Integrand size = 26, antiderivative size = 93 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi +c^2 \pi x^2}}-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b c \log (x)}{\pi ^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}} \]
b*c*ln(x)/Pi^(3/2)+1/2*b*c*ln(c^2*x^2+1)/Pi^(3/2)+(-a-b*arcsinh(c*x))/Pi/x /(Pi*c^2*x^2+Pi)^(1/2)-2*c^2*x*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-2 a-4 a c^2 x^2-2 \left (b+2 b c^2 x^2\right ) \text {arcsinh}(c x)+2 b c x \sqrt {1+c^2 x^2} \log (x)+b c x \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2} x \sqrt {1+c^2 x^2}} \]
(-2*a - 4*a*c^2*x^2 - 2*(b + 2*b*c^2*x^2)*ArcSinh[c*x] + 2*b*c*x*Sqrt[1 + c^2*x^2]*Log[x] + b*c*x*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*Pi^(3/2)*x* Sqrt[1 + c^2*x^2])
Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6219, 25, 27, 354, 86, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {2 c^2 x^2+1}{\pi ^2 x \left (c^2 x^2+1\right )}dx-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\pi } b c \int \frac {2 c^2 x^2+1}{\pi ^2 x \left (c^2 x^2+1\right )}dx-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {2 c^2 x^2+1}{x \left (c^2 x^2+1\right )}dx}{\pi ^{3/2}}-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b c \int \frac {2 c^2 x^2+1}{x^2 \left (c^2 x^2+1\right )}dx^2}{2 \pi ^{3/2}}-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {b c \int \left (\frac {c^2}{c^2 x^2+1}+\frac {1}{x^2}\right )dx^2}{2 \pi ^{3/2}}-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )}{2 \pi ^{3/2}}\) |
-((a + b*ArcSinh[c*x])/(Pi*x*Sqrt[Pi + c^2*Pi*x^2])) - (2*c^2*x*(a + b*Arc Sinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*(Log[x^2] + Log[1 + c^2*x^2] ))/(2*Pi^(3/2))
3.1.97.3.1 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[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && 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. \(243\) vs. \(2(85)=170\).
Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.62
| method | result | size |
| default | \(a \left (-\frac {1}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )-\frac {b \left (2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{4} c^{4}-2 \sqrt {c^{2} x^{2}+1}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x c +\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}} x \left (c^{2} x^{2}+1\right )}\) | \(244\) |
| parts | \(a \left (-\frac {1}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )-\frac {b \left (2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{4} c^{4}-2 \sqrt {c^{2} x^{2}+1}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x c +\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}} x \left (c^{2} x^{2}+1\right )}\) | \(244\) |
a*(-1/Pi/x/(Pi*c^2*x^2+Pi)^(1/2)-2/Pi*c^2*x/(Pi*c^2*x^2+Pi)^(1/2))-b*(2*ln ((c*x+(c^2*x^2+1)^(1/2))^4-1)*x^4*c^4-2*(c^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2 +1)^(1/2))^4-1)*x^3*c^3+2*ln((c*x+(c^2*x^2+1)^(1/2))^4-1)*x^2*c^2-(c^2*x^2 +1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^4-1)*x*c+arcsinh(c*x))*(2*c^3*x^3+2*c ^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))/Pi^(3/2)/x/(c^2*x^2+1)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^2*c^4*x^6 + 2*pi^2 *c^2*x^4 + pi^2*x^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a/(c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {1}{2} \, b c {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {3}{2}}} + \frac {2 \, \log \left (x\right )}{\pi ^{\frac {3}{2}}}\right )} - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} b \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} a \]
1/2*b*c*(log(c^2*x^2 + 1)/pi^(3/2) + 2*log(x)/pi^(3/2)) - (2*c^2*x/(pi*sqr t(pi + pi*c^2*x^2)) + 1/(pi*sqrt(pi + pi*c^2*x^2)*x))*b*arcsinh(c*x) - (2* c^2*x/(pi*sqrt(pi + pi*c^2*x^2)) + 1/(pi*sqrt(pi + pi*c^2*x^2)*x))*a
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]