Integrand size = 26, antiderivative size = 270 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c d \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \]
-1/2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x^2+3/2*c^2*d*(a+b*arcsinh(c*x ))*(c^2*d*x^2+d)^(1/2)-1/2*b*c*d*(c^2*d*x^2+d)^(1/2)/x/(c^2*x^2+1)^(1/2)-b *c^3*d*x*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3*c^2*d*(a+b*arcsinh(c*x))* arctanh(c*x+(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/2*b *c^2*d*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^( 1/2)+3/2*b*c^2*d*polylog(2,c*x+(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/(c^2 *x^2+1)^(1/2)
Time = 3.47 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=a \left (c^2 d-\frac {d}{2 x^2}\right ) \sqrt {d+c^2 d x^2}+\frac {3}{2} a c^2 d^{3/2} \log (x)-\frac {3}{2} a c^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 d \sqrt {d+c^2 d x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^2 d \sqrt {d+c^2 d x^2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{8 \sqrt {1+c^2 x^2}} \]
a*(c^2*d - d/(2*x^2))*Sqrt[d + c^2*d*x^2] + (3*a*c^2*d^(3/2)*Log[x])/2 - ( 3*a*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/2 + (b*c^2*d*Sqrt[d + c^2*d*x^2]*(-(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])]))/Sqrt[1 + c^2*x^2] + (b*c^2*d*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Cs ch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSi nh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4* PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Ta nh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.81, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6222, 244, 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.
\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\) |
\(\Big \downarrow \) 6222 |
\(\displaystyle \frac {3}{2} c^2 d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{x}dx+\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {c^2 x^2+1}{x^2}dx}{2 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {3}{2} c^2 d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{x}dx+\frac {b c d \sqrt {c^2 d x^2+d} \int \left (c^2+\frac {1}{x^2}\right )dx}{2 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} c^2 d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6221 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int 1dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6231 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {i \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {i \sqrt {c^2 d x^2+d} \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {i \sqrt {c^2 d x^2+d} \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {3}{2} c^2 d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {b c d \left (c^2 x-\frac {1}{x}\right ) \sqrt {c^2 d x^2+d}}{2 \sqrt {c^2 x^2+1}}\) |
(b*c*d*(-x^(-1) + c^2*x)*Sqrt[d + c^2*d*x^2])/(2*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*x^2) + (3*c^2*d*(-((b*c*x*Sqrt [d + c^2*d*x^2])/Sqrt[1 + c^2*x^2]) + Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c *x]) + (I*Sqrt[d + c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSin h[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]] ))/Sqrt[1 + c^2*x^2]))/2
3.2.34.3.1 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[((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[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 + 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]
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 = 0.24 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07
method | result | size |
default | \(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-2 c^{3} x^{3}+3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(289\) |
parts | \(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-2 c^{3} x^{3}+3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(289\) |
a*(-1/2/d/x^2*(c^2*d*x^2+d)^(5/2)+3/2*c^2*(1/3*(c^2*d*x^2+d)^(3/2)+d*((c^2 *d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x))))+1/2*b *(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x^2*(2*arcsinh(c*x)*(c^2*x^2+1)^( 1/2)*x^2*c^2+3*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-3*arcsinh( c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*x^2*c^2-2*c^3*x^3+3*polylog(2,c*x+(c^2*x^ 2+1)^(1/2))*x^2*c^2-3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*x^2*c^2-arcsinh(c* x)*(c^2*x^2+1)^(1/2)-c*x)*d
\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
-1/2*(3*c^2*d^(3/2)*arcsinh(1/(c*abs(x))) - (c^2*d*x^2 + d)^(3/2)*c^2 - 3* sqrt(c^2*d*x^2 + d)*c^2*d + (c^2*d*x^2 + d)^(5/2)/(d*x^2))*a + b*integrate ((c^2*d*x^2 + d)^(3/2)*log(c*x + sqrt(c^2*x^2 + 1))/x^3, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^3} \,d x \]