Integrand size = 23, antiderivative size = 362 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {b c^2 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
-1/3*(a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/d/x^3+2/3*e*(a+b*arcsec(c*x))*(e*x^ 2+d)^(1/2)/d^2/x+1/9*b*c*(2*c^2*d-5*e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d ^2/(c^2*x^2)^(1/2)+1/9*b*c*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(c^2*x^ 2)^(1/2)-1/9*b*c^2*(2*c^2*d-5*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^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2 /d)^(1/2)+2/9*b*(c^2*d-3*e)*(c^2*d+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^2/(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 = 6.52 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.69 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2-5 e x^2\right )-3 a \left (d-2 e x^2\right )-3 b \left (d-2 e x^2\right ) \sec ^{-1}(c x)\right )}{9 d^2 x^3}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (2 c^2 d-5 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (-c^4 d^2+2 c^2 d e+3 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(Sqrt[d + e*x^2]*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 - 5*e*x^2) - 3*a*(d - 2*e*x^2) - 3*b*(d - 2*e*x^2)*ArcSec[c*x]))/(9*d^2*x^3) - ((I/9) *b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2*c^2*d - 5*e)*El lipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(-(c^4*d^2) + 2*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d ^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 0.71 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {5761, 27, 442, 25, 445, 27, 399, 323, 323, 321, 331, 330, 327}
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.
| \(\displaystyle \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 5761 |
| \(\displaystyle -\frac {b c x \int -\frac {\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{3 d^2 x^4 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{x^4 \sqrt {c^2 x^2-1}}dx}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}-\frac {1}{3} \int -\frac {\left (c^2 d-6 e\right ) e x^2+d \left (2 c^2 d-5 e\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \int \frac {\left (c^2 d-6 e\right ) e x^2+d \left (2 c^2 d-5 e\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (-\left (\left (2 c^2 d-5 e\right ) x^2 c^2\right )+d c^2-6 e\right )}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \int \frac {-\left (\left (2 c^2 d-5 e\right ) x^2 c^2\right )+d c^2-6 e}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \left (2 c^2 d-5 e\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \left (2 c^2 d-5 e\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (2 c^2 d-5 e\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (2 c^2 d-5 e\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {b c x \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d^2 \sqrt {c^2 x^2}}\) |
-1/3*(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(d*x^3) + (2*e*Sqrt[d + e*x^2]* (a + b*ArcSec[c*x]))/(3*d^2*x) + (b*c*x*((d*Sqrt[-1 + c^2*x^2]*Sqrt[d + e* x^2])/(3*x^3) + (((2*c^2*d - 5*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/x + e*(-((c*(2*c^2*d - 5*e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin [c*x], -(e/(c^2*d))])/(e*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d])) + (2*(c^ 2*d - 3*e)*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[Arc Sin[c*x], -(e/(c^2*d))])/(c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])))/3))/(3 *d^2*Sqrt[c^2*x^2])
3.2.39.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]))
\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}d x\]
Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.54 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {{\left (6 \, a c d e x^{2} - 3 \, a c d^{2} + 3 \, {\left (2 \, b c d e x^{2} - b c d^{2}\right )} \operatorname {arcsec}\left (c x\right ) + {\left (b c d^{2} + {\left (2 \, b c^{3} d^{2} - 5 \, b c d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left ({\left (2 \, b c^{6} d^{2} - 5 \, b c^{4} d e\right )} x^{3} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (2 \, b c^{6} d^{2} - {\left (5 \, b c^{4} - b c^{2}\right )} d e - 6 \, b e^{2}\right )} x^{3} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{9 \, c d^{3} x^{3}} \]
1/9*((6*a*c*d*e*x^2 - 3*a*c*d^2 + 3*(2*b*c*d*e*x^2 - b*c*d^2)*arcsec(c*x) + (b*c*d^2 + (2*b*c^3*d^2 - 5*b*c*d*e)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) + ((2*b*c^6*d^2 - 5*b*c^4*d*e)*x^3*elliptic_e(arcsin(c*x), -e/(c^2*d) ) - (2*b*c^6*d^2 - (5*b*c^4 - b*c^2)*d*e - 6*b*e^2)*x^3*elliptic_f(arcsin( c*x), -e/(c^2*d)))*sqrt(-d))/(c*d^3*x^3)
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]