\(\int \frac {a+b \text {arcsinh}(c x)}{(\pi +c^2 \pi x^2)^{5/2}} \, dx\) [116]

Optimal result

Integrand size = 23, antiderivative size = 108 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {b \log \left (1+c^2 x^2\right )}{3 c \pi ^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {6 a c x+4 a c^3 x^3+b \sqrt {1+c^2 x^2}+2 b c x \left (3+2 c^2 x^2\right ) \text {arcsinh}(c x)-2 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \] Input:

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

Output:

(6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*c*x*(3 + 2*c^2*x^2)*Arc 
Sinh[c*x] - 2*b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(6*c*Pi^(5/2)*(1 + c 
^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6203, 241, 6202, 240}

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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 241

\(\displaystyle \frac {2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6202

\(\displaystyle \frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b c \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2}}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c}\right )}{3 \pi }+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\)

Input:

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

Output:

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

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 241

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

rule 6202

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, 
 c^2*d] && GtQ[n, 0]
 

rule 6203

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + 
c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(92)=184\).

Time = 0.91 (sec) , antiderivative size = 619, normalized size of antiderivative = 5.73

method	result	size
default	\(a \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {4 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c}+\frac {2 b \,c^{7} x^{8}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {2 b \,c^{5} \operatorname {arcsinh}\left (x c \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b \,c^{4} \operatorname {arcsinh}\left (x c \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {20 b \,c^{3} \operatorname {arcsinh}\left (x c \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {17 b \,c^{2} \operatorname {arcsinh}\left (x c \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {4 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {22 b c \,\operatorname {arcsinh}\left (x c \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {4 b \,\operatorname {arcsinh}\left (x c \right ) x}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b c \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}} c}\)	\(619\)
parts	\(a \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {4 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c}+\frac {2 b \,c^{7} x^{8}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {2 b \,c^{5} \operatorname {arcsinh}\left (x c \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b \,c^{4} \operatorname {arcsinh}\left (x c \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {20 b \,c^{3} \operatorname {arcsinh}\left (x c \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {17 b \,c^{2} \operatorname {arcsinh}\left (x c \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {4 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {22 b c \,\operatorname {arcsinh}\left (x c \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {4 b \,\operatorname {arcsinh}\left (x c \right ) x}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b c \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}} c}\)	\(619\)

Input:

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

Output:

a*(1/3/Pi*x/(Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(Pi*c^2*x^2+Pi)^(1/2))+4/3*b/ 
Pi^(5/2)/c*arcsinh(x*c)+2/3*b/Pi^(5/2)*c^7/(3*c^2*x^2+4)/(c^2*x^2+1)^2*x^8 
-2/3*b/Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1)*x^6-2*b/Pi^(5/2)*c^5/(3*c^2* 
x^2+4)/(c^2*x^2+1)^2*arcsinh(x*c)*x^6+2*b/Pi^(5/2)*c^4/(3*c^2*x^2+4)/(c^2* 
x^2+1)^(3/2)*arcsinh(x*c)*x^5+8/3*b/Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1) 
^2*x^6-2*b/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^2*x^2+1)*x^4-20/3*b/Pi^(5/2)*c^3/ 
(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(x*c)*x^4+17/3*b/Pi^(5/2)*c^2/(3*c^2*x^ 
2+4)/(c^2*x^2+1)^(3/2)*arcsinh(x*c)*x^3+4*b/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^ 
2*x^2+1)^2*x^4-3/2*b/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)*x^2-22/3*b/Pi^(5 
/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(x*c)*x^2+4*b/Pi^(5/2)/(3*c^2*x^2 
+4)/(c^2*x^2+1)^(3/2)*arcsinh(x*c)*x+8/3*b/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x 
^2+1)^2*x^2-8/3*b/Pi^(5/2)/c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(x*c)+2/3* 
b/Pi^(5/2)/c/(3*c^2*x^2+4)/(c^2*x^2+1)^2-2/3*b/Pi^(5/2)/c*ln(1+(x*c+(c^2*x 
^2+1)^(1/2))^2)
 

Fricas [F]

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

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

Output:

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^3*c^6*x^6 + 3*pi^3 
*c^4*x^4 + 3*pi^3*c^2*x^2 + pi^3), x)
 

Sympy [F]

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

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

Output:

(Integral(a/(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*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))/p 
i**(5/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {1}{\pi ^{\frac {5}{2}} c^{4} x^{2} + \pi ^{\frac {5}{2}} c^{2}} - \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {5}{2}} c^{2}}\right )} + \frac {1}{3} \, b {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} \] Input:

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

Output:

1/6*b*c*(1/(pi^(5/2)*c^4*x^2 + pi^(5/2)*c^2) - 2*log(c^2*x^2 + 1)/(pi^(5/2 
)*c^2)) + 1/3*b*(x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 2*x/(pi^2*sqrt(pi + pi*c 
^2*x^2)))*arcsinh(c*x) + 1/3*a*(x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 2*x/(pi^2 
*sqrt(pi + pi*c^2*x^2)))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 3*sqrt(c**2*x**2 + 1)*a*c*x + 3*int(a 
sinh(c*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**5*x**4 + 6*int(asinh(c*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**3*x**2 + 3*int(asinh(c*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 - 2*a*c**4*x**4 - 4*a 
*c**2*x**2 - 2*a)/(3*sqrt(pi)*c*pi**2*(c**4*x**4 + 2*c**2*x**2 + 1))