Integrand size = 26, antiderivative size = 89 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=-b c \sqrt {\pi } x+\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-2 \sqrt {\pi } (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-b \sqrt {\pi } \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+b \sqrt {\pi } \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
-b*c*x*Pi^(1/2)-2*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))*Pi^(1/ 2)-b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*Pi^(1/2)+b*polylog(2,c*x+(c^2*x^2+1 )^(1/2))*Pi^(1/2)+(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\sqrt {\pi } \left (a \sqrt {1+c^2 x^2}+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )\right ) \]
Sqrt[Pi]*(a*Sqrt[1 + c^2*x^2] + a*Log[x] - a*Log[Pi*(1 + Sqrt[1 + c^2*x^2] )] + b*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^( -ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^ (-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6221, 24, 6231, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x} \, dx\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\sqrt {\pi } b c \int 1dx+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\pi } \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \sqrt {\pi } \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle i \sqrt {\pi } \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle i \sqrt {\pi } \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle i \sqrt {\pi } \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\) |
-(b*c*Sqrt[Pi]*x) + Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + I*Sqrt[Pi ]*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]])
3.1.59.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.92
| method | result | size |
| default | \(a \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )+\sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \sqrt {\pi }\, b +\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -b c x \sqrt {\pi }+b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }-b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\) | \(171\) |
| parts | \(a \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )+\sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \sqrt {\pi }\, b +\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -b c x \sqrt {\pi }+b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }-b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\) | \(171\) |
a*((Pi*c^2*x^2+Pi)^(1/2)-Pi^(1/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))) +(c^2*x^2+1)^(1/2)*arcsinh(c*x)*Pi^(1/2)*b+arcsinh(c*x)*ln(1-c*x-(c^2*x^2+ 1)^(1/2))*Pi^(1/2)*b-arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*Pi^(1/2)*b-b *c*x*Pi^(1/2)+b*polylog(2,c*x+(c^2*x^2+1)^(1/2))*Pi^(1/2)-b*polylog(2,-c*x -(c^2*x^2+1)^(1/2))*Pi^(1/2)
\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \]
sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1)/x, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x, x))
\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
-(sqrt(pi)*arcsinh(1/(c*abs(x))) - sqrt(pi + pi*c^2*x^2))*a + b*integrate( sqrt(pi + pi*c^2*x^2)*log(c*x + sqrt(c^2*x^2 + 1))/x, x)
Exception generated. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x} \,d x \]