Integrand size = 12, antiderivative size = 381 \[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}-\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{3 b^4}-\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{b^4}+\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{b^4}-\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \sec ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \sec ^{-1}(a+b x)^2-\frac {2 i a \sec ^{-1}(a+b x) \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )}{b^4}-\frac {4 i a^3 \sec ^{-1}(a+b x) \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )}{b^4}+\frac {\log (a+b x)}{3 b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {i a \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^4}+\frac {2 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^4} \]
-a*x/b^3+1/12*(b*x+a)^2/b^4-1/4*a^4*arcsec(b*x+a)^2/b^4+1/4*x^4*arcsec(b*x +a)^2-2*I*a^3*polylog(2,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^4+2*I*a^3 *polylog(2,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^4+1/3*ln(b*x+a)/b^4+3 *a^2*ln(b*x+a)/b^4+I*a*polylog(2,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b ^4-2*I*a*arcsec(b*x+a)*arctan(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/b^4-4*I*a ^3*arcsec(b*x+a)*arctan(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/b^4-I*a*polylog (2,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^4-1/3*(b*x+a)*arcsec(b*x+a)*(1 -1/(b*x+a)^2)^(1/2)/b^4-3*a^2*(b*x+a)*arcsec(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/ b^4+a*(b*x+a)^2*arcsec(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4-1/6*(b*x+a)^3*arcs ec(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4
Time = 8.39 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.75 \[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\frac {\left (1-\frac {a}{a+b x}\right )^3 \left (24 a \left (2+\left (1+2 a^2\right ) \sec ^{-1}(a+b x)^2\right )+\frac {2+(-2+24 a) \sec ^{-1}(a+b x)+3 \left (1-4 a+12 a^2\right ) \sec ^{-1}(a+b x)^2}{-1+\sqrt {1-\frac {1}{(a+b x)^2}}}+16 \left (1+9 a^2\right ) \log \left (\frac {1}{a+b x}\right )-24 a \left (1+2 a^2\right ) \left (\left (\pi -2 \sec ^{-1}(a+b x)\right ) \left (\log \left (1-i e^{-i \sec ^{-1}(a+b x)}\right )-\log \left (1+i e^{-i \sec ^{-1}(a+b x)}\right )\right )-\pi \log \left (\cot \left (\frac {1}{4} \left (\pi +2 \sec ^{-1}(a+b x)\right )\right )\right )+2 i \left (\operatorname {PolyLog}\left (2,-i e^{-i \sec ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,i e^{-i \sec ^{-1}(a+b x)}\right )\right )\right )-\frac {3 \sec ^{-1}(a+b x)^2}{\left (\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )^4}+\frac {4 \sec ^{-1}(a+b x) \left (1+6 a \sec ^{-1}(a+b x)\right ) \sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\left (\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )^3}+\frac {8 \left (2 \sec ^{-1}(a+b x)+18 a^2 \sec ^{-1}(a+b x)+6 a^3 \sec ^{-1}(a+b x)^2+3 a \left (2+\sec ^{-1}(a+b x)^2\right )\right ) \sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}-\frac {3 \sec ^{-1}(a+b x)^2}{\left (\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )^4}+\frac {4 \sec ^{-1}(a+b x) \left (1-6 a \sec ^{-1}(a+b x)\right ) \sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\left (\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )^3}-\frac {2+(2-24 a) \sec ^{-1}(a+b x)+3 \left (1-4 a+12 a^2\right ) \sec ^{-1}(a+b x)^2}{\left (\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )^2}-\frac {8 \left (-2 \sec ^{-1}(a+b x)-18 a^2 \sec ^{-1}(a+b x)+6 a^3 \sec ^{-1}(a+b x)^2+3 a \left (2+\sec ^{-1}(a+b x)^2\right )\right ) \sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\cos \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}\right )}{48 b^4 \left (-1+\frac {a}{a+b x}\right )^3} \]
((1 - a/(a + b*x))^3*(24*a*(2 + (1 + 2*a^2)*ArcSec[a + b*x]^2) + (2 + (-2 + 24*a)*ArcSec[a + b*x] + 3*(1 - 4*a + 12*a^2)*ArcSec[a + b*x]^2)/(-1 + Sq rt[1 - (a + b*x)^(-2)]) + 16*(1 + 9*a^2)*Log[(a + b*x)^(-1)] - 24*a*(1 + 2 *a^2)*((Pi - 2*ArcSec[a + b*x])*(Log[1 - I/E^(I*ArcSec[a + b*x])] - Log[1 + I/E^(I*ArcSec[a + b*x])]) - Pi*Log[Cot[(Pi + 2*ArcSec[a + b*x])/4]] + (2 *I)*(PolyLog[2, (-I)/E^(I*ArcSec[a + b*x])] - PolyLog[2, I/E^(I*ArcSec[a + b*x])])) - (3*ArcSec[a + b*x]^2)/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2])^4 + (4*ArcSec[a + b*x]*(1 + 6*a*ArcSec[a + b*x])*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2])^3 + (8*(2*ArcSe c[a + b*x] + 18*a^2*ArcSec[a + b*x] + 6*a^3*ArcSec[a + b*x]^2 + 3*a*(2 + A rcSec[a + b*x]^2))*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[A rcSec[a + b*x]/2]) - (3*ArcSec[a + b*x]^2)/(Cos[ArcSec[a + b*x]/2] + Sin[A rcSec[a + b*x]/2])^4 + (4*ArcSec[a + b*x]*(1 - 6*a*ArcSec[a + b*x])*Sin[Ar cSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^3 - (2 + (2 - 24*a)*ArcSec[a + b*x] + 3*(1 - 4*a + 12*a^2)*ArcSec[a + b*x]^2)/(C os[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^2 - (8*(-2*ArcSec[a + b*x] - 18*a^2*ArcSec[a + b*x] + 6*a^3*ArcSec[a + b*x]^2 + 3*a*(2 + ArcSec[a + b*x]^2))*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])))/(48*b^4*(-1 + a/(a + b*x))^3)
