Integrand size = 21, antiderivative size = 186 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} i b d^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
-1/2*I*b*d^2*arccsc(c*x)^2+d*e*x^2*(a+b*arcsec(c*x))+1/4*e^2*x^4*(a+b*arcs ec(c*x))+b*d^2*arccsc(c*x)*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)-b*d^2*arccs c(c*x)*ln(1/x)-d^2*(a+b*arcsec(c*x))*ln(1/x)-1/2*I*b*d^2*polylog(2,(I/c/x+ (1-1/c^2/x^2)^(1/2))^2)-1/6*b*e*(6*c^2*d+e)*x*(1-1/c^2/x^2)^(1/2)/c^3-1/12 *b*e^2*x^3*(1-1/c^2/x^2)^(1/2)/c
Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=a d e x^2+\frac {1}{4} a e^2 x^4-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+c^2 x^2\right )}{12 c^3}+\frac {1}{4} b e^2 x^4 \sec ^{-1}(c x)+\frac {b d e x \left (-\sqrt {1-\frac {1}{c^2 x^2}}+c x \sec ^{-1}(c x)\right )}{c}+a d^2 \log (x)+\frac {1}{2} i b d^2 \left (\sec ^{-1}(c x) \left (\sec ^{-1}(c x)+2 i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right ) \]
a*d*e*x^2 + (a*e^2*x^4)/4 - (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + c^2*x^2))/ (12*c^3) + (b*e^2*x^4*ArcSec[c*x])/4 + (b*d*e*x*(-Sqrt[1 - 1/(c^2*x^2)] + c*x*ArcSec[c*x]))/c + a*d^2*Log[x] + (I/2)*b*d^2*(ArcSec[c*x]*(ArcSec[c*x] + (2*I)*Log[1 + E^((2*I)*ArcSec[c*x])]) + PolyLog[2, -E^((2*I)*ArcSec[c*x ])])
Time = 0.79 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5763, 5231, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx\) |
| \(\Big \downarrow \) 5763 |
| \(\displaystyle -\int \left (\frac {d}{x^2}+e\right )^2 x^5 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle -\frac {b \int -\frac {e \left (\frac {4 d}{x^2}+e\right ) x^4-4 d^2 \log \left (\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}}{c}-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+d e x^2 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {e \left (\frac {4 d}{x^2}+e\right ) x^4-4 d^2 \log \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}}{4 c}-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+d e x^2 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b \int \left (\frac {e \left (\frac {4 d}{x^2}+e\right ) x^4}{\sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 d^2 \log \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )d\frac {1}{x}}{4 c}-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+d e x^2 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+d e x^2 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \arccos \left (\frac {1}{c x}\right )\right )+\frac {b \left (-2 i c d^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )-2 i c d^2 \arcsin \left (\frac {1}{c x}\right )^2+4 c d^2 \arcsin \left (\frac {1}{c x}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )-4 c d^2 \log \left (\frac {1}{x}\right ) \arcsin \left (\frac {1}{c x}\right )-\frac {2}{3} e x \sqrt {1-\frac {1}{c^2 x^2}} \left (\frac {e}{c^2}+6 d\right )-\frac {1}{3} e^2 x^3 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c}\) |
d*e*x^2*(a + b*ArcCos[1/(c*x)]) + (e^2*x^4*(a + b*ArcCos[1/(c*x)]))/4 - d^ 2*(a + b*ArcCos[1/(c*x)])*Log[x^(-1)] + (b*((-2*e*(6*d + e/c^2)*Sqrt[1 - 1 /(c^2*x^2)]*x)/3 - (e^2*Sqrt[1 - 1/(c^2*x^2)]*x^3)/3 - (2*I)*c*d^2*ArcSin[ 1/(c*x)]^2 + 4*c*d^2*ArcSin[1/(c*x)]*Log[1 - E^((2*I)*ArcSin[1/(c*x)])] - 4*c*d^2*ArcSin[1/(c*x)]*Log[x^(-1)] - (2*I)*c*d^2*PolyLog[2, E^((2*I)*ArcS in[1/(c*x)])]))/(4*c)
3.1.89.3.1 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_.) + ArcCos[(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]}, Simp [(a + b*ArcCos[c*x]) u, x] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCos[x/c])^n/x^( m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
