Integrand size = 23, antiderivative size = 108 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {b \log \left (1+c^2 x^2\right )}{3 c \pi ^{5/2}} \]
1/6*b/c/Pi^(5/2)/(c^2*x^2+1)+1/3*x*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+Pi)^( 3/2)-1/3*b*ln(c^2*x^2+1)/c/Pi^(5/2)+2/3*x*(a+b*arcsinh(c*x))/Pi^2/(Pi*c^2* x^2+Pi)^(1/2)
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {6 a c x+4 a c^3 x^3+b \sqrt {1+c^2 x^2}+2 b c x \left (3+2 c^2 x^2\right ) \text {arcsinh}(c x)-2 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \]
(6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*c*x*(3 + 2*c^2*x^2)*Arc Sinh[c*x] - 2*b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(6*c*Pi^(5/2)*(1 + c ^2*x^2)^(3/2))
Time = 0.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6203, 241, 6202, 240}
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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle \frac {2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{3 \pi }-\frac {b c \int \frac {x}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle \frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b c \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2}}\right )}{3 \pi }+\frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c}\right )}{3 \pi }+\frac {b}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )}\) |
b/(6*c*Pi^(5/2)*(1 + c^2*x^2)) + (x*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2* Pi*x^2)^(3/2)) + (2*((x*(a + b*ArcSinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*Log[1 + c^2*x^2])/(2*c*Pi^(3/2))))/(3*Pi)
3.2.6.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(92)=184\).
Time = 0.20 (sec) , antiderivative size = 619, normalized size of antiderivative = 5.73
| method | result | size |
| default | \(a \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {4 b \,\operatorname {arcsinh}\left (c x \right )}{3 c \,\pi ^{\frac {5}{2}}}+\frac {2 b \,c^{7} x^{8}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {2 b \,c^{5} \operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b \,c^{4} \operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {20 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {17 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {4 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {22 b c \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {4 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b c \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c \,\pi ^{\frac {5}{2}}}\) | \(619\) |
| parts | \(a \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {4 b \,\operatorname {arcsinh}\left (c x \right )}{3 c \,\pi ^{\frac {5}{2}}}+\frac {2 b \,c^{7} x^{8}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {2 b \,c^{5} \operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b \,c^{4} \operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{5} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {20 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {17 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {4 b \,c^{3} x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )}-\frac {22 b c \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {4 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b c \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b}{3 \pi ^{\frac {5}{2}} c \left (3 c^{2} x^{2}+4\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {2 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c \,\pi ^{\frac {5}{2}}}\) | \(619\) |
a*(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))+4/3*b/ c/Pi^(5/2)*arcsinh(c*x)+2/3*b/Pi^(5/2)*c^7/(3*c^2*x^2+4)/(c^2*x^2+1)^2*x^8 -2/3*b/Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1)*x^6-2*b/Pi^(5/2)*c^5/(3*c^2* x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^6+2*b/Pi^(5/2)*c^4/(3*c^2*x^2+4)/(c^2* x^2+1)^(3/2)*arcsinh(c*x)*x^5+8/3*b/Pi^(5/2)*c^5/(3*c^2*x^2+4)/(c^2*x^2+1) ^2*x^6-2*b/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^2*x^2+1)*x^4-20/3*b/Pi^(5/2)*c^3/ (3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^4+17/3*b/Pi^(5/2)*c^2/(3*c^2*x^ 2+4)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)*x^3+4*b/Pi^(5/2)*c^3/(3*c^2*x^2+4)/(c^ 2*x^2+1)^2*x^4-3/2*b/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)*x^2-22/3*b/Pi^(5 /2)*c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)*x^2+4*b/Pi^(5/2)/(3*c^2*x^2 +4)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)*x+8/3*b/Pi^(5/2)*c/(3*c^2*x^2+4)/(c^2*x ^2+1)^2*x^2-8/3*b/Pi^(5/2)/c/(3*c^2*x^2+4)/(c^2*x^2+1)^2*arcsinh(c*x)+2/3* b/Pi^(5/2)/c/(3*c^2*x^2+4)/(c^2*x^2+1)^2-2/3*b/c/Pi^(5/2)*ln(1+(c*x+(c^2*x ^2+1)^(1/2))^2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^3*c^6*x^6 + 3*pi^3 *c^4*x^4 + 3*pi^3*c^2*x^2 + pi^3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
(Integral(a/(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*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))/p i**(5/2)
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {1}{\pi ^{\frac {5}{2}} c^{4} x^{2} + \pi ^{\frac {5}{2}} c^{2}} - \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {5}{2}} c^{2}}\right )} + \frac {1}{3} \, b {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} \]
1/6*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)) + 1/3*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*(x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 2*x/(pi^2 *sqrt(pi + pi*c^2*x^2)))
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]