Integrand size = 26, antiderivative size = 108 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {1}{4} b c^3 \pi ^{3/2} x^2+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 c \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{4 b}+b c \pi ^{3/2} \log (x) \]
-1/4*b*c^3*Pi^(3/2)*x^2-(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/x+3/4*c*P i^(3/2)*(a+b*arcsinh(c*x))^2/b+b*c*Pi^(3/2)*ln(x)+3/2*c^2*Pi*x*(a+b*arcsin h(c*x))*(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\pi ^{3/2} \left (-8 a \sqrt {1+c^2 x^2}+4 a c^2 x^2 \sqrt {1+c^2 x^2}+6 b c x \text {arcsinh}(c x)^2-b c x \cosh (2 \text {arcsinh}(c x))+8 b c x \log (c x)+2 \text {arcsinh}(c x) \left (6 a c x-4 b \sqrt {1+c^2 x^2}+b c x \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 x} \]
(Pi^(3/2)*(-8*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 6*b*c* x*ArcSinh[c*x]^2 - b*c*x*Cosh[2*ArcSinh[c*x]] + 8*b*c*x*Log[c*x] + 2*ArcSi nh[c*x]*(6*a*c*x - 4*b*Sqrt[1 + c^2*x^2] + b*c*x*Sinh[2*ArcSinh[c*x]])))/( 8*x)
Time = 0.54 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6222, 244, 2009, 6200, 15, 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.
\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6222 |
\(\displaystyle 3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\pi ^{3/2} b c \int \frac {c^2 x^2+1}{x}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle 3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\pi ^{3/2} b c \int \left (x c^2+\frac {1}{x}\right )dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle 3 \pi c^2 \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} \sqrt {\pi } b c \int xdx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle 3 \pi c^2 \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{4} \sqrt {\pi } b c x^2\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle 3 \pi c^2 \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} \sqrt {\pi } b c x^2\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\) |
-(((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x) + 3*c^2*Pi*(-1/4*(b*c* Sqrt[Pi]*x^2) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[P i]*(a + b*ArcSinh[c*x])^2)/(4*b*c)) + b*c*Pi^(3/2)*((c^2*x^2)/2 + Log[x])
3.1.68.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 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]) /; FreeQ[{a, b, c, d, e , f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(92)=184\).
Time = 0.17 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.93
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right )}{8 x}\) | \(208\) |
parts | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right )}{8 x}\) | \(208\) |
-a/Pi/x*(Pi*c^2*x^2+Pi)^(5/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(3/2)+3/2*a*c^2*Pi*x *(Pi*c^2*x^2+Pi)^(1/2)+3/2*a*c^2*Pi^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x ^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/8*b*Pi^(3/2)*(4*arcsinh(c*x)*(c^2*x^2+1)^(1 /2)*x^2*c^2-2*c^3*x^3+6*arcsinh(c*x)^2*x*c-8*arcsinh(c*x)*c*x+8*ln((c*x+(c ^2*x^2+1)^(1/2))^2-1)*x*c-8*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-c*x)/x
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(pi*a*c^2*x^2 + pi*a + (pi*b*c^2*x^2 + pi*b )*arcsinh(c*x))/x^2, x)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\pi ^{\frac {3}{2}} \left (\int a c^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
pi**(3/2)*(Integral(a*c**2*sqrt(c**2*x**2 + 1), x) + Integral(a*sqrt(c**2* x**2 + 1)/x**2, x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**2, x))
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^2} \,d x \]