\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx\) [89]

Optimal result

Integrand size = 26, antiderivative size = 196 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=-\frac {b c \pi ^{5/2}}{110 x^{10}}-\frac {23 b c^3 \pi ^{5/2}}{792 x^8}-\frac {113 b c^5 \pi ^{5/2}}{4158 x^6}-\frac {b c^7 \pi ^{5/2}}{924 x^4}+\frac {2 b c^9 \pi ^{5/2}}{693 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}+\frac {8}{693} b c^{11} \pi ^{5/2} \log (x) \] Output:

-1/110*b*c*Pi^(5/2)/x^10-23/792*b*c^3*Pi^(5/2)/x^8-113/4158*b*c^5*Pi^(5/2) 
/x^6-1/924*b*c^7*Pi^(5/2)/x^4+2/693*b*c^9*Pi^(5/2)/x^2-1/11*(Pi*c^2*x^2+Pi 
)^(7/2)*(a+b*arcsinh(c*x))/Pi/x^11+4/99*c^2*(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arc 
sinh(c*x))/Pi/x^9-8/693*c^4*(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arcsinh(c*x))/Pi/x^ 
7+8/693*b*c^11*Pi^(5/2)*ln(x)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.21 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\frac {\pi ^{5/2} \left (-15876 b c x-50715 b c^3 x^3-47460 b c^5 x^5-1890 b c^7 x^7+5040 b c^9 x^9-59048 b c^{11} x^{11}-158760 a \sqrt {1+c^2 x^2}-405720 a c^2 x^2 \sqrt {1+c^2 x^2}-284760 a c^4 x^4 \sqrt {1+c^2 x^2}-7560 a c^6 x^6 \sqrt {1+c^2 x^2}+10080 a c^8 x^8 \sqrt {1+c^2 x^2}-20160 a c^{10} x^{10} \sqrt {1+c^2 x^2}-2520 b \left (1+c^2 x^2\right )^{7/2} \left (63-28 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+20160 b c^{11} x^{11} \log (x)\right )}{1746360 x^{11}} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^12,x]
 

Output:

(Pi^(5/2)*(-15876*b*c*x - 50715*b*c^3*x^3 - 47460*b*c^5*x^5 - 1890*b*c^7*x 
^7 + 5040*b*c^9*x^9 - 59048*b*c^11*x^11 - 158760*a*Sqrt[1 + c^2*x^2] - 405 
720*a*c^2*x^2*Sqrt[1 + c^2*x^2] - 284760*a*c^4*x^4*Sqrt[1 + c^2*x^2] - 756 
0*a*c^6*x^6*Sqrt[1 + c^2*x^2] + 10080*a*c^8*x^8*Sqrt[1 + c^2*x^2] - 20160* 
a*c^10*x^10*Sqrt[1 + c^2*x^2] - 2520*b*(1 + c^2*x^2)^(7/2)*(63 - 28*c^2*x^ 
2 + 8*c^4*x^4)*ArcSinh[c*x] + 20160*b*c^11*x^11*Log[x]))/(1746360*x^11)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6219, 27, 1578, 1195, 2009}

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.

\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx\)

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int -\frac {\pi ^2 \left (c^2 x^2+1\right )^3 \left (8 c^4 x^4-28 c^2 x^2+63\right )}{693 x^{11}}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{693} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^3 \left (8 c^4 x^4-28 c^2 x^2+63\right )}{x^{11}}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {\pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^3 \left (8 c^4 x^4-28 c^2 x^2+63\right )}{x^{12}}dx^2}{1386}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {\pi ^{5/2} b c \int \left (\frac {8 c^{10}}{x^2}-\frac {4 c^8}{x^4}+\frac {3 c^6}{x^6}+\frac {113 c^4}{x^8}+\frac {161 c^2}{x^{10}}+\frac {63}{x^{12}}\right )dx^2}{1386}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{11 \pi x^{11}}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{99 \pi x^9}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{693 \pi x^7}+\frac {\pi ^{5/2} b c \left (8 c^{10} \log \left (x^2\right )+\frac {4 c^8}{x^2}-\frac {3 c^6}{2 x^4}-\frac {113 c^4}{3 x^6}-\frac {161 c^2}{4 x^8}-\frac {63}{5 x^{10}}\right )}{1386}\)

Input:

Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^12,x]
 

Output:

-1/11*((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(Pi*x^11) + (4*c^2*(P 
i + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(99*Pi*x^9) - (8*c^4*(Pi + c^2 
*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(693*Pi*x^7) + (b*c*Pi^(5/2)*(-63/(5* 
x^10) - (161*c^2)/(4*x^8) - (113*c^4)/(3*x^6) - (3*c^6)/(2*x^4) + (4*c^8)/ 
x^2 + 8*c^10*Log[x^2]))/1386
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3923\) vs. \(2(160)=320\).

