\(\int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx\) [393]

Optimal result

Integrand size = 24, antiderivative size = 16 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\log (a+b \text {arcsinh}(c x))}{b c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, 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 = {5782} \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\log (a+b \text {arcsinh}(c x))}{b c} \]

[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 5782

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

Rubi steps \begin{align*} \text {integral}& = \frac {\log (a+b \text {arcsinh}(c x))}{b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\log (a+b \text {arcsinh}(c x))}{b c} \]

[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] (verified)

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{b c}\)	\(17\)
default	\(\frac {\ln \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{b c}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\log \left (b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a\right )}{b c} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge c = 0 \\\frac {\operatorname {asinh}{\left (c x \right )}}{a c} & \text {for}\: b = 0 \\\frac {x}{a} & \text {for}\: c = 0 \\\frac {\log {\left (\frac {a}{b} + \operatorname {asinh}{\left (c x \right )} \right )}}{b c} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/a, Eq(b, 0) & Eq(c, 0)), (asinh(c*x)/(a*c), Eq(b, 0)), (x/a, Eq(c, 0)), (log(a/b + asinh(c*x))/(b
*c), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\log \left (b \operatorname {arsinh}\left (c x\right ) + a\right )}{b c} \]

[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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\ln \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{b\,c} \]

[In]

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

[Out]

log(a + b*asinh(c*x))/(b*c)