\(\int \frac {x (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{5/2}} \, dx\) [115]

Optimal result

Integrand size = 24, antiderivative size = 75 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b x}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \arctan (c x)}{6 c^2 \pi ^{5/2}} \] Output:

1/6*b*x/c/Pi^(5/2)/(c^2*x^2+1)-1/3*(a+b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x^2+P 
i)^(3/2)+1/6*b*arctan(c*x)/c^2/Pi^(5/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-2 a+b c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)+b \left (1+c^2 x^2\right )^{3/2} \arctan (c x)}{6 c^2 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \] Input:

Integrate[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]
 

Output:

(-2*a + b*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x] + b*(1 + c^2*x^2)^(3/2) 
*ArcTan[c*x])/(6*c^2*Pi^(5/2)*(1 + c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6213, 215, 216}

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.

\(\displaystyle \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {b \int \frac {1}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2} c}-\frac {a+b \text {arcsinh}(c x)}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {b \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 \pi ^{5/2} c}-\frac {a+b \text {arcsinh}(c x)}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 \pi ^{5/2} c}-\frac {a+b \text {arcsinh}(c x)}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\)

Input:

Int[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]
 

Output:

-1/3*(a + b*ArcSinh[c*x])/(c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (b*(x/(2*(1 + 
 c^2*x^2)) + ArcTan[c*x]/(2*c)))/(3*c*Pi^(5/2))
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 216

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

rule 6213

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] - Simp[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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.61

method	result	size
default	\(-\frac {a}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+b \left (-\frac {-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {i \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )}{6 \pi ^{\frac {5}{2}} c^{2}}-\frac {i \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )}{6 \pi ^{\frac {5}{2}} c^{2}}\right )\)	\(121\)
parts	\(-\frac {a}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+b \left (-\frac {-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {i \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )}{6 \pi ^{\frac {5}{2}} c^{2}}-\frac {i \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )}{6 \pi ^{\frac {5}{2}} c^{2}}\right )\)	\(121\)

Input:

int(x*(a+b*arcsinh(x*c))/(Pi*c^2*x^2+Pi)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/Pi/c^2/(Pi*c^2*x^2+Pi)^(3/2)+b*(-1/6/Pi^(5/2)/(c^2*x^2+1)^(3/2)*(-( 
c^2*x^2+1)^(1/2)*x*c+2*arcsinh(x*c))/c^2+1/6*I/Pi^(5/2)/c^2*ln(x*c+(c^2*x^ 
2+1)^(1/2)+I)-1/6*I/Pi^(5/2)/c^2*ln(x*c+(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. 165 vs. \(2 (63) = 126\).

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.20 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {\sqrt {\pi } {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 4 \, \sqrt {\pi + \pi c^{2} x^{2}} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\sqrt {c^{2} x^{2} + 1} b c x - 2 \, a\right )}}{12 \, {\left (\pi ^{3} c^{6} x^{4} + 2 \, \pi ^{3} c^{4} x^{2} + \pi ^{3} c^{2}\right )}} \] Input:

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="fricas" 
)
 

Output:

-1/12*(sqrt(pi)*(b*c^4*x^4 + 2*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)) + 4*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)*(sqrt(c^2*x 
^2 + 1)*b*c*x - 2*a))/(pi^3*c^6*x^4 + 2*pi^3*c^4*x^2 + pi^3*c^2)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \] Input:

integrate(x*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(5/2),x)
 

Output:

(Integral(a*x/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 
+ 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x*asinh(c*x)/(c**4*x**4*sqrt( 
c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x 
))/pi**(5/2)
 

Maxima [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="maxima" 
)
 

Output:

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

Giac [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)*x/(pi + pi*c^2*x^2)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \] Input:

int((x*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(5/2),x)
 

Output:

int((x*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-\sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{6} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{2}}{3 \sqrt {\pi }\, c^{2} \pi ^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int(x*(a+b*asinh(c*x))/(Pi*c^2*x^2+Pi)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

( - sqrt(c**2*x**2 + 1)*a + 3*int((asinh(c*x)*x)/(sqrt(c**2*x**2 + 1)*c**4 
*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**6*x 
**4 + 6*int((asinh(c*x)*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x* 
*2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**4*x**2 + 3*int((asinh(c*x 
)*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sq 
rt(c**2*x**2 + 1)),x)*b*c**2)/(3*sqrt(pi)*c**2*pi**2*(c**4*x**4 + 2*c**2*x 
**2 + 1))