\(\int \frac {x (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [94]

Optimal result

Integrand size = 24, antiderivative size = 45 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {a+b \text {arcsinh}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \arctan (c x)}{c^2 \pi ^{3/2}} \]

[Out]

b*arctan(c*x)/c^2/Pi^(3/2)+(-a-b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x^2+Pi)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, 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, 209} \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {b \arctan (c x)}{\pi ^{3/2} c^2}-\frac {a+b \text {arcsinh}(c x)}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }} \]

[In]

Int[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

-((a + b*ArcSinh[c*x])/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (b*ArcTan[c*x])/(c^2*Pi^(3/2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 {a+b \text {arcsinh}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{c \pi ^{3/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \arctan (c x)}{c^2 \pi ^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-a-b \text {arcsinh}(c x)+b \sqrt {1+c^2 x^2} \arctan (c x)}{c^2 \pi ^{3/2} \sqrt {1+c^2 x^2}} \]

[In]

Integrate[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

(-a - b*ArcSinh[c*x] + b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(c^2*Pi^(3/2)*Sqrt[1 + c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.29

method	result	size
default	\(-\frac {a}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c^{2} \pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}+\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{c^{2} \pi ^{\frac {3}{2}}}-\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{c^{2} \pi ^{\frac {3}{2}}}\right )\)	\(103\)
parts	\(-\frac {a}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c^{2} \pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}+\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{c^{2} \pi ^{\frac {3}{2}}}-\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{c^{2} \pi ^{\frac {3}{2}}}\right )\)	\(103\)

[In]

int(x*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-a/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)+b*(-1/c^2/Pi^(3/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)+I/c^2/Pi^(3/2)*ln(c*x+(c^2*x
^2+1)^(1/2)+I)-I/c^2/Pi^(3/2)*ln(c*x+(c^2*x^2+1)^(1/2)-I))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (41) = 82\).

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {\sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} a}{2 \, {\left (\pi ^{2} c^{4} x^{2} + \pi ^{2} c^{2}\right )}} \]

[In]

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]

[In]

integrate(x*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(3/2),x)

[Out]

(Integral(a*x/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x*asinh(c*x)/(c**2*x**2*s
qrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x))/pi**(3/2)

Maxima [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="maxima")

[Out]

b*(integrate(1/(sqrt(c^2*x^2 + 1)*x), x)/(pi^(3/2)*c^2) - log(c*x + sqrt(c^2*x^2 + 1))/(pi^(3/2)*sqrt(c^2*x^2
+ 1)*c^2) - integrate(1/(pi^(3/2)*c^5*x^4 + pi^(3/2)*c^3*x^2 + (pi^(3/2)*c^4*x^3 + pi^(3/2)*c^2*x)*sqrt(c^2*x^
2 + 1)), x)) - a/(pi*sqrt(pi + pi*c^2*x^2)*c^2)

Giac [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)*x/(pi + pi*c^2*x^2)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]

[In]

int((x*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2),x)

[Out]

int((x*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2), x)