\(\int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 71 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{d x}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{\sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {5917, 29} \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{d x}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \log (x)}{\sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 5917

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Dist[b*c*(n/(f*(m + 1)))*Simp[
(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x}-b c \log (x)\right )}{\sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(63)=126\).

Time = 1.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.93

method	result	size
default	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(208\)
parts	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(208\)

[In]

int((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.73 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\left [-\frac {b c \sqrt {-d} x \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} a}{2 \, d x}, \frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {-c^{2} d x^{2} + d} a}{d x}\right ] \]

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*(b*c*sqrt(-d)*x*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sq
rt(-d) - d)/(c^2*x^4 - x^2)) + 2*sqrt(-c^2*d*x^2 + d)*b*log(c*x + sqrt(c^2*x^2 - 1)) + 2*sqrt(-c^2*d*x^2 + d)*
a)/(d*x), (b*c*sqrt(d)*x*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1
)*d*x^2 + d)) - sqrt(-c^2*d*x^2 + d)*b*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(-c^2*d*x^2 + d)*a)/(d*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {{\left (c^{2} d \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} \sqrt {d} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right )\right )} b c}{2 \, d} - \frac {\sqrt {-c^{2} d x^{2} + d} b \operatorname {arcosh}\left (c x\right )}{d x} - \frac {\sqrt {-c^{2} d x^{2} + d} a}{d x} \]

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*acosh(c*x))/(x^2*(d - c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*acosh(c*x))/(x^2*(d - c^2*d*x^2)^(1/2)), x)