3.393 \(\int \frac{1}{\sqrt{1+c^2 x^2} (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=16 \[ \frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{b c} \]

[Out]

Log[a + b*ArcSinh[c*x]]/(b*c)

________________________________________________________________________________________

Rubi [A]  time = 0.0497027, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {5673} \[ \frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Log[a + b*ArcSinh[c*x]]/(b*c)

Rule 5673

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

Rubi steps

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

Mathematica [A]  time = 0.0254949, size = 16, normalized size = 1. \[ \frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Log[a + b*ArcSinh[c*x]]/(b*c)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }{bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

[Out]

ln(a+b*arcsinh(c*x))/b/c

________________________________________________________________________________________

Maxima [A]  time = 1.08688, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (b \operatorname{arsinh}\left (c x\right ) + a\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*arcsinh(c*x) + a)/(b*c)

________________________________________________________________________________________

Fricas [A]  time = 2.13455, size = 63, normalized size = 3.94 \begin{align*} \frac{\log \left (b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*asinh(c*x))/(c**2*x**2+1)**(1/2),x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.41217, size = 39, normalized size = 2.44 \begin{align*} \frac{\log \left ({\left | b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a \right |}\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*log(c*x + sqrt(c^2*x^2 + 1)) + a))/(b*c)