\(\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx\) [85]

Optimal result

Integrand size = 23, antiderivative size = 25 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783} \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 \sqrt {\pi } b c} \]

[In]

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

[Out]

(a + b*ArcSinh[c*x])^2/(2*b*c*Sqrt[Pi])

Rule 5783

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {\pi }} \]

[In]

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

[Out]

(a + b*ArcSinh[c*x])^2/(2*b*c*Sqrt[Pi])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(21)=42\).

Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c \sqrt {\pi }}\)	\(53\)
parts	\(\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c \sqrt {\pi }}\)	\(53\)

[In]

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

[Out]

a*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/2*b/c/Pi^(1/2)*arcsinh(c*x)^2

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)/sqrt(pi + pi*c^2*x^2), x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).

Time = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\begin {cases} a \left (\begin {cases} \frac {\log {\left (2 \pi c^{2} x + 2 \sqrt {\pi } \sqrt {\pi c^{2} x^{2} + \pi } \sqrt {c^{2}} \right )}}{\sqrt {\pi } \sqrt {c^{2}}} & \text {for}\: \pi c^{2} \neq 0 \\\frac {x}{\sqrt {\pi }} & \text {otherwise} \end {cases}\right ) & \text {for}\: b = 0 \\\frac {a x}{\sqrt {\pi }} & \text {for}\: c = 0 \\\frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{2 \sqrt {\pi } b c} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*Piecewise((log(2*pi*c**2*x + 2*sqrt(pi)*sqrt(pi*c**2*x**2 + pi)*sqrt(c**2))/(sqrt(pi)*sqrt(c**2))
, Ne(pi*c**2, 0)), (x/sqrt(pi), True)), Eq(b, 0)), (a*x/sqrt(pi), Eq(c, 0)), ((a + b*asinh(c*x))**2/(2*sqrt(pi
)*b*c), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {b \operatorname {arsinh}\left (c x\right )^{2}}{2 \, \sqrt {\pi } c} + \frac {a \operatorname {arsinh}\left (c x\right )}{\sqrt {\pi } c} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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