Integrand size = 26, antiderivative size = 134 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {4}{3} b c \pi ^{3/2} x-\frac {1}{9} b c^3 \pi ^{3/2} x^3+\pi \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-2 \pi ^{3/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-b \pi ^{3/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+b \pi ^{3/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
-4/3*b*c*Pi^(3/2)*x-1/9*b*c^3*Pi^(3/2)*x^3+1/3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b* arcsinh(c*x))-2*Pi^(3/2)*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2)) -b*Pi^(3/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+b*Pi^(3/2)*polylog(2,c*x+(c^ 2*x^2+1)^(1/2))+Pi*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.37 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.34 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{9} \pi ^{3/2} \left (3 a \sqrt {1+c^2 x^2} \left (4+c^2 x^2\right )-b \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )+9 a \log (x)-9 a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+9 b \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )\right ) \]
(Pi^(3/2)*(3*a*Sqrt[1 + c^2*x^2]*(4 + c^2*x^2) - b*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) + 9*a*Log[x] - 9*a*Log[Pi*(1 + Sqrt[1 + c^ 2*x^2])] + 9*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])] + PolyLo g[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])])))/9
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6223, 2009, 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.
| \(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \pi \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{3} \pi ^{3/2} b c \int \left (c^2 x^2+1\right )dx+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \pi \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \pi \left (\sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\sqrt {\pi } b c \int 1dx+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \pi \left (\sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \pi \left (\sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \pi \left (\sqrt {\pi } \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \pi \left (i \sqrt {\pi } \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \pi \left (i \sqrt {\pi } \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \pi \left (i \sqrt {\pi } \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \pi \left (i \sqrt {\pi } \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\sqrt {\pi } (-b) c x\right )+\frac {1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} \pi ^{3/2} b c \left (\frac {c^2 x^3}{3}+x\right )\) |
-1/3*(b*c*Pi^(3/2)*(x + (c^2*x^3)/3)) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*Ar cSinh[c*x]))/3 + Pi*(-(b*c*Sqrt[Pi]*x) + Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcS inh[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]]))
3.1.67.3.1 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_.)*((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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*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]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -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 = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(a \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {3}{2}} x^{2} c^{2}}{3}+\frac {4 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {3}{2}}}{3}-\frac {4 b c \,\pi ^{\frac {3}{2}} x}{3}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{3}}{9}+b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {3}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {3}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(228\) |
| parts | \(a \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {3}{2}} x^{2} c^{2}}{3}+\frac {4 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {3}{2}}}{3}-\frac {4 b c \,\pi ^{\frac {3}{2}} x}{3}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{3}}{9}+b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {3}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {3}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(228\) |
a*(1/3*(Pi*c^2*x^2+Pi)^(3/2)+Pi*((Pi*c^2*x^2+Pi)^(1/2)-Pi^(1/2)*arctanh(Pi ^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))))+1/3*b*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*Pi^(3 /2)*x^2*c^2+4/3*b*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*Pi^(3/2)-4/3*b*c*Pi^(3/2) *x-1/9*b*c^3*Pi^(3/2)*x^3+b*Pi^(3/2)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/ 2))-b*Pi^(3/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+b*Pi^(3/2)*polylog (2,c*x+(c^2*x^2+1)^(1/2))-b*Pi^(3/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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, x)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\pi ^{\frac {3}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int a c^{2} x \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x, x) + Integral(a*c**2*x*sqrt(c **2*x**2 + 1), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x, x) + Inte gral(b*c**2*x*sqrt(c**2*x**2 + 1)*asinh(c*x), x))
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
-1/3*(3*pi^(3/2)*arcsinh(1/(c*abs(x))) - 3*pi*sqrt(pi + pi*c^2*x^2) - (pi + pi*c^2*x^2)^(3/2))*a + b*integrate((pi + pi*c^2*x^2)^(3/2)*log(c*x + sqr t(c^2*x^2 + 1))/x, x)
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x} \,d x \]