Integrand size = 12, antiderivative size = 244 \[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \sec ^{-1}(a+b x)^2}{a}-\frac {\sec ^{-1}(a+b x)^2}{x}-\frac {2 i b \sec ^{-1}(a+b x) \log \left (1-\frac {a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 i b \sec ^{-1}(a+b x) \log \left (1-\frac {a e^{i \sec ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,\frac {a e^{i \sec ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]
-b*arcsec(b*x+a)^2/a-arcsec(b*x+a)^2/x-2*I*b*arcsec(b*x+a)*ln(1-a*(1/(b*x+ a)+I*(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)+2*I*b*arc sec(b*x+a)*ln(1-a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))/ a/(-a^2+1)^(1/2)-2*b*polylog(2,a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))/(1-(- a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)+2*b*polylog(2,a*(1/(b*x+a)+I*(1-1/(b*x+a)^ 2)^(1/2))/(1+(-a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(686\) vs. \(2(244)=488\).
Time = 1.73 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.81 \[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {\frac {(a+b x) \sec ^{-1}(a+b x)^2}{x}+\frac {2 b \left (2 \sec ^{-1}(a+b x) \text {arctanh}\left (\frac {(-1+a) \cot \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )-2 \arccos \left (\frac {1}{a}\right ) \text {arctanh}\left (\frac {(1+a) \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos \left (\frac {1}{a}\right )-2 i \text {arctanh}\left (\frac {(-1+a) \cot \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+2 i \text {arctanh}\left (\frac {(1+a) \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{-\frac {1}{2} i \sec ^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 i \left (\text {arctanh}\left (\frac {(-1+a) \cot \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )-\text {arctanh}\left (\frac {(1+a) \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{\frac {1}{2} i \sec ^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )-\left (\arccos \left (\frac {1}{a}\right )-2 i \text {arctanh}\left (\frac {(1+a) \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (i+i a+\sqrt {-1+a^2}\right ) \left (-i+\tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}\right )-\left (\arccos \left (\frac {1}{a}\right )+2 i \text {arctanh}\left (\frac {(1+a) \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (-i-i a+\sqrt {-1+a^2}\right ) \left (i+\tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}\right )\right )\right )}{\sqrt {-1+a^2}}}{a} \]
-((((a + b*x)*ArcSec[a + b*x]^2)/x + (2*b*(2*ArcSec[a + b*x]*ArcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] - 2*ArcCos[a^(-1)]*ArcTanh[(( 1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] + (ArcCos[a^(-1)] - (2*I)*A rcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] + (2*I)*ArcTanh[( (1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[Sqrt[-1 + a^2]/(Sqrt[ 2]*Sqrt[a]*E^((I/2)*ArcSec[a + b*x])*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[ a^(-1)] + (2*I)*(ArcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] - ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]]))*Log[(Sqrt[-1 + a^2]*E^((I/2)*ArcSec[a + b*x]))/(Sqrt[2]*Sqrt[a]*Sqrt[-((b*x)/(a + b*x) )])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sq rt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] - (ArcC os[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2] ])*Log[((-1 + a)*(-I - I*a + Sqrt[-1 + a^2])*(I + Tan[ArcSec[a + b*x]/2])) /(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] + I*(-PolyLog[2, (( 1 - I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))/(a* (-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] + PolyLog[2, ((1 + I*Sq rt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))])))/Sqrt[-1 + a^2])/a)
Time = 0.74 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5781, 4926, 3042, 4679, 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.
\(\displaystyle \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 5781 |
\(\displaystyle b \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^2 x^2}d\sec ^{-1}(a+b x)\) |
\(\Big \downarrow \) 4926 |
\(\displaystyle b \left (-2 \int -\frac {\sec ^{-1}(a+b x)}{b x}d\sec ^{-1}(a+b x)-\frac {\sec ^{-1}(a+b x)^2}{b x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \left (-2 \int \frac {\sec ^{-1}(a+b x)}{a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )}d\sec ^{-1}(a+b x)-\frac {\sec ^{-1}(a+b x)^2}{b x}\right )\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle b \left (-2 \int \left (\frac {\sec ^{-1}(a+b x)}{a}+\frac {\sec ^{-1}(a+b x)}{a \left (\frac {a}{a+b x}-1\right )}\right )d\sec ^{-1}(a+b x)-\frac {\sec ^{-1}(a+b x)^2}{b x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b \left (-\frac {\sec ^{-1}(a+b x)^2}{b x}-2 \left (\frac {\operatorname {PolyLog}\left (2,\frac {a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{i \sec ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {i \sec ^{-1}(a+b x) \log \left (1-\frac {a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {i \sec ^{-1}(a+b x) \log \left (1-\frac {a e^{i \sec ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {\sec ^{-1}(a+b x)^2}{2 a}\right )\right )\) |
b*(-(ArcSec[a + b*x]^2/(b*x)) - 2*(ArcSec[a + b*x]^2/(2*a) + (I*ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (I*ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqr t[1 - a^2])]/(a*Sqrt[1 - a^2]) - PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])]/(a*Sqrt[1 - a^2])))
3.1.32.3.1 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, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 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 = 1.64 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.38
method | result | size |
derivativedivides | \(b \left (-\frac {\left (b x +a \right ) \operatorname {arcsec}\left (b x +a \right )^{2}}{a b x}-\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {arcsec}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}+\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {arcsec}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}\right )\) | \(336\) |
default | \(b \left (-\frac {\left (b x +a \right ) \operatorname {arcsec}\left (b x +a \right )^{2}}{a b x}-\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {arcsec}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}+\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {arcsec}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}\right )\) | \(336\) |
b*(-(b*x+a)*arcsec(b*x+a)^2/a/b/x-2*I*(-a^2+1)^(1/2)/a/(a^2-1)*arcsec(b*x+ a)*ln((-a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1 )^(1/2)))+2*I*(-a^2+1)^(1/2)/a/(a^2-1)*arcsec(b*x+a)*ln((a*(1/(b*x+a)+I*(1 -1/(b*x+a)^2)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2)))-2*(-a^2+1)^(1/ 2)/a/(a^2-1)*dilog((-a*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))+(-a^2+1)^(1/2)+ 1)/(1+(-a^2+1)^(1/2)))+2*(-a^2+1)^(1/2)/a/(a^2-1)*dilog((a*(1/(b*x+a)+I*(1 -1/(b*x+a)^2)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2))))
\[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcsec}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {\operatorname {asec}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcsec}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
-1/4*(4*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 4*x*integrate((2*s qrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a)*log(b*x + a )^2 - (b^3*x^3 + 2*a*b^2*x^2 + (a^2 - 1)*b*x - (b^3*x^3 + 3*a*b^2*x^2 + a^ 3 + (3*a^2 - 1)*b*x - a)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3* x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*x^2), x) - log(b^2*x^2 + 2*a*b*x + a^2)^2)/x
\[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcsec}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}^2}{x^2} \,d x \]