\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx\) [93]

Optimal result

Integrand size = 26, antiderivative size = 75 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {b x^2}{4 c \sqrt {\pi }}+\frac {x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^2 \pi }-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {\pi }} \] Output:

-1/4*b*x^2/c/Pi^(1/2)+1/2*x*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/c^2/P 
i-1/4*(a+b*arcsinh(c*x))^2/b/c^3/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {4 a c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)^2-b \cosh (2 \text {arcsinh}(c x))+\text {arcsinh}(c x) (-4 a+2 b \sinh (2 \text {arcsinh}(c x)))}{8 c^3 \sqrt {\pi }} \] Input:

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

Output:

(4*a*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x]^2 - b*Cosh[2*ArcSinh[c*x]] + 
 ArcSinh[c*x]*(-4*a + 2*b*Sinh[2*ArcSinh[c*x]]))/(8*c^3*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6227, 15, 6198}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6227

\(\displaystyle -\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{2 c^2}-\frac {b \int xdx}{2 \sqrt {\pi } c}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{2 c^2}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}-\frac {b x^2}{4 \sqrt {\pi } c}\)

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{4 \sqrt {\pi } b c^3}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}-\frac {b x^2}{4 \sqrt {\pi } c}\)

Input:

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

Output:

-1/4*(b*x^2)/(c*Sqrt[Pi]) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])) 
/(2*c^2*Pi) - (a + b*ArcSinh[c*x])^2/(4*b*c^3*Sqrt[Pi])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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]
 

rule 6227

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43

method	result	size
default	\(\frac {a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,c^{2}}-\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \left (-2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )^{2}+1\right )}{4 \sqrt {\pi }\, c^{3}}\)	\(107\)
parts	\(\frac {a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,c^{2}}-\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \left (-2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )^{2}+1\right )}{4 \sqrt {\pi }\, c^{3}}\)	\(107\)

Input:

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

Output:

1/2*a*x/Pi/c^2*(Pi*c^2*x^2+Pi)^(1/2)-1/2*a/c^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+ 
(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)-1/4*b/Pi^(1/2)*(-2*arcsinh(x*c)*(c^2 
*x^2+1)^(1/2)*x*c+c^2*x^2+arcsinh(x*c)^2+1)/c^3
 

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} + 1}}{2 c^{2}} - \frac {\log {\left (2 c^{2} x + 2 \sqrt {c^{2} x^{2} + 1} \sqrt {c^{2}} \right )}}{2 c^{2} \sqrt {c^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{2}}{4 c} + \frac {x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} - \frac {\operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{3}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \] Input:

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

Output:

a*Piecewise((x*sqrt(c**2*x**2 + 1)/(2*c**2) - log(2*c**2*x + 2*sqrt(c**2*x 
**2 + 1)*sqrt(c**2))/(2*c**2*sqrt(c**2)), Ne(c**2, 0)), (x**3/3, True))/sq 
rt(pi) + b*Piecewise((-x**2/(4*c) + x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2*c* 
*2) - asinh(c*x)**2/(4*c**3), Ne(c, 0)), (0, True))/sqrt(pi)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a c x +2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3}-\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a}{2 \sqrt {\pi }\, c^{3}} \] Input:

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

Output:

(sqrt(c**2*x**2 + 1)*a*c*x + 2*int((asinh(c*x)*x**2)/sqrt(c**2*x**2 + 1),x 
)*b*c**3 - log(sqrt(c**2*x**2 + 1) + c*x)*a)/(2*sqrt(pi)*c**3)