Integrand size = 26, antiderivative size = 113 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {b c \sqrt {\pi }}{20 x^4}-\frac {b c^3 \sqrt {\pi }}{30 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {2 c^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 \pi x^3}-\frac {2}{15} b c^5 \sqrt {\pi } \log (x) \] Output:
-1/20*b*c*Pi^(1/2)/x^4-1/30*b*c^3*Pi^(1/2)/x^2-1/5*(Pi*c^2*x^2+Pi)^(3/2)*( a+b*arcsinh(c*x))/Pi/x^5+2/15*c^2*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x)) /Pi/x^3-2/15*b*c^5*Pi^(1/2)*ln(x)
Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {\sqrt {\pi } \left (9 b c x+6 b c^3 x^3-50 b c^5 x^5+36 a \sqrt {1+c^2 x^2}+12 a c^2 x^2 \sqrt {1+c^2 x^2}-24 a c^4 x^4 \sqrt {1+c^2 x^2}-12 b \sqrt {1+c^2 x^2} \left (-3-c^2 x^2+2 c^4 x^4\right ) \text {arcsinh}(c x)+24 b c^5 x^5 \log (x)\right )}{180 x^5} \] Input:
Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^6,x]
Output:
-1/180*(Sqrt[Pi]*(9*b*c*x + 6*b*c^3*x^3 - 50*b*c^5*x^5 + 36*a*Sqrt[1 + c^2 *x^2] + 12*a*c^2*x^2*Sqrt[1 + c^2*x^2] - 24*a*c^4*x^4*Sqrt[1 + c^2*x^2] - 12*b*Sqrt[1 + c^2*x^2]*(-3 - c^2*x^2 + 2*c^4*x^4)*ArcSinh[c*x] + 24*b*c^5* x^5*Log[x]))/x^5
Time = 0.61 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 1433, 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.
| \(\displaystyle \int \frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x^6} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {-2 c^4 x^4+c^2 x^2+3}{15 x^5}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{15 \pi x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \sqrt {\pi } b c \int \frac {-2 c^4 x^4+c^2 x^2+3}{x^5}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{15 \pi x^3}\) |
| \(\Big \downarrow \) 1433 |
| \(\displaystyle \frac {1}{15} \sqrt {\pi } b c \int \left (-\frac {2 c^4}{x}+\frac {c^2}{x^3}+\frac {3}{x^5}\right )dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{15 \pi x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{15 \pi x^3}+\frac {1}{15} \sqrt {\pi } b c \left (-2 c^4 \log (x)-\frac {c^2}{2 x^2}-\frac {3}{4 x^4}\right )\) |
Input:
Int[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^6,x]
Output:
-1/5*((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(Pi*x^5) + (2*c^2*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(15*Pi*x^3) + (b*c*Sqrt[Pi]*(-3/ (4*x^4) - c^2/(2*x^2) - 2*c^4*Log[x]))/15
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
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])
Leaf count of result is larger than twice the leaf count of optimal. \(956\) vs. \(2(93)=186\).