Time = 2.52 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.24
| method | result | size |
| parts | \(a \left (\frac {e^{2} x^{4}}{4}+d e \,x^{2}+d^{2} \ln \left (x \right )\right )+b \left (\frac {i d^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}+\frac {e \left (12 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+3 \,\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-12 i c^{2} d -2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e c x -2 i e \right )}{12 c^{4}}-d^{2} \operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i d^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(231\) |
| derivativedivides | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \left (\frac {i c^{4} d^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}+\frac {e \left (12 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+3 \,\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-12 i c^{2} d -2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e c x -2 i e \right )}{12}-\ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right ) c^{4} d^{2} \operatorname {arcsec}\left (c x \right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right ) c^{4} d^{2}}{2}\right )}{c^{4}}\) | \(242\) |
| default | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \left (\frac {i c^{4} d^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}+\frac {e \left (12 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+3 \,\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-12 i c^{2} d -2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e c x -2 i e \right )}{12}-\ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right ) c^{4} d^{2} \operatorname {arcsec}\left (c x \right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right ) c^{4} d^{2}}{2}\right )}{c^{4}}\) | \(242\) |
a*(1/4*e^2*x^4+d*e*x^2+d^2*ln(x))+b*(1/2*I*d^2*arcsec(c*x)^2+1/12/c^4*e*(1 2*c^4*d*arcsec(c*x)*x^2+3*arcsec(c*x)*e*c^4*x^4-12*((c^2*x^2-1)/c^2/x^2)^( 1/2)*c^3*d*x-((c^2*x^2-1)/c^2/x^2)^(1/2)*e*c^3*x^3-12*I*c^2*d-2*((c^2*x^2- 1)/c^2/x^2)^(1/2)*e*c*x-2*I*e)-d^2*arcsec(c*x)*ln(1+(1/c/x+I*(1-1/c^2/x^2) ^(1/2))^2)+1/2*I*d^2*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2))
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x} \,d x } \]
integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d ^2)*arcsec(c*x))/x, x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x} \,d x } \]
1/4*a*e^2*x^4 + a*d*e*x^2 + a*d^2*log(x) - 1/8*(-2*I*b*c^4*e^2*x^4*log(c) - 4*I*b*c^4*d^2*log(-c*x + 1)*log(x) - 4*I*b*c^4*d^2*log(x)^2 - 4*I*b*c^4* d^2*dilog(c*x) - 4*I*b*c^4*d^2*dilog(-c*x) + I*(b*e^2*(x^2/c^2 + log(c*x + 1)/c^4 + log(c*x - 1)/c^4) + 4*b*d*e*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2 ) + 32*b*d^2*integrate(1/4*log(x)/(c^2*x^3 - x), x))*c^4 + 8*c^4*integrate (1/4*(b*e^2*x^4 + 4*b*d*e*x^2 + 4*b*d^2*log(x))*sqrt(c*x + 1)*sqrt(c*x - 1 )/(c^2*x^3 - x), x) + (-8*I*b*c^4*d*e*log(c) - I*b*c^2*e^2)*x^2 - 2*(b*c^4 *e^2*x^4 + 4*b*c^4*d*e*x^2 + 4*b*c^4*d^2*log(x))*arctan(sqrt(c*x + 1)*sqrt (c*x - 1)) + (I*b*c^4*e^2*x^4 + 4*I*b*c^4*d*e*x^2 + 4*I*b*c^4*d^2*log(x))* log(c^2*x^2) + (-4*I*b*c^4*d^2*log(x) - 4*I*b*c^2*d*e - I*b*e^2)*log(c*x + 1) + (-4*I*b*c^2*d*e - I*b*e^2)*log(c*x - 1) - 2*(I*b*c^4*e^2*x^4 + 4*I*b *c^4*d*e*x^2 + 4*I*b*c^4*d^2*log(c))*log(x))/c^4
Exception generated. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Limit: Max order reached or unable to make series expan sion Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]