Integrand size = 26, antiderivative size = 153 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {b c}{6 \pi ^{3/2} x^2}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi +c^2 \pi x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi +c^2 \pi x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac {b c^3 \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}} \]
-1/6*b*c/Pi^(3/2)/x^2-5/3*b*c^3*ln(x)/Pi^(3/2)-1/2*b*c^3*ln(c^2*x^2+1)/Pi^ (3/2)+1/3*(-a-b*arcsinh(c*x))/Pi/x^3/(Pi*c^2*x^2+Pi)^(1/2)+4/3*c^2*(a+b*ar csinh(c*x))/Pi/x/(Pi*c^2*x^2+Pi)^(1/2)+8/3*c^4*x*(a+b*arcsinh(c*x))/Pi/(Pi *c^2*x^2+Pi)^(1/2)
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-2 a+8 a c^2 x^2+16 a c^4 x^4-b c x \sqrt {1+c^2 x^2}-16 b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-1+4 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)-10 b c^3 x^3 \sqrt {1+c^2 x^2} \log (x)-3 b c^3 x^3 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}} \]
(-2*a + 8*a*c^2*x^2 + 16*a*c^4*x^4 - b*c*x*Sqrt[1 + c^2*x^2] - 16*b*c^3*x^ 3*Sqrt[1 + c^2*x^2] + 2*b*(-1 + 4*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] - 10*b *c^3*x^3*Sqrt[1 + c^2*x^2]*Log[x] - 3*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(6*Pi^(3/2)*x^3*Sqrt[1 + c^2*x^2])
Time = 0.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6219, 27, 1578, 1195, 2009}
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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {-8 c^4 x^4-4 c^2 x^2+1}{3 \pi ^2 x^3 \left (c^2 x^2+1\right )}dx+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {-8 c^4 x^4-4 c^2 x^2+1}{x^3 \left (c^2 x^2+1\right )}dx}{3 \pi ^{3/2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c \int \frac {-8 c^4 x^4-4 c^2 x^2+1}{x^4 \left (c^2 x^2+1\right )}dx^2}{6 \pi ^{3/2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c \int \left (-\frac {3 c^4}{c^2 x^2+1}-\frac {5 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 \pi ^{3/2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi c^2 x^2+\pi }}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \text {arcsinh}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 \pi \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \left (-5 c^2 \log \left (x^2\right )-3 c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )}{6 \pi ^{3/2}}\) |
-1/3*(a + b*ArcSinh[c*x])/(Pi*x^3*Sqrt[Pi + c^2*Pi*x^2]) + (4*c^2*(a + b*A rcSinh[c*x]))/(3*Pi*x*Sqrt[Pi + c^2*Pi*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x ]))/(3*Pi*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*(-x^(-2) - 5*c^2*Log[x^2] - 3*c^2* Log[1 + c^2*x^2]))/(6*Pi^(3/2))
3.1.99.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(132)=264\).
Time = 0.17 (sec) , antiderivative size = 604, normalized size of antiderivative = 3.95
| method | result | size |
| default | \(a \left (-\frac {1}{3 \pi \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 c^{2} \left (-\frac {1}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3}\right )+\frac {16 b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {3}{2}}}-\frac {32 b \,x^{8} c^{11}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {56 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {8 b x \,\operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {4 b \,c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}+\frac {8 b \,\operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {4 b \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x \sqrt {c^{2} x^{2}+1}}+\frac {b c}{6 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{3} \sqrt {c^{2} x^{2}+1}}-\frac {5 b \,c^{3} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3 \pi ^{\frac {3}{2}}}-\frac {b \,c^{3} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}}}\) | \(604\) |
| parts | \(a \left (-\frac {1}{3 \pi \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 c^{2} \left (-\frac {1}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3}\right )+\frac {16 b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {3}{2}}}-\frac {32 b \,x^{8} c^{11}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {56 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {8 b x \,\operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {4 b \,c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}+\frac {8 b \,\operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {4 b \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x \sqrt {c^{2} x^{2}+1}}+\frac {b c}{6 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{3} \sqrt {c^{2} x^{2}+1}}-\frac {5 b \,c^{3} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3 \pi ^{\frac {3}{2}}}-\frac {b \,c^{3} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}}}\) | \(604\) |
a*(-1/3/Pi/x^3/(Pi*c^2*x^2+Pi)^(1/2)-4/3*c^2*(-1/Pi/x/(Pi*c^2*x^2+Pi)^(1/2 )-2/Pi*c^2*x/(Pi*c^2*x^2+Pi)^(1/2)))+16/3*b*c^3/Pi^(3/2)*arcsinh(c*x)-32/3 *b/Pi^(3/2)/(8*c^2*x^2-1)*x^8/(c^2*x^2+1)*c^11+32/3*b/Pi^(3/2)/(8*c^2*x^2- 1)*x^6*c^9-64/3*b/Pi^(3/2)/(8*c^2*x^2-1)*x^6/(c^2*x^2+1)*c^9+32/3*b/Pi^(3/ 2)/(8*c^2*x^2-1)*x^4*c^7-64/3*b/Pi^(3/2)/(8*c^2*x^2-1)*x^4/(c^2*x^2+1)*arc sinh(c*x)*c^7+64/3*b/Pi^(3/2)/(8*c^2*x^2-1)*x^3/(c^2*x^2+1)^(1/2)*arcsinh( c*x)*c^6-32/3*b/Pi^(3/2)/(8*c^2*x^2-1)*x^4/(c^2*x^2+1)*c^7-56/3*b/Pi^(3/2) /(8*c^2*x^2-1)*x^2/(c^2*x^2+1)*arcsinh(c*x)*c^5+8*b/Pi^(3/2)/(8*c^2*x^2-1) *x/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^4-4/3*b/Pi^(3/2)/(8*c^2*x^2-1)*c^3+8/3 *b/Pi^(3/2)/(8*c^2*x^2-1)/(c^2*x^2+1)*arcsinh(c*x)*c^3-4*b/Pi^(3/2)/(8*c^2 *x^2-1)/x/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^2+1/6*b/Pi^(3/2)/(8*c^2*x^2-1)/ x^2*c+1/3*b/Pi^(3/2)/(8*c^2*x^2-1)/x^3/(c^2*x^2+1)^(1/2)*arcsinh(c*x)-5/3* b*c^3/Pi^(3/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)-b*c^3/Pi^(3/2)*ln(1+(c*x+(c ^2*x^2+1)^(1/2))^2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^2*c^4*x^8 + 2*pi^2 *c^2*x^6 + pi^2*x^4), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a}{c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} + x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} + x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
(Integral(a/(c**2*x**6*sqrt(c**2*x**2 + 1) + x**4*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**2*x**6*sqrt(c**2*x**2 + 1) + x**4*sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
1/3*(8*c^4*x/(pi*sqrt(pi + pi*c^2*x^2)) + 4*c^2/(pi*sqrt(pi + pi*c^2*x^2)* x) - 1/(pi*sqrt(pi + pi*c^2*x^2)*x^3))*a + b*integrate(log(c*x + sqrt(c^2* x^2 + 1))/((pi + pi*c^2*x^2)^(3/2)*x^4), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]