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.56

method	result	size
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{f x +e} d^{2} f^{2} x^{2}+6 \,{\mathrm e}^{f x +e} c d \,f^{2} x +d^{2} x^{2} f^{2}-2 d^{2} f x \,{\mathrm e}^{2 f x +2 e}+3 \,{\mathrm e}^{f x +e} c^{2} f^{2}+2 c d \,f^{2} x -2 c d f \,{\mathrm e}^{2 f x +2 e}-2 d^{2} f x \,{\mathrm e}^{f x +e}+c^{2} f^{2}-2 c d f \,{\mathrm e}^{f x +e}-2 \,{\mathrm e}^{2 f x +2 e} d^{2}-4 \,{\mathrm e}^{f x +e} d^{2}-2 d^{2}\right )}{3 f^{3} a^{2} \left (1+{\mathrm e}^{f x +e}\right )^{3}}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}-\frac {4 d c \ln \left (1+{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}\)	\(313\)

[In]

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

[Out]

-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*ex
p(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))

Fricas [B] (verification not implemented)

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 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\text {Too large to display} \]

[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))

Sympy [F]

\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\int \frac {c^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[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

Maxima [F]

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

[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))

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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