Time = 0.62 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5781, 25, 4926, 3042, 4678, 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.
| \(\displaystyle \int x^3 \sec ^{-1}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 5781 |
| \(\displaystyle \frac {\int b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2d\sec ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2d\sec ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4926 |
| \(\displaystyle \frac {\frac {1}{4} b^4 x^4 \sec ^{-1}(a+b x)^2-\frac {1}{2} \int b^4 x^4 \sec ^{-1}(a+b x)d\sec ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} b^4 x^4 \sec ^{-1}(a+b x)^2-\frac {1}{2} \int \sec ^{-1}(a+b x) \left (a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^4d\sec ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4678 |
| \(\displaystyle \frac {\frac {1}{4} b^4 x^4 \sec ^{-1}(a+b x)^2-\frac {1}{2} \int \left (\sec ^{-1}(a+b x) a^4-4 (a+b x) \sec ^{-1}(a+b x) a^3+6 (a+b x)^2 \sec ^{-1}(a+b x) a^2-4 (a+b x)^3 \sec ^{-1}(a+b x) a+(a+b x)^4 \sec ^{-1}(a+b x)\right )d\sec ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} b^4 x^4 \sec ^{-1}(a+b x)^2+\frac {1}{2} \left (-\frac {1}{2} a^4 \sec ^{-1}(a+b x)^2-8 i a^3 \sec ^{-1}(a+b x) \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )-6 a^2 \log \left (\frac {1}{a+b x}\right )-6 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)-4 i a \sec ^{-1}(a+b x) \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )+2 i a \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )-2 a (a+b x)+\frac {1}{6} (a+b x)^2-\frac {2}{3} \log \left (\frac {1}{a+b x}\right )+2 a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)-\frac {1}{3} (a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)-\frac {2}{3} (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)\right )}{b^4}\) |
((b^4*x^4*ArcSec[a + b*x]^2)/4 + (-2*a*(a + b*x) + (a + b*x)^2/6 - (2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x])/3 - 6*a^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x] + 2*a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(- 2)]*ArcSec[a + b*x] - ((a + b*x)^3*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x ])/3 - (a^4*ArcSec[a + b*x]^2)/2 - (4*I)*a*ArcSec[a + b*x]*ArcTan[E^(I*Arc Sec[a + b*x])] - (8*I)*a^3*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])] - (2*Log[(a + b*x)^(-1)])/3 - 6*a^2*Log[(a + b*x)^(-1)] + (2*I)*a*PolyLog[2 , (-I)*E^(I*ArcSec[a + b*x])] + (4*I)*a^3*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] - (2*I)*a*PolyLog[2, I*E^(I*ArcSec[a + b*x])] - (4*I)*a^3*PolyLog[2 , I*E^(I*ArcSec[a + b*x])])/2)/b^4
3.1.27.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*Sec[(c _.) + (d_.)*(x_)])^(n_.)*Tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e + f* x)^m*((a + b*Sec[c + d*x])^(n + 1)/(b*d*(n + 1))), x] - Simp[f*(m/(b*d*(n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Sec[c + d*x])^(n + 1), x], x] /; FreeQ [{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcSec[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d^(m + 1) Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d *e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]], x] /; FreeQ[{a, b, c, d, e , f}, x] && IGtQ[p, 0] && IntegerQ[m]
Time = 1.44 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(\frac {-\operatorname {arcsec}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arcsec}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsec}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}+\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-3 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a^{2} \left (b x +a \right )+\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )^{2}-\frac {\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{3}}{6}-3 i a^{2} \operatorname {arcsec}\left (b x +a \right )-\frac {\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}{3}+i \operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a -\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{12}-\frac {\ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right )}{3}+\frac {2 \ln \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-3 \ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a^{2}+6 \ln \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-2 \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3} \operatorname {arcsec}\left (b x +a \right )+2 \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3} \operatorname {arcsec}\left (b x +a \right )-\frac {i \operatorname {arcsec}\left (b x +a \right )}{3}-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3}-\ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a \,\operatorname {arcsec}\left (b x +a \right )+\ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a \,\operatorname {arcsec}\left (b x +a \right )-i \operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a +2 i \operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3}}{b^{4}}\) | \(673\) |
