Integrand size = 25, antiderivative size = 204 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c \pi ^{5/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c \pi ^{5/2}}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c \pi ^{5/2}} \]
1/3*b*(a+b*arcsinh(c*x))/c/Pi^(5/2)/(c^2*x^2+1)+2/3*(a+b*arcsinh(c*x))^2/c /Pi^(5/2)+1/3*x*(a+b*arcsinh(c*x))^2/Pi/(Pi*c^2*x^2+Pi)^(3/2)-4/3*b*(a+b*a rcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/Pi^(5/2)-2/3*b^2*polylog(2, -(c*x+(c^2*x^2+1)^(1/2))^2)/c/Pi^(5/2)-1/3*b^2*x/Pi^(5/2)/(c^2*x^2+1)^(1/2 )+2/3*x*(a+b*arcsinh(c*x))^2/Pi^2/(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.80 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {3 a^2 c x-b^2 c x+2 a^2 c^3 x^3-b^2 c^3 x^3+a b \sqrt {1+c^2 x^2}-b^2 \left (-3 c x-2 c^3 x^3+2 \sqrt {1+c^2 x^2}+2 c^2 x^2 \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-b \text {arcsinh}(c x) \left (-6 a c x-4 a c^3 x^3-b \sqrt {1+c^2 x^2}+4 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )-2 a b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )-2 a b c^2 x^2 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+2 b^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )}{3 c \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \]
(3*a^2*c*x - b^2*c*x + 2*a^2*c^3*x^3 - b^2*c^3*x^3 + a*b*Sqrt[1 + c^2*x^2] - b^2*(-3*c*x - 2*c^3*x^3 + 2*Sqrt[1 + c^2*x^2] + 2*c^2*x^2*Sqrt[1 + c^2* x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(-6*a*c*x - 4*a*c^3*x^3 - b*Sqrt[1 + c^2*x^2] + 4*b*(1 + c^2*x^2)^(3/2)*Log[1 + E^(-2*ArcSinh[c*x])]) - 2*a*b* Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] - 2*a*b*c^2*x^2*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + 2*b^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])])/ (3*c*Pi^(5/2)*(1 + c^2*x^2)^(3/2))
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {6203, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6213, 208}
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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{\pi ^{3/2}}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 6212 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }-\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {2 b c \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {2 b c \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 \pi ^{5/2}}\) |
(x*(a + b*ArcSinh[c*x])^2)/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (2*b*c*((b*x)/ (2*c*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(2*c^2*(1 + c^2*x^2))))/(3* Pi^(5/2)) + (2*((x*(a + b*ArcSinh[c*x])^2)/(Pi*Sqrt[Pi + c^2*Pi*x^2]) + (( 2*I)*b*(((-1/2*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)*(((a + b*ArcSinh[c*x]) *Log[1 + E^(2*ArcSinh[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4))) /(c*Pi^(3/2))))/(3*Pi)
3.3.57.3.1 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)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp [b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSinh[ c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x] ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1728\) vs. \(2(192)=384\).
Time = 0.26 (sec) , antiderivative size = 1729, normalized size of antiderivative = 8.48
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1729\) |
| parts | \(\text {Expression too large to display}\) | \(1729\) |
4*a*b/Pi^(5/2)*c^4/(3*c^2*x^2+4)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)*x^5-4*a*b/ Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^6+4/3*a*b/Pi^(5/2) /c/(3*c^2*x^2+4)/(c^2*x^2+1)^2-3*b^2/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)* arcsinh(c*x)*x^2-4/3*b^2/Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1)*arcsinh(c* x)*x^6+2*b^2/Pi^(5/2)*c^4/(3*c^2*x^2+4)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)^2*x ^5-22/3*b^2/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)^2*x^2+16/3 *b^2/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^2+4/3*b^2/Pi^(5 /2)*c^7/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^8-2*b^2/Pi^(5/2)*c^5/(3 *c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)^2*x^6+16/3*b^2/Pi^(5/2)*c^5/(3*c^2* x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^6-20/3*b^2/Pi^(5/2)*c^3/(3*c^2*x^2+4)/ (c^2*x^2+1)^2*arcsinh(c*x)^2*x^4+8*b^2/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^2*x^2 +1)^2*arcsinh(c*x)*x^4-4*b^2/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^2*x^2+1)*arcsin h(c*x)*x^4+17/3*b^2/Pi^(5/2)*c^2/(3*c^2*x^2+4)/(c^2*x^2+1)^(3/2)*arcsinh(c *x)^2*x^3+6*b^2/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^2*x^2+1)^2*x^4+a^2*(1/3/Pi*x /(Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(Pi*c^2*x^2+Pi)^(1/2))+8/3*a*b/c/Pi^(5/2 )*arcsinh(c*x)-4/3*a*b/c/Pi^(5/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-2/3*b^2* polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/Pi^(5/2)-4/3*b^2/Pi^(5/2)/(3*c^2*x ^2+4)/(c^2*x^2+1)^(3/2)*x-4/3*b^2/c/Pi^(5/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x ^2+1)^(1/2))^2)+4/3*b^2/Pi^(5/2)/c/(3*c^2*x^2+4)/(c^2*x^2+1)^2+4/3*b^2/c/P i^(5/2)*arcsinh(c*x)^2+34/3*a*b/Pi^(5/2)*c^2/(3*c^2*x^2+4)/(c^2*x^2+1)^...
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(pi^3*c^6*x^6 + 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 + pi^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
(Integral(a**2/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b**2*asinh(c*x)**2/(c**4*x**4* sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1 )), x) + Integral(2*a*b*asinh(c*x)/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2 *x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x))/pi**(5/2)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*b*c*(1/(pi^(5/2)*c^4*x^2 + pi^(5/2)*c^2) - 2*log(c^2*x^2 + 1)/(pi^(5 /2)*c^2)) + 2/3*a*b*(x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 2*x/(pi^2*sqrt(pi + pi*c^2*x^2)))*arcsinh(c*x) + 1/3*a^2*(x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 2*x /(pi^2*sqrt(pi + pi*c^2*x^2))) + b^2*integrate(log(c*x + sqrt(c^2*x^2 + 1) )^2/(pi + pi*c^2*x^2)^(5/2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]