\(\int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx\) [84]

Optimal result

Integrand size = 24, antiderivative size = 42 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {b x}{c \sqrt {\pi }}+\frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^2 \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 8} \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi c^2}-\frac {b x}{\sqrt {\pi } c} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5798

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

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {-b c x+a \sqrt {1+c^2 x^2}+b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c^2 \sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {a \sqrt {\pi \,c^{2} x^{2}+\pi }}{\pi \,c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )-c x \sqrt {c^{2} x^{2}+1}\right )}{c^{2} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}}\)	\(72\)
parts	\(\frac {a \sqrt {\pi \,c^{2} x^{2}+\pi }}{\pi \,c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )-c x \sqrt {c^{2} x^{2}+1}\right )}{c^{2} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}}\)	\(72\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (38) = 76\).

Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.29 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (a c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b c x + a\right )}}{\pi c^{4} x^{2} + \pi c^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.43 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} + 1}}{c^{2}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x}{c} + \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c^{2}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

a*Piecewise((sqrt(c**2*x**2 + 1)/c**2, Ne(c**2, 0)), (x**2/2, True))/sqrt(pi) + b*Piecewise((-x/c + sqrt(c**2*
x**2 + 1)*asinh(c*x)/c**2, Ne(c, 0)), (0, True))/sqrt(pi)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {b x}{\sqrt {\pi } c} + \frac {\sqrt {\pi + \pi c^{2} x^{2}} b \operatorname {arsinh}\left (c x\right )}{\pi c^{2}} + \frac {\sqrt {\pi + \pi c^{2} x^{2}} a}{\pi c^{2}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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