Integrand size = 12, antiderivative size = 56 \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61 \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {a c x \left (-2 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )+2 b c x \left (-b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right ) \sec ^{-1}(c x)+b^2 c^2 x^2 \sec ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]
(a*c*x*(-2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) + 2*b*c*x*(-(b*Sqrt[1 - 1/(c^2 *x^2)]) + a*c*x)*ArcSec[c*x] + b^2*c^2*x^2*ArcSec[c*x]^2 + 2*b^2*Log[c*x]) /(2*c^2)
Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5745, 4909, 3042, 4672, 25, 3042, 3956}
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.
\(\displaystyle \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx\) |
\(\Big \downarrow \) 5745 |
\(\displaystyle \frac {\int c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2d\sec ^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 4909 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \int c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )d\sec ^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \int \left (a+b \sec ^{-1}(c x)\right ) \csc \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \left (b \int -c \sqrt {1-\frac {1}{c^2 x^2}} xd\sec ^{-1}(c x)+c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )\right )}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \left (c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-b \int c \sqrt {1-\frac {1}{c^2 x^2}} xd\sec ^{-1}(c x)\right )}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \left (c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-b \int \tan \left (\sec ^{-1}(c x)\right )d\sec ^{-1}(c x)\right )}{c^2}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {\frac {1}{2} c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^2-b \left (c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )+b \log \left (\frac {1}{c x}\right )\right )}{c^2}\) |
((c^2*x^2*(a + b*ArcSec[c*x])^2)/2 - b*(c*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*A rcSec[c*x]) + b*Log[1/(c*x)]))/c^2
3.1.17.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b _.)*(x_)]^(p_.), x_Symbol] :> Simp[(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; FreeQ[{ a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /c^(m + 1) Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x ]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] | | LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(52)=104\).
Time = 0.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.20
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(123\) |
derivativedivides | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(124\) |
default | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(124\) |
1/2*a^2*x^2+b^2/c^2*(1/2*c^2*x^2*arcsec(c*x)^2-arcsec(c*x)*c*x*((c^2*x^2-1 )/c^2/x^2)^(1/2)-ln(1/c/x))+2*a*b/c^2*(1/2*c^2*x^2*arcsec(c*x)-1/2/((c^2*x ^2-1)/c^2/x^2)^(1/2)/c/x*(c^2*x^2-1))
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (52) = 104\).
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.98 \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {b^{2} c^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + a^{2} c^{2} x^{2} + 4 \, a b c^{2} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \left (x\right ) + 2 \, {\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arcsec}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \]
1/2*(b^2*c^2*x^2*arcsec(c*x)^2 + a^2*c^2*x^2 + 4*a*b*c^2*arctan(-c*x + sqr t(c^2*x^2 - 1)) + 2*b^2*log(x) + 2*(a*b*c^2*x^2 - a*b*c^2)*arcsec(c*x) - 2 *sqrt(c^2*x^2 - 1)*(b^2*arcsec(c*x) + a*b))/c^2
\[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int x \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \]
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.55 \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a b - {\left (\frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsec}\left (c x\right )}{c} - \frac {\log \left (x\right )}{c^{2}}\right )} b^{2} \]
1/2*b^2*x^2*arcsec(c*x)^2 + 1/2*a^2*x^2 + (x^2*arcsec(c*x) - x*sqrt(-1/(c^ 2*x^2) + 1)/c)*a*b - (x*sqrt(-1/(c^2*x^2) + 1)*arcsec(c*x)/c - log(x)/c^2) *b^2
Leaf count of result is larger than twice the leaf count of optimal. 2181 vs. \(2 (52) = 104\).
Time = 0.45 (sec) , antiderivative size = 2181, normalized size of antiderivative = 38.95 \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\text {Too large to display} \]
1/2*(b^2*arccos(1/(c*x))^2/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) + 2*a*b*arccos(1/(c*x))/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) - 2*b^2*(1/(c^2*x^2) - 1)*arccos(1/(c*x))^2/((c^3 + 2*c^3*(1/(c^2 *x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/( c*x) + 1)^2) - 2*b^2*log(2)/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) + 2*b^2*log(2/(c*x) + 2)/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c* x) + 1)^4) - 2*b^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^3 + 2 *c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) - 2*b^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^3 + 2*c^ 3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1 )^4) - 4*b^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))/((c^3 + 2*c^3*(1/(c^2* x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c *x) + 1)) + a^2/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c ^2*x^2) - 1)^2/(1/(c*x) + 1)^4) - 4*a*b*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/ ((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/ (1/(c*x) + 1)^4)*(1/(c*x) + 1)^2) + b^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x) )^2/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1 )^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4) - 4*b^2*(1/(c^2*x^2) - 1)*log(2)/...
Timed out. \[ \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]