Time = 1.30 (sec) , antiderivative size = 3924, normalized size of antiderivative = 20.02

method	result	size
default	\(\text {Expression too large to display}\)	\(3924\)
parts	\(\text {Expression too large to display}\)	\(3924\)

Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))/x^12,x,method=_RETURNVERBOSE)
 

Output:

-16/693*b*Pi^(5/2)*c^11*arcsinh(x*c)+8/693*b*Pi^(5/2)*c^11*ln((x*c+(c^2*x^ 
2+1)^(1/2))^2-1)-12*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^14+2079*c^12*x^ 
12+13629*c^10*x^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^4+24255*c^2*x^2 
+3969)*x^11*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^22-1385/3*b*Pi^(5/2)/(1848*c^ 
16*x^16+3465*c^14*x^14+2079*c^12*x^12+13629*c^10*x^10+49995*c^8*x^8+76923* 
c^6*x^6+60445*c^4*x^4+24255*c^2*x^2+3969)*x^9*(c^2*x^2+1)^(1/2)*arcsinh(x* 
c)*c^20-12902333/41580*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^14+2079*c^12 
*x^12+13629*c^10*x^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^4+24255*c^2* 
x^2+3969)*x^2*(c^2*x^2+1)*c^13+280*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^ 
14+2079*c^12*x^12+13629*c^10*x^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^ 
4+24255*c^2*x^2+3969)*x^2*arcsinh(x*c)*c^13-3400361/1485*b*Pi^(5/2)/(1848* 
c^16*x^16+3465*c^14*x^14+2079*c^12*x^12+13629*c^10*x^10+49995*c^8*x^8+7692 
3*c^6*x^6+60445*c^4*x^4+24255*c^2*x^2+3969)/x^2*(c^2*x^2+1)*c^9-16298009/1 
1880*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^14+2079*c^12*x^12+13629*c^10*x 
^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^4+24255*c^2*x^2+3969)*(c^2*x^2 
+1)*c^11+504/11*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^14+2079*c^12*x^12+1 
3629*c^10*x^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^4+24255*c^2*x^2+396 
9)*arcsinh(x*c)*c^11-128/693*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x^14+207 
9*c^12*x^12+13629*c^10*x^10+49995*c^8*x^8+76923*c^6*x^6+60445*c^4*x^4+2425 
5*c^2*x^2+3969)*x^20*c^31+16/231*b*Pi^(5/2)/(1848*c^16*x^16+3465*c^14*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (160) = 320\).

Time = 0.16 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.15 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=-\frac {120 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (8 \, \pi ^{2} b c^{12} x^{12} + 4 \, \pi ^{2} b c^{10} x^{10} - \pi ^{2} b c^{8} x^{8} + 116 \, \pi ^{2} b c^{6} x^{6} + 274 \, \pi ^{2} b c^{4} x^{4} + 224 \, \pi ^{2} b c^{2} x^{2} + 63 \, \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 480 \, \sqrt {\pi } {\left (\pi ^{2} b c^{13} x^{13} + \pi ^{2} b c^{11} x^{11}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (960 \, \pi ^{2} a c^{12} x^{12} + 480 \, \pi ^{2} a c^{10} x^{10} - 120 \, \pi ^{2} a c^{8} x^{8} + 13920 \, \pi ^{2} a c^{6} x^{6} + 32880 \, \pi ^{2} a c^{4} x^{4} + 26880 \, \pi ^{2} a c^{2} x^{2} + 7560 \, \pi ^{2} a - {\left (240 \, \pi ^{2} b c^{9} x^{9} - 90 \, \pi ^{2} b c^{7} x^{7} - \pi ^{2} {\left (240 \, b c^{9} - 90 \, b c^{7} - 2260 \, b c^{5} - 2415 \, b c^{3} - 756 \, b c\right )} x^{11} - 2260 \, \pi ^{2} b c^{5} x^{5} - 2415 \, \pi ^{2} b c^{3} x^{3} - 756 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{83160 \, {\left (c^{2} x^{13} + x^{11}\right )}} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^12,x, algorithm="fric 
as")
 

Output:

