Integrand size = 26, antiderivative size = 80 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]
1/2*(a+b*arcsinh(c*x))^2/b/c^3/Pi^(3/2)+1/2*b*ln(c^2*x^2+1)/c^3/Pi^(3/2)-x *(a+b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\frac {2 a c x}{\sqrt {1+c^2 x^2}}+\left (2 a-\frac {2 b c x}{\sqrt {1+c^2 x^2}}\right ) \text {arcsinh}(c x)+b \text {arcsinh}(c x)^2+b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]
((-2*a*c*x)/Sqrt[1 + c^2*x^2] + (2*a - (2*b*c*x)/Sqrt[1 + c^2*x^2])*ArcSin h[c*x] + b*ArcSinh[c*x]^2 + b*Log[1 + c^2*x^2])/(2*c^3*Pi^(3/2))
Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6225, 240, 6198}
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.
\(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6225 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}+\frac {b \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2} c}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{3/2} b c^3}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}\) |
-((x*(a + b*ArcSinh[c*x]))/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (a + b*ArcSin h[c*x])^2/(2*b*c^3*Pi^(3/2)) + (b*Log[1 + c^2*x^2])/(2*c^3*Pi^(3/2))
3.1.93.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^( m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(70)=140\).
Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10
method | result | size |
default | \(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\) | \(168\) |
parts | \(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\) | \(168\) |
-a*x/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)+a/Pi/c^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi* c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+b*(1/2/c^3/Pi^(3/2)*arcsinh(c*x)^2-2/c^3 /Pi^(3/2)*arcsinh(c*x)+1/Pi^(3/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*arcsin h(c*x)/c^3/(c^2*x^2+1)+1/c^3/Pi^(3/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2))
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*x^2*arcsinh(c*x) + a*x^2)/(pi^2*c^4*x^4 + 2*pi^2*c^2*x^2 + pi^2), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{2}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a*x**2/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x**2*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
-a*(x/(pi*sqrt(pi + pi*c^2*x^2)*c^2) - arcsinh(c*x)/(pi^(3/2)*c^3)) + b*in tegrate(x^2*log(c*x + sqrt(c^2*x^2 + 1))/(pi + pi*c^2*x^2)^(3/2), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]