Integrand size = 26, antiderivative size = 137 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}+\frac {b \arctan (c x)}{c^6 \pi ^{3/2}} \]
5/3*b*x/c^5/Pi^(3/2)-1/9*b*x^3/c^3/Pi^(3/2)+1/3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b *arcsinh(c*x))/c^6/Pi^3+b*arctan(c*x)/c^6/Pi^(3/2)+(-a-b*arcsinh(c*x))/c^6 /Pi/(Pi*c^2*x^2+Pi)^(1/2)-2*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)/c^6/P i^2
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-24 a-12 a c^2 x^2+3 a c^4 x^4+15 b c x \sqrt {1+c^2 x^2}-b c^3 x^3 \sqrt {1+c^2 x^2}+3 b \left (-8-4 c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)+9 b \sqrt {1+c^2 x^2} \arctan (c x)}{9 c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}} \]
(-24*a - 12*a*c^2*x^2 + 3*a*c^4*x^4 + 15*b*c*x*Sqrt[1 + c^2*x^2] - b*c^3*x ^3*Sqrt[1 + c^2*x^2] + 3*b*(-8 - 4*c^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9*b*S qrt[1 + c^2*x^2]*ArcTan[c*x])/(9*c^6*Pi^(3/2)*Sqrt[1 + c^2*x^2])
Time = 0.46 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 1467, 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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {-c^4 x^4+4 c^2 x^2+8}{3 c^6 \pi ^2 \left (c^2 x^2+1\right )}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {-c^4 x^4+4 c^2 x^2+8}{c^2 x^2+1}dx}{3 \pi ^{3/2} c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {b \int \left (-c^2 x^2+\frac {3}{c^2 x^2+1}+5\right )dx}{3 \pi ^{3/2} c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \left (\frac {3 \arctan (c x)}{c}-\frac {1}{3} c^2 x^3+5 x\right )}{3 \pi ^{3/2} c^5}\) |
-((a + b*ArcSinh[c*x])/(c^6*Pi*Sqrt[Pi + c^2*Pi*x^2])) - (2*Sqrt[Pi + c^2* Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b *ArcSinh[c*x]))/(3*c^6*Pi^3) + (b*(5*x - (c^2*x^3)/3 + (3*ArcTan[c*x])/c)) /(3*c^5*Pi^(3/2))
3.1.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -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])
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.04
| method | result | size |
| default | \(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )-\frac {i b \left (3 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-i x^{5} c^{5}-12 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+14 i x^{3} c^{3}-9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}-24 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+15 i c x -9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 \pi ^{\frac {3}{2}} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(279\) |
| parts | \(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )-\frac {i b \left (3 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-i x^{5} c^{5}-12 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+14 i x^{3} c^{3}-9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}-24 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+15 i c x -9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 \pi ^{\frac {3}{2}} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(279\) |
a*(1/3*x^4/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)-4/3/c^2*(x^2/Pi/c^2/(Pi*c^2*x^2+Pi )^(1/2)+2/Pi/c^4/(Pi*c^2*x^2+Pi)^(1/2)))-1/9*I*b*(3*I*arcsinh(c*x)*(c^2*x^ 2+1)^(1/2)*x^4*c^4-I*x^5*c^5-12*I*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+1 4*I*x^3*c^3-9*ln(c*x+(c^2*x^2+1)^(1/2)+I)*x^2*c^2+9*ln(c*x+(c^2*x^2+1)^(1/ 2)-I)*x^2*c^2-24*I*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+15*I*c*x-9*ln(c*x+(c^2*x ^2+1)^(1/2)+I)+9*ln(c*x+(c^2*x^2+1)^(1/2)-I))/Pi^(3/2)/c^6/(c^2*x^2+1)
Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.43 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {9 \, \sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) - 6 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - 4 \, b c^{2} x^{2} - 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 15 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 24 \, a\right )}}{18 \, {\left (\pi ^{2} c^{8} x^{2} + \pi ^{2} c^{6}\right )}} \]
-1/18*(9*sqrt(pi)*(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)) - 6*sqrt(pi + pi*c^2*x^2)*(b*c^4 *x^4 - 4*b*c^2*x^2 - 8*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 2*sqrt(pi + pi*c^ 2*x^2)*(3*a*c^4*x^4 - 12*a*c^2*x^2 - (b*c^3*x^3 - 15*b*c*x)*sqrt(c^2*x^2 + 1) - 24*a))/(pi^2*c^8*x^2 + pi^2*c^6)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{5}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{5} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a*x**5/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x**5*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{5}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
1/3*a*(x^4/(pi*sqrt(pi + pi*c^2*x^2)*c^2) - 4*x^2/(pi*sqrt(pi + pi*c^2*x^2 )*c^4) - 8/(pi*sqrt(pi + pi*c^2*x^2)*c^6)) + 1/3*b*((sqrt(pi)*c^4*x^4 - 4* sqrt(pi)*c^2*x^2 - 8*sqrt(pi))*log(c*x + sqrt(c^2*x^2 + 1))/(pi^2*sqrt(c^2 *x^2 + 1)*c^6) - integrate((sqrt(pi)*c^4*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt (pi))/(sqrt(c^2*x^2 + 1)*x), x)/(pi^2*c^6) + 3*integrate(1/3*(sqrt(pi)*c^4 *x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt(pi))/(pi^2*c^9*x^4 + pi^2*c^7*x^2 + (pi ^2*c^8*x^3 + pi^2*c^6*x)*sqrt(c^2*x^2 + 1)), x))
Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]