Time = 1.07 (sec) , antiderivative size = 957, normalized size of antiderivative = 8.47
| method | result | size |
| default | \(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,x^{5}}+\frac {2 c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,x^{3}}\right )+\frac {4 b \sqrt {\pi }\, c^{5} \operatorname {arcsinh}\left (x c \right )}{15}-\frac {2 b \sqrt {\pi }\, x^{8} c^{13}}{15 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {2 b \sqrt {\pi }\, x^{6} \left (c^{2} x^{2}+1\right ) c^{11}}{15 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {2 b \sqrt {\pi }\, x^{6} \operatorname {arcsinh}\left (x c \right ) c^{11}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}+\frac {2 b \sqrt {\pi }\, x^{5} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{10}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}-\frac {b \sqrt {\pi }\, x^{6} c^{11}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {b \sqrt {\pi }\, x^{4} \left (c^{2} x^{2}+1\right ) c^{9}}{225 c^{6} x^{6}+75 c^{4} x^{4}-225 c^{2} x^{2}-135}-\frac {2 b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (x c \right ) c^{9}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{8}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {17 b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{7}}{30 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {2 b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (x c \right ) c^{7}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}-\frac {16 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{6}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {7 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{5}}{20 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {6 b \sqrt {\pi }\, \operatorname {arcsinh}\left (x c \right ) c^{5}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {6 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{4}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x}+\frac {3 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{2}}+\frac {18 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{3}}+\frac {9 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{20 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{4}}+\frac {9 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{5}}-\frac {2 b \sqrt {\pi }\, c^{5} \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{15}\) | \(957\) |
| parts | \(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,x^{5}}+\frac {2 c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,x^{3}}\right )+\frac {4 b \sqrt {\pi }\, c^{5} \operatorname {arcsinh}\left (x c \right )}{15}-\frac {2 b \sqrt {\pi }\, x^{8} c^{13}}{15 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {2 b \sqrt {\pi }\, x^{6} \left (c^{2} x^{2}+1\right ) c^{11}}{15 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {2 b \sqrt {\pi }\, x^{6} \operatorname {arcsinh}\left (x c \right ) c^{11}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}+\frac {2 b \sqrt {\pi }\, x^{5} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{10}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}-\frac {b \sqrt {\pi }\, x^{6} c^{11}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {b \sqrt {\pi }\, x^{4} \left (c^{2} x^{2}+1\right ) c^{9}}{225 c^{6} x^{6}+75 c^{4} x^{4}-225 c^{2} x^{2}-135}-\frac {2 b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (x c \right ) c^{9}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{8}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {17 b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{7}}{30 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {2 b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (x c \right ) c^{7}}{15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9}-\frac {16 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{6}}{3 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {7 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{5}}{20 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}+\frac {6 b \sqrt {\pi }\, \operatorname {arcsinh}\left (x c \right ) c^{5}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right )}-\frac {6 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{4}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x}+\frac {3 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{2}}+\frac {18 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{3}}+\frac {9 b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{20 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{4}}+\frac {9 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )}{5 \left (15 c^{6} x^{6}+5 c^{4} x^{4}-15 c^{2} x^{2}-9\right ) x^{5}}-\frac {2 b \sqrt {\pi }\, c^{5} \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{15}\) | \(957\) |
Input:
int((Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(x*c))/x^6,x,method=_RETURNVERBOSE)
Output:
a*(-1/5/Pi/x^5*(Pi*c^2*x^2+Pi)^(3/2)+2/15/Pi*c^2/x^3*(Pi*c^2*x^2+Pi)^(3/2) )+4/15*b*Pi^(1/2)*c^5*arcsinh(x*c)-2/15*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-1 5*c^2*x^2-9)*x^8*c^13+2/15*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)* x^6*(c^2*x^2+1)*c^11-2*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)*x^6* arcsinh(x*c)*c^11+2*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)*x^5*(c^ 2*x^2+1)^(1/2)*arcsinh(x*c)*c^10-1/5*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c ^2*x^2-9)*x^6*c^11+1/15*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)*x^4 *(c^2*x^2+1)*c^9-2/3*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)*x^4*ar csinh(x*c)*c^9-1/3*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)*x^3*(c^2 *x^2+1)^(1/2)*arcsinh(x*c)*c^8-17/30*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c ^2*x^2-9)*x^2*(c^2*x^2+1)*c^7+2*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^ 2-9)*x^2*arcsinh(x*c)*c^7-16/3*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x^2 -9)*x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^6-7/20*b*Pi^(1/2)/(15*c^6*x^6+5*c^4 *x^4-15*c^2*x^2-9)*(c^2*x^2+1)*c^5+6/5*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15 *c^2*x^2-9)*arcsinh(x*c)*c^5-6/5*b*Pi^(1/2)/(15*c^6*x^6+5*c^4*x^4-15*c^2*x ^2-9)/x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^4+3/5*b*Pi^(1/2)/(15*c^6*x^6+5*c^ 4*x^4-15*c^2*x^2-9)/x^2*(c^2*x^2+1)*c^3+18/5*b*Pi^(1/2)/(15*c^6*x^6+5*c^4* x^4-15*c^2*x^2-9)/x^3*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^2+9/20*b*Pi^(1/2)/( 15*c^6*x^6+5*c^4*x^4-15*c^2*x^2-9)/x^4*(c^2*x^2+1)*c+9/5*b*Pi^(1/2)/(15*c^ 6*x^6+5*c^4*x^4-15*c^2*x^2-9)/x^5*(c^2*x^2+1)^(1/2)*arcsinh(x*c)-2/15*b...