-1/83160*(120*sqrt(pi + pi*c^2*x^2)*(8*pi^2*b*c^12*x^12 + 4*pi^2*b*c^10*x^ 
10 - pi^2*b*c^8*x^8 + 116*pi^2*b*c^6*x^6 + 274*pi^2*b*c^4*x^4 + 224*pi^2*b 
*c^2*x^2 + 63*pi^2*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 480*sqrt(pi)*(pi^2*b* 
c^13*x^13 + pi^2*b*c^11*x^11)*log((pi + pi*c^2*x^6 + pi*c^2*x^2 + pi*x^4 + 
 sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 + 1)*(x^4 - 1))/(c^2*x^4 + x^ 
2)) + sqrt(pi + pi*c^2*x^2)*(960*pi^2*a*c^12*x^12 + 480*pi^2*a*c^10*x^10 - 
 120*pi^2*a*c^8*x^8 + 13920*pi^2*a*c^6*x^6 + 32880*pi^2*a*c^4*x^4 + 26880* 
pi^2*a*c^2*x^2 + 7560*pi^2*a - (240*pi^2*b*c^9*x^9 - 90*pi^2*b*c^7*x^7 - p 
i^2*(240*b*c^9 - 90*b*c^7 - 2260*b*c^5 - 2415*b*c^3 - 756*b*c)*x^11 - 2260 
*pi^2*b*c^5*x^5 - 2415*pi^2*b*c^3*x^3 - 756*pi^2*b*c*x)*sqrt(c^2*x^2 + 1)) 
)/(c^2*x^13 + x^11)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\text {Timed out} \] Input:

integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**12,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.09 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\frac {1}{83160} \, {\left (960 \, \pi ^{\frac {5}{2}} c^{10} \log \left (x\right ) + \frac {240 \, \pi ^{\frac {5}{2}} c^{8} x^{8} - 90 \, \pi ^{\frac {5}{2}} c^{6} x^{6} - 2260 \, \pi ^{\frac {5}{2}} c^{4} x^{4} - 2415 \, \pi ^{\frac {5}{2}} c^{2} x^{2} - 756 \, \pi ^{\frac {5}{2}}}{x^{10}}\right )} b c - \frac {1}{693} \, b {\left (\frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{4}}{\pi x^{7}} - \frac {28 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{2}}{\pi x^{9}} + \frac {63 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi x^{11}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{693} \, a {\left (\frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{4}}{\pi x^{7}} - \frac {28 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{2}}{\pi x^{9}} + \frac {63 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi x^{11}}\right )} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^12,x, algorithm="maxi 
ma")
 

Output:

1/83160*(960*pi^(5/2)*c^10*log(x) + (240*pi^(5/2)*c^8*x^8 - 90*pi^(5/2)*c^ 
6*x^6 - 2260*pi^(5/2)*c^4*x^4 - 2415*pi^(5/2)*c^2*x^2 - 756*pi^(5/2))/x^10 
)*b*c - 1/693*b*(8*(pi + pi*c^2*x^2)^(7/2)*c^4/(pi*x^7) - 28*(pi + pi*c^2* 
x^2)^(7/2)*c^2/(pi*x^9) + 63*(pi + pi*c^2*x^2)^(7/2)/(pi*x^11))*arcsinh(c* 
x) - 1/693*a*(8*(pi + pi*c^2*x^2)^(7/2)*c^4/(pi*x^7) - 28*(pi + pi*c^2*x^2 
)^(7/2)*c^2/(pi*x^9) + 63*(pi + pi*c^2*x^2)^(7/2)/(pi*x^11))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^12,x, algorithm="giac 
")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^{12}} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^12,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^12, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{12}} \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (-8 \sqrt {c^{2} x^{2}+1}\, a \,c^{10} x^{10}+4 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}-3 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}-113 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-161 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-63 \sqrt {c^{2} x^{2}+1}\, a +693 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{12}}d x \right ) b \,x^{11}+1386 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{10}}d x \right ) b \,c^{2} x^{11}+693 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{8}}d x \right ) b \,c^{4} x^{11}+8 a \,c^{11} x^{11}\right )}{693 x^{11}} \] Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x))/x^12,x)
 

Output:

(sqrt(pi)*pi**2*( - 8*sqrt(c**2*x**2 + 1)*a*c**10*x**10 + 4*sqrt(c**2*x**2 
 + 1)*a*c**8*x**8 - 3*sqrt(c**2*x**2 + 1)*a*c**6*x**6 - 113*sqrt(c**2*x**2 
 + 1)*a*c**4*x**4 - 161*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - 63*sqrt(c**2*x** 
2 + 1)*a + 693*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**12,x)*b*x**11 + 138 
6*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**10,x)*b*c**2*x**11 + 693*int((sq 
rt(c**2*x**2 + 1)*asinh(c*x))/x**8,x)*b*c**4*x**11 + 8*a*c**11*x**11))/(69 
3*x**11)