| default | \(\frac {-\operatorname {arcsec}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arcsec}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsec}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}+\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-3 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a^{2} \left (b x +a \right )+\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )^{2}-\frac {\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{3}}{6}-3 i a^{2} \operatorname {arcsec}\left (b x +a \right )-\frac {\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}{3}+i \operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a -\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{12}-\frac {\ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right )}{3}+\frac {2 \ln \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-3 \ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a^{2}+6 \ln \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-2 \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3} \operatorname {arcsec}\left (b x +a \right )+2 \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3} \operatorname {arcsec}\left (b x +a \right )-\frac {i \operatorname {arcsec}\left (b x +a \right )}{3}-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3}-\ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a \,\operatorname {arcsec}\left (b x +a \right )+\ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a \,\operatorname {arcsec}\left (b x +a \right )-i \operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a +2 i \operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{3}}{b^{4}}\) | \(673\) |
1/b^4*(-arcsec(b*x+a)^2*a^3*(b*x+a)+3/2*arcsec(b*x+a)^2*a^2*(b*x+a)^2-arcs ec(b*x+a)^2*a*(b*x+a)^3+1/4*arcsec(b*x+a)^2*(b*x+a)^4-3*arcsec(b*x+a)*(((b *x+a)^2-1)/(b*x+a)^2)^(1/2)*a^2*(b*x+a)+arcsec(b*x+a)*(((b*x+a)^2-1)/(b*x+ a)^2)^(1/2)*a*(b*x+a)^2-1/6*arcsec(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)* (b*x+a)^3-3*I*a^2*arcsec(b*x+a)-1/3*arcsec(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2 )^(1/2)*(b*x+a)+I*dilog(1+I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a-(b*x+a) *a+1/12*(b*x+a)^2-1/3*ln(1+(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)+2/3*ln(1 /(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))-3*ln(1+(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2 ))^2)*a^2+6*ln(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))*a^2-2*ln(1+I*(1/(b*x+a)+ I*(1-1/(b*x+a)^2)^(1/2)))*a^3*arcsec(b*x+a)+2*ln(1-I*(1/(b*x+a)+I*(1-1/(b* x+a)^2)^(1/2)))*a^3*arcsec(b*x+a)-1/3*I*arcsec(b*x+a)-2*I*dilog(1-I*(1/(b* x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a^3-ln(1+I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/ 2)))*a*arcsec(b*x+a)+ln(1-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a*arcsec( b*x+a)-I*dilog(1-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a+2*I*dilog(1+I*(1 /(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a^3)
\[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcsec}\left (b x + a\right )^{2} \,d x } \]
\[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\int x^{3} \operatorname {asec}^{2}{\left (a + b x \right )}\, dx \]
\[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcsec}\left (b x + a\right )^{2} \,d x } \]
1/4*x^4*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 1/16*x^4*log(b^2*x ^2 + 2*a*b*x + a^2)^2 - integrate(1/4*(2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x^4*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 4*(b^3*x^6 + 3*a*b^ 2*x^5 + (3*a^2 - 1)*b*x^4 + (a^3 - a)*x^3)*log(b*x + a)^2 - (b^3*x^6 + 2*a *b^2*x^5 + (a^2 - 1)*b*x^4 + 4*(b^3*x^6 + 3*a*b^2*x^5 + (3*a^2 - 1)*b*x^4 + (a^3 - a)*x^3)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^3 + 3* a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a), x)
Exception generated. \[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int x^3 \sec ^{-1}(a+b x)^2 \, dx=\int x^3\,{\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]