\(\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx\) [109]

Optimal result

Integrand size = 24, antiderivative size = 53 \[ \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \[ \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 5892

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{2 d \left (c^{2} x^{2}-1\right ) c}\)	\(89\)
parts	\(\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{2 d \left (c^{2} x^{2}-1\right ) c}\)	\(89\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arccosh(c*x))/(-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)

Mupad [F(-1)]

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

[In]

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

[Out]

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