\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [103]

Optimal result

Integrand size = 26, antiderivative size = 80 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\frac {2 a c x}{\sqrt {1+c^2 x^2}}+\left (2 a-\frac {2 b c x}{\sqrt {1+c^2 x^2}}\right ) \text {arcsinh}(c x)+b \text {arcsinh}(c x)^2+b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 80, 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 = {6225, 240, 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.

\(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6225

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}+\frac {b \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2} c}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{3/2} b c^3}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

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 6225

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(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]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(70)=140\).

Time = 0.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10

method	result	size
default	\(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2 \pi ^{\frac {3}{2}} c^{3}}-\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c^{3}}+\frac {\left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}} c^{3}}\right )\)	\(168\)
parts	\(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2 \pi ^{\frac {3}{2}} c^{3}}-\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c^{3}}+\frac {\left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}} c^{3}}\right )\)	\(168\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(pi + pi*c^2*x^2)*(b*x^2*arcsinh(c*x) + a*x^2)/(pi^2*c^4*x^4 
+ 2*pi^2*c^2*x^2 + pi^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-a*(x/(pi*sqrt(pi + pi*c^2*x^2)*c^2) - arcsinh(c*x)/(pi^(3/2)*c^3)) + b*in 
tegrate(x^2*log(c*x + sqrt(c^2*x^2 + 1))/(pi + pi*c^2*x^2)^(3/2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c^{2} x^{2}+1}\, a c x +\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{5} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3}+\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} x^{2}+\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a -a \,c^{2} x^{2}-a}{\sqrt {\pi }\, c^{3} \pi \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - sqrt(c**2*x**2 + 1)*a*c*x + int((asinh(c*x)*x**2)/(sqrt(c**2*x**2 + 1) 
*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**5*x**2 + int((asinh(c*x)*x**2)/( 
sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**3 + log(sqrt( 
c**2*x**2 + 1) + c*x)*a*c**2*x**2 + log(sqrt(c**2*x**2 + 1) + c*x)*a - a*c 
**2*x**2 - a)/(sqrt(pi)*c**3*pi*(c**2*x**2 + 1))