Integrand size = 19, antiderivative size = 193 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}-\frac {b c \left (3 c^2 d+2 e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \]
1/4*(-a-b*arcsec(c*x))/e/(e*x^2+d)^2+1/4*b*c*x*arctan((c^2*x^2-1)^(1/2))/d ^2/e/(c^2*x^2)^(1/2)-1/8*b*c*(3*c^2*d+2*e)*x*arctan(e^(1/2)*(c^2*x^2-1)^(1 /2)/(c^2*d+e)^(1/2))/d^2/(c^2*d+e)^(3/2)/e^(1/2)/(c^2*x^2)^(1/2)-1/8*b*c*x *(c^2*x^2-1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)/(c^2*x^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.00 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}-\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} x}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \sec ^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {4 b \arcsin \left (\frac {1}{c x}\right )}{d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (-\frac {16 d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 i d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (-\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}\right ) \]
((-4*a)/(e*(d + e*x^2)^2) - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d + e) *(d + e*x^2)) - (4*b*ArcSec[c*x])/(e*(d + e*x^2)^2) - (4*b*ArcSin[1/(c*x)] )/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(-16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(Sq rt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b* (3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[ e]) - (b*(3*c^2*d + 2*e)*Log[((16*I)*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(-Sqrt [e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3 *c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e] ))/16
Time = 0.39 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5759, 354, 114, 27, 174, 73, 218}
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.
| \(\displaystyle \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5759 |
| \(\displaystyle \frac {b c x \int \frac {1}{x \sqrt {c^2 x^2-1} \left (e x^2+d\right )^2}dx}{4 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b c x \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^2}dx^2}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {b c x \left (-\frac {\int -\frac {-e x^2 c^2+2 d c^2+2 e}{2 x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx^2}{d \left (c^2 d+e\right )}-\frac {e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {\int \frac {2 \left (d c^2+e\right )-c^2 e x^2}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx^2}{2 d \left (c^2 d+e\right )}-\frac {e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {2 \left (c^2 d+e\right ) \int \frac {1}{x^2 \sqrt {c^2 x^2-1}}dx^2}{d}-\frac {e \left (3 c^2 d+2 e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx^2}{d}}{2 d \left (c^2 d+e\right )}-\frac {e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {4 \left (c^2 d+e\right ) \int \frac {1}{\frac {x^4}{c^2}+\frac {1}{c^2}}d\sqrt {c^2 x^2-1}}{c^2 d}-\frac {2 e \left (3 c^2 d+2 e\right ) \int \frac {1}{\frac {e x^4}{c^2}+d+\frac {e}{c^2}}d\sqrt {c^2 x^2-1}}{c^2 d}}{2 d \left (c^2 d+e\right )}-\frac {e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {4 \arctan \left (\sqrt {c^2 x^2-1}\right ) \left (c^2 d+e\right )}{d}-\frac {2 \sqrt {e} \left (3 c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{d \sqrt {c^2 d+e}}}{2 d \left (c^2 d+e\right )}-\frac {e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
-1/4*(a + b*ArcSec[c*x])/(e*(d + e*x^2)^2) + (b*c*x*(-((e*Sqrt[-1 + c^2*x^ 2])/(d*(c^2*d + e)*(d + e*x^2))) + ((4*(c^2*d + e)*ArcTan[Sqrt[-1 + c^2*x^ 2]])/d - (2*Sqrt[e]*(3*c^2*d + 2*e)*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sq rt[c^2*d + e]])/(d*Sqrt[c^2*d + e]))/(2*d*(c^2*d + e))))/(8*e*Sqrt[c^2*x^2 ])
3.2.6.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSec[c*x])/(2*e*(p + 1))), x ] - Simp[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2])) Int[(d + e*x^2)^(p + 1)/(x*S qrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(168)=336\).
Time = 5.55 (sec) , antiderivative size = 894, normalized size of antiderivative = 4.63
| method | result | size |
| parts | \(-\frac {a}{4 e \left (e \,x^{2}+d \right )^{2}}+\frac {b \left (-\frac {c^{6} \operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {c \sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e \right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(894\) |
| derivativedivides | \(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (c e x +\sqrt {-c^{2} d e}\right ) \left (-c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(899\) |
| default | \(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (c e x +\sqrt {-c^{2} d e}\right ) \left (-c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(899\) |
-1/4*a/e/(e*x^2+d)^2+b/c^2*(-1/4*c^6/e/(c^2*e*x^2+c^2*d)^2*arcsec(c*x)+1/1 6*c*(c^2*x^2-1)^(1/2)*(4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)* c^4*d*e*x^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^4*d^2-3*l n(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c *e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2-3*ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e) /e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2-3*l n(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e *x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2-3*ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e) ^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2+4*arcta n(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*e^2*c^2*x^2+4*arctan(1/(c^2*x^ 2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^2*d*e+2*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e )^(1/2)*c^2*d*e-2*ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d* e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2)))*e^2*c^2*x^2-2*ln(-2*((c^2*x^2-1 )^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^ (1/2)))*c^2*d*e-2*ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e )^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*e^2*c^2*x^2-2*ln(2*((c^2*x^2-1)^( 1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2 )))*c^2*d*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d^2/(-(c^2*d+e)/e)^(1/2)/(c^2*d +e)/(-c*e*x+(-c^2*d*e)^(1/2))/(c*e*x+(-c^2*d*e)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (165) = 330\).
Time = 0.49 (sec) , antiderivative size = 888, normalized size of antiderivative = 4.60 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d e - e^{2}} \log \left (\frac {c^{2} e x^{2} - c^{2} d + 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]
[-1/16*(4*a*c^4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^ 2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sq rt(-c^2*d*e - e^2)*log((c^2*e*x^2 - c^2*d + 2*sqrt(-c^2*d*e - e^2)*sqrt(c^ 2*x^2 - 1) - 2*e)/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2) *arcsec(c*x) - 8*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)* x^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c^2*d^3*e + b*d^2*e^2 + (b*c^ 2*d^2*e^2 + b*d*e^3)*x^2)*sqrt(c^2*x^2 - 1))/(c^4*d^6*e + 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2 *c^2*d^4*e^3 + d^3*e^4)*x^2), -1/8*(2*a*c^4*d^4 + 4*a*c^2*d^3*e + 2*a*d^2* e^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^ 2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(c^2*d*e + e^2)*arctan(sqrt(c^2*d*e + e^2)*s qrt(c^2*x^2 - 1)/(c^2*d + e)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)* arcsec(c*x) - 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x ^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (b*c^2*d^3*e + b*d^2*e^2 + (b*c^2*d ^2*e^2 + b*d*e^3)*x^2)*sqrt(c^2*x^2 - 1))/(c^4*d^6*e + 2*c^2*d^5*e^2 + d^4 *e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^ 2*d^4*e^3 + d^3*e^4)*x^2)]
Timed out. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
1/4*(4*(c^2*e^3*x^4 + 2*c^2*d*e^2*x^2 + c^2*d^2*e)*integrate(1/4*x*e^(1/2* log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*e^3*x^6 + (2*c^2*d*e^2 - e^3)*x^4 - d^2*e + (c^2*d^2*e - 2*d*e^2)*x^2 + (c^2*e^3*x^6 + (2*c^2*d*e^2 - e^3)*x^4 - d^2*e + (c^2*d^2*e - 2*d*e^2)*x^2)*e^(log(c*x + 1) + log(c*x - 1))), x) - arctan(sqrt(c*x + 1)*sqrt(c*x - 1)))*b/(e^3*x^4 + 2*d*e^2*x^2 + d^2*e) - 1/4*a/(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)
Exception generated. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]