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (93) = 186\).
Time = 0.12 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {4 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, b c^{6} x^{6} + b c^{4} x^{4} - 4 \, b c^{2} x^{2} - 3 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 4 \, \sqrt {\pi } {\left (b c^{7} x^{7} + b c^{5} x^{5}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} - \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (8 \, a c^{6} x^{6} + 4 \, a c^{4} x^{4} - 16 \, a c^{2} x^{2} - {\left (2 \, b c^{3} x^{3} - {\left (2 \, b c^{3} + 3 \, b c\right )} x^{5} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 12 \, a\right )}}{60 \, {\left (c^{2} x^{7} + x^{5}\right )}} \] Input:
integrate((pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="frica s")
Output:
1/60*(4*sqrt(pi + pi*c^2*x^2)*(2*b*c^6*x^6 + b*c^4*x^4 - 4*b*c^2*x^2 - 3*b )*log(c*x + sqrt(c^2*x^2 + 1)) + 4*sqrt(pi)*(b*c^7*x^7 + b*c^5*x^5)*log((p i + 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)*(8*a*c^6 *x^6 + 4*a*c^4*x^4 - 16*a*c^2*x^2 - (2*b*c^3*x^3 - (2*b*c^3 + 3*b*c)*x^5 + 3*b*c*x)*sqrt(c^2*x^2 + 1) - 12*a))/(c^2*x^7 + x^5)
\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{6}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{6}}\, dx\right ) \] Input:
integrate((pi*c**2*x**2+pi)**(1/2)*(a+b*asinh(c*x))/x**6,x)
Output:
sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1)/x**6, x) + Integral(b*sqrt(c**2*x **2 + 1)*asinh(c*x)/x**6, x))
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {1}{60} \, {\left (8 \, \sqrt {\pi } c^{4} \log \left (x\right ) + \frac {2 \, \sqrt {\pi } c^{2} x^{2} + 3 \, \sqrt {\pi }}{x^{4}}\right )} b c + \frac {1}{15} \, b {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}}{\pi x^{3}} - \frac {3 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi x^{5}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}}{\pi x^{3}} - \frac {3 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi x^{5}}\right )} \] Input:
integrate((pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="maxim a")
Output:
-1/60*(8*sqrt(pi)*c^4*log(x) + (2*sqrt(pi)*c^2*x^2 + 3*sqrt(pi))/x^4)*b*c + 1/15*b*(2*(pi + pi*c^2*x^2)^(3/2)*c^2/(pi*x^3) - 3*(pi + pi*c^2*x^2)^(3/ 2)/(pi*x^5))*arcsinh(c*x) + 1/15*a*(2*(pi + pi*c^2*x^2)^(3/2)*c^2/(pi*x^3) - 3*(pi + pi*c^2*x^2)^(3/2)/(pi*x^5))
Exception generated. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x))/x^6,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
Timed out. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^6} \,d x \] Input:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2))/x^6,x)
Output:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2))/x^6, x)
\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {\sqrt {\pi }\, \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a +15 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{6}}d x \right ) b \,x^{5}-2 a \,c^{5} x^{5}\right )}{15 x^{5}} \] Input:
int((Pi*c^2*x^2+Pi)^(1/2)*(a+b*asinh(c*x))/x^6,x)
Output:
(sqrt(pi)*(2*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - sqrt(c**2*x**2 + 1)*a*c**2* x**2 - 3*sqrt(c**2*x**2 + 1)*a + 15*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x **6,x)*b*x**5 - 2*a*c**5*x**5))/(15*x**5)