\(\int \frac {x (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^2} \, dx\) [40]

Optimal result

Integrand size = 23, antiderivative size = 61 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \]

[Out]

1/2*(a+b*arccosh(c*x))/c^2/d^2/(-c^2*x^2+1)-1/2*b*x/c/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5914, 39} \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a+b \text {arccosh}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

-1/2*(b*x)/(c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(2*c^2*d^2*(1 - c^2*x^2))

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2} \\ & = -\frac {b x}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a+b c x \sqrt {-1+c x} \sqrt {1+c x}+b \text {arccosh}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]

[In]

Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

(a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*ArcCosh[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {c x}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\right )}{d^{2}}}{c^{2}}\)	\(64\)
default	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {c x}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\right )}{d^{2}}}{c^{2}}\)	\(64\)
parts	\(-\frac {a}{2 d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {c x}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\right )}{d^{2} c^{2}}\)	\(66\)

[In]

int(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-1/2*a/d^2/(c^2*x^2-1)+b/d^2*(-1/2/(c^2*x^2-1)*arccosh(c*x)-1/2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c*x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {a c^{2} x^{2} + \sqrt {c^{2} x^{2} - 1} b c x + b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]

[In]

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

-1/2*(a*c^2*x^2 + sqrt(c^2*x^2 - 1)*b*c*x + b*log(c*x + sqrt(c^2*x^2 - 1)))/(c^4*d^2*x^2 - c^2*d^2)

Sympy [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

[In]

integrate(x*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**2,x)

[Out]

(Integral(a*x/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x*acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d
**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (52) = 104\).

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.20 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {1}{4} \, {\left ({\left (\frac {\sqrt {c^{2} x^{2} - 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {c^{2} x^{2} - 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]

[In]

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

-1/4*((sqrt(c^2*x^2 - 1)*c^2*d^2/(c^7*d^4*x + c^6*d^4) + sqrt(c^2*x^2 - 1)*c^2*d^2/(c^7*d^4*x - c^6*d^4))*c^2
+ 2*arccosh(c*x)/(c^4*d^2*x^2 - c^2*d^2))*b - 1/2*a/(c^4*d^2*x^2 - c^2*d^2)

Giac [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x/(c^2*d*x^2 - d)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

[In]

int((x*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^2,x)

[Out]

int((x*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^2, x)