Integrand size = 25, antiderivative size = 104 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}} \]
(a+b*arcsinh(c*x))^2/c/Pi^(3/2)-2*b*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+ 1)^(1/2))^2)/c/Pi^(3/2)-b^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/Pi^(3/ 2)+x*(a+b*arcsinh(c*x))^2/Pi/(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-b^2 \left (-c x+\sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2+2 b \text {arcsinh}(c x) \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )+a \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )\right )+b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2} \sqrt {1+c^2 x^2}} \]
(-(b^2*(-(c*x) + Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2) + 2*b*ArcSinh[c*x]*(a* c*x - b*Sqrt[1 + c^2*x^2]*Log[1 + E^(-2*ArcSinh[c*x])]) + a*(a*c*x - b*Sqr t[1 + c^2*x^2]*Log[1 + c^2*x^2]) + b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2 *ArcSinh[c*x])])/(c*Pi^(3/2)*Sqrt[1 + c^2*x^2])
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6202, 6212, 3042, 26, 4201, 2620, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{\pi ^{3/2}}\) |
| \(\Big \downarrow \) 6212 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 b \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{\pi ^{3/2} c}\) |
(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*Ar cSinh[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/(c*Pi^(3/2))
3.3.56.3.1 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[(((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_.)*(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]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(114)=228\).
Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.34
| method | result | size |
| default | \(\frac {a^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2}}{c \,\pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )\) | \(243\) |
| parts | \(\frac {a^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2}}{c \,\pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )\) | \(243\) |
a^2/Pi*x/(Pi*c^2*x^2+Pi)^(1/2)+b^2*(-1/Pi^(3/2)*(c^2*x^2-c*x*(c^2*x^2+1)^( 1/2)+1)*arcsinh(c*x)^2/c/(c^2*x^2+1)+2/c/Pi^(3/2)*arcsinh(c*x)^2-2/c/Pi^(3 /2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-1/c/Pi^(3/2)*polylog(2,-( c*x+(c^2*x^2+1)^(1/2))^2))+2*a*b*(2/c/Pi^(3/2)*arcsinh(c*x)-1/Pi^(3/2)*(c^ 2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*arcsinh(c*x)/c/(c^2*x^2+1)-1/c/Pi^(3/2)*ln( 1+(c*x+(c^2*x^2+1)^(1/2))^2))
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(pi^2*c^4*x^4 + 2*pi^2*c^2*x^2 + pi^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a**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**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x* *2 + 1)), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x))/pi**(3/2)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
b^2*integrate(log(c*x + sqrt(c^2*x^2 + 1))^2/(pi + pi*c^2*x^2)^(3/2), x) + 2*a*b*x*arcsinh(c*x)/(pi*sqrt(pi + pi*c^2*x^2)) + a^2*x/(pi*sqrt(pi + pi* c^2*x^2)) - a*b*log(x^2 + 1/c^2)/(pi^(3/2)*c)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]