Integrand size = 26, antiderivative size = 162 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {b c}{2 \pi ^{3/2} x}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b c^2 \arctan (c x)}{\pi ^{3/2}}+\frac {3 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{3/2}}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{3/2}}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{3/2}} \]
-1/2*b*c/Pi^(3/2)/x+b*c^2*arctan(c*x)/Pi^(3/2)+3*c^2*(a+b*arcsinh(c*x))*ar ctanh(c*x+(c^2*x^2+1)^(1/2))/Pi^(3/2)+3/2*b*c^2*polylog(2,-c*x-(c^2*x^2+1) ^(1/2))/Pi^(3/2)-3/2*b*c^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))/Pi^(3/2)-3/2*c ^2*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+Pi)^(1/2)+1/2*(-a-b*arcsinh(c*x))/Pi/ x^2/(Pi*c^2*x^2+Pi)^(1/2)
Time = 2.93 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.66 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\frac {8 a c^2}{\sqrt {1+c^2 x^2}}-\frac {4 a \sqrt {1+c^2 x^2}}{x^2}-\frac {8 b c^2 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+16 b c^2 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 b c^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-b c^2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-12 b c^2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+12 b c^2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-12 a c^2 \log (x)+12 a c^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )-12 b c^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+12 b c^2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-b c^2 \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 b c^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{8 \pi ^{3/2}} \]
((-8*a*c^2)/Sqrt[1 + c^2*x^2] - (4*a*Sqrt[1 + c^2*x^2])/x^2 - (8*b*c^2*Arc Sinh[c*x])/Sqrt[1 + c^2*x^2] + 16*b*c^2*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*b *c^2*Coth[ArcSinh[c*x]/2] - b*c^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 12 *b*c^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*b*c^2*ArcSinh[c*x]*Log [1 + E^(-ArcSinh[c*x])] - 12*a*c^2*Log[x] + 12*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] - 12*b*c^2*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*b*c^2*PolyLog[2, E^(-ArcSinh[c*x])] - b*c^2*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*b*c^2* Tanh[ArcSinh[c*x]/2])/(8*Pi^(3/2))
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6224, 264, 216, 6226, 216, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 \pi x^2+\pi \right )^{3/2}}dx+\frac {b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx}{2 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 \pi x^2+\pi \right )^{3/2}}dx+\frac {b c \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )}{2 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 \pi x^2+\pi \right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 \pi x^2+\pi }}dx}{\pi }-\frac {b c \int \frac {1}{c^2 x^2+1}dx}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 \pi x^2+\pi }}dx}{\pi }+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{\pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \arctan (c x)}{\pi ^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 \pi ^{3/2}}\) |
-1/2*(a + b*ArcSinh[c*x])/(Pi*x^2*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*(-x^(-1) - c*ArcTan[c*x]))/(2*Pi^(3/2)) - (3*c^2*((a + b*ArcSinh[c*x])/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*ArcTan[c*x])/Pi^(3/2) + (I*((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]]))/Pi^(3/2)))/2
3.1.98.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -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 + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ b*c*(n/(2*f*(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] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || 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 = 0.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(a \left (-\frac {1}{2 \pi \,x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {3 c^{2} \left (\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 \pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {2 c^{2} \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}\right )\) | \(224\) |
| parts | \(a \left (-\frac {1}{2 \pi \,x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {3 c^{2} \left (\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 \pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {2 c^{2} \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}+\frac {3 c^{2} \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {3}{2}}}\right )\) | \(224\) |
a*(-1/2/Pi/x^2/(Pi*c^2*x^2+Pi)^(1/2)-3/2*c^2*(1/Pi/(Pi*c^2*x^2+Pi)^(1/2)-1 /Pi^(3/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))))+b*(-1/2/Pi^(3/2)/(c^2* x^2+1)^(1/2)*(3*arcsinh(c*x)*c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x))/x ^2+2*c^2/Pi^(3/2)*arctan(c*x+(c^2*x^2+1)^(1/2))+3/2*c^2/Pi^(3/2)*dilog(1+c *x+(c^2*x^2+1)^(1/2))+3/2*c^2/Pi^(3/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^( 1/2))+3/2*c^2/Pi^(3/2)*dilog(c*x+(c^2*x^2+1)^(1/2)))
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^2*c^4*x^7 + 2*pi^2 *c^2*x^5 + pi^2*x^3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a}{c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a/(c**2*x**5*sqrt(c**2*x**2 + 1) + x**3*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**2*x**5*sqrt(c**2*x**2 + 1) + x**3*sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
1/2*(3*c^2*arcsinh(1/(c*abs(x)))/pi^(3/2) - 3*c^2/(pi*sqrt(pi + pi*c^2*x^2 )) - 1/(pi*sqrt(pi + pi*c^2*x^2)*x^2))*a + b*integrate(log(c*x + sqrt(c^2* x^2 + 1))/((pi + pi*c^2*x^2)^(3/2)*x^3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]