Integrand size = 23, antiderivative size = 631 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
(-a-b*arcsec(c*x))/d/x/(e*x^2+d)^(3/2)-4/3*e*x*(a+b*arcsec(c*x))/d^2/(e*x^ 2+d)^(3/2)-8/3*e*x*(a+b*arcsec(c*x))/d^3/(e*x^2+d)^(1/2)-b*c*e*(c^2*x^2-1) ^(1/2)/d^2/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)-4/3*b*c*e^2*x^2*(c^2* x^2-1)^(1/2)/d^3/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)+b*c*(c^2*d+2*e) *(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^3/(c^2*d+e)/(c^2*x^2)^(1/2)+4/3*b*c^2 *e*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^ 3/(c^2*d+e)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-b*c^2*(c^2 *d+2*e)*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/ 2)/d^3/(c^2*d+e)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)+b*c^2 *x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^ 2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)+8/3*b*e*x*EllipticF(c* x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^3/(c^2*x^2)^(1/ 2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 7.59 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\frac {-a \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (3 c^2 d \left (d+e x^2\right )+e \left (3 d+2 e x^2\right )\right )-b \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right ) \sec ^{-1}(c x)}{3 d^3 \left (c^2 d+e\right ) x \left (d+e x^2\right )^{3/2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (3 c^2 d+2 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (3 c^4 d^2+11 c^2 d e+8 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^3 \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(-(a*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)) + b*c*Sqrt[1 - 1/(c^2*x ^2)]*x*(d + e*x^2)*(3*c^2*d*(d + e*x^2) + e*(3*d + 2*e*x^2)) - b*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)*ArcSec[c*x])/(3*d^3*(c^2*d + e)*x*(d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c ^2*d*(3*c^2*d + 2*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (3 *c^4*d^2 + 11*c^2*d*e + 8*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2 *d))]))/(Sqrt[-c^2]*d^3*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 1.67 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5761, 27, 7293, 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.
| \(\displaystyle \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5761 |
| \(\displaystyle -\frac {b c x \int -\frac {8 e^2 x^4+12 d e x^2+3 d^2}{3 d^3 x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {c^2 x^2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {8 e^2 x^4+12 d e x^2+3 d^2}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b c x \int \left (\frac {3 d^2}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}+\frac {12 e d}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}+\frac {8 e^2 x^2}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}\right )dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}+\frac {b c x \left (\frac {8 e \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {3 c d \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {4 c e \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{\sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}-\frac {3 c \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{\sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}-\frac {4 e^2 x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {3 d e \sqrt {c^2 x^2-1}}{x \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {3 \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{x \left (c^2 d+e\right )}\right )}{3 d^3 \sqrt {c^2 x^2}}\) |
-((a + b*ArcSec[c*x])/(d*x*(d + e*x^2)^(3/2))) - (4*e*x*(a + b*ArcSec[c*x] ))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (b*c*x*((-3*d*e*Sqrt[-1 + c^2*x^2])/((c^2*d + e)*x*Sqrt[d + e*x ^2]) - (4*e^2*x*Sqrt[-1 + c^2*x^2])/((c^2*d + e)*Sqrt[d + e*x^2]) + (3*(c^ 2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/((c^2*d + e)*x) + (4*c*e*Sq rt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/((c^ 2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (3*c*(c^2*d + 2*e)*Sqrt [1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/((c^2* d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (3*c*d*Sqrt[1 - c^2*x^2]* Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(Sqrt[-1 + c^2*x ^2]*Sqrt[d + e*x^2]) + (8*e*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*Elliptic F[ArcSin[c*x], -(e/(c^2*d))])/(c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])))/(3* d^3*Sqrt[c^2*x^2])
3.2.60.3.1 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_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim p[(a + b*ArcSec[c*x]) u, x] - Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) | | (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, a c^{3} d^{4} + 3 \, a c d^{3} e + 8 \, {\left (a c^{3} d^{2} e^{2} + a c d e^{3}\right )} x^{4} + 12 \, {\left (a c^{3} d^{3} e + a c d^{2} e^{2}\right )} x^{2} + {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + 8 \, {\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{4} + 12 \, {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + {\left (3 \, b c^{3} d^{2} e^{2} + 2 \, b c d e^{3}\right )} x^{4} + {\left (6 \, b c^{3} d^{3} e + 5 \, b c d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left ({\left (3 \, b c^{6} d^{2} e^{2} + 2 \, b c^{4} d e^{3}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + 2 \, b c^{4} d^{2} e^{2}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + 2 \, b c^{4} d^{3} e\right )} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (3 \, b c^{6} d^{2} e^{2} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{2} e^{2} + 8 \, b d e^{3}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{3} e + 8 \, b d^{2} e^{2}\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left ({\left (c^{3} d^{5} e^{2} + c d^{4} e^{3}\right )} x^{5} + 2 \, {\left (c^{3} d^{6} e + c d^{5} e^{2}\right )} x^{3} + {\left (c^{3} d^{7} + c d^{6} e\right )} x\right )}} \]
-1/3*((3*a*c^3*d^4 + 3*a*c*d^3*e + 8*(a*c^3*d^2*e^2 + a*c*d*e^3)*x^4 + 12* (a*c^3*d^3*e + a*c*d^2*e^2)*x^2 + (3*b*c^3*d^4 + 3*b*c*d^3*e + 8*(b*c^3*d^ 2*e^2 + b*c*d*e^3)*x^4 + 12*(b*c^3*d^3*e + b*c*d^2*e^2)*x^2)*arcsec(c*x) - (3*b*c^3*d^4 + 3*b*c*d^3*e + (3*b*c^3*d^2*e^2 + 2*b*c*d*e^3)*x^4 + (6*b*c ^3*d^3*e + 5*b*c*d^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) - (((3*b *c^6*d^2*e^2 + 2*b*c^4*d*e^3)*x^5 + 2*(3*b*c^6*d^3*e + 2*b*c^4*d^2*e^2)*x^ 3 + (3*b*c^6*d^4 + 2*b*c^4*d^3*e)*x)*elliptic_e(arcsin(c*x), -e/(c^2*d)) - ((3*b*c^6*d^2*e^2 + (2*b*c^4 + 9*b*c^2)*d*e^3 + 8*b*e^4)*x^5 + 2*(3*b*c^6 *d^3*e + (2*b*c^4 + 9*b*c^2)*d^2*e^2 + 8*b*d*e^3)*x^3 + (3*b*c^6*d^4 + (2* b*c^4 + 9*b*c^2)*d^3*e + 8*b*d^2*e^2)*x)*elliptic_f(arcsin(c*x), -e/(c^2*d )))*sqrt(-d))/((c^3*d^5*e^2 + c*d^4*e^3)*x^5 + 2*(c^3*d^6*e + c*d^5*e^2)*x ^3 + (c^3*d^7 + c*d^6*e)*x)
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]