Integrand size = 12, antiderivative size = 89 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {4 b \sqrt {1-\frac {1}{c^2 x^2}} x}{45 c^5}-\frac {2 b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{45 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right ) \]
1/6*x^6*(a+b*arcsec(c*x))-4/45*b*x*(1-1/c^2/x^2)^(1/2)/c^5-2/45*b*x^3*(1-1 /c^2/x^2)^(1/2)/c^3-1/30*b*x^5*(1-1/c^2/x^2)^(1/2)/c
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.81 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a x^6}{6}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (-\frac {4 x}{45 c^5}-\frac {2 x^3}{45 c^3}-\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \sec ^{-1}(c x) \]
(a*x^6)/6 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-4*x)/(45*c^5) - (2*x^3)/(4 5*c^3) - x^5/(30*c)) + (b*x^6*ArcSec[c*x])/6
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5743, 803, 803, 746}
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.
\(\displaystyle \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx\) |
\(\Big \downarrow \) 5743 |
\(\displaystyle \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \int \frac {x^4}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{6 c}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (\frac {4 \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{5 c^2}+\frac {1}{5} x^5 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{3 c^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{5 c^2}+\frac {1}{5} x^5 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (\frac {1}{5} x^5 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {4 \left (\frac {2 x \sqrt {1-\frac {1}{c^2 x^2}}}{3 c^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{5 c^2}\right )}{6 c}\) |
-1/6*(b*((Sqrt[1 - 1/(c^2*x^2)]*x^5)/5 + (4*((2*Sqrt[1 - 1/(c^2*x^2)]*x)/( 3*c^2) + (Sqrt[1 - 1/(c^2*x^2)]*x^3)/3))/(5*c^2)))/c + (x^6*(a + b*ArcSec[ c*x]))/6
3.1.2.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) /a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(d*x)^(m + 1)*((a + b*ArcSec[c*x])/(d*(m + 1))), x] - Simp[b*(d/(c*(m + 1 ))) Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {x^{6} a}{6}+\frac {b \left (\frac {c^{6} x^{6} \operatorname {arcsec}\left (c x \right )}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}}\) | \(79\) |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arcsec}\left (c x \right )}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}}\) | \(83\) |
default | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arcsec}\left (c x \right )}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}}\) | \(83\) |
1/6*x^6*a+b/c^6*(1/6*c^6*x^6*arcsec(c*x)-1/90*(c^2*x^2-1)*(3*c^4*x^4+4*c^2 *x^2+8)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x)
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {15 \, b c^{6} x^{6} \operatorname {arcsec}\left (c x\right ) + 15 \, a c^{6} x^{6} - {\left (3 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]
1/90*(15*b*c^6*x^6*arcsec(c*x) + 15*a*c^6*x^6 - (3*b*c^4*x^4 + 4*b*c^2*x^2 + 8*b)*sqrt(c^2*x^2 - 1))/c^6
Time = 2.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.72 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {asec}{\left (c x \right )}}{6} - \frac {b \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]
a*x**6/6 + b*x**6*asec(c*x)/6 - b*Piecewise((x**4*sqrt(c**2*x**2 - 1)/(5*c ) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5) , Abs(c**2*x**2) > 1), (I*x**4*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt( -c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), True))/(6* c)
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsec}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \]
1/6*a*x^6 + 1/90*(15*x^6*arcsec(c*x) - (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5) *b
Leaf count of result is larger than twice the leaf count of optimal. 3862 vs. \(2 (75) = 150\).
Time = 0.32 (sec) , antiderivative size = 3862, normalized size of antiderivative = 43.39 \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/90*c*(15*b*arccos(1/(c*x))/(c^7 + 6*c^7*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^ 2 + 15*c^7*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 20*c^7*(1/(c^2*x^2) - 1)^ 3/(1/(c*x) + 1)^6 + 15*c^7*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 6*c^7*(1/ (c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + c^7*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^ 12) + 15*a/(c^7 + 6*c^7*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 15*c^7*(1/(c^2 *x^2) - 1)^2/(1/(c*x) + 1)^4 + 20*c^7*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 15*c^7*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 6*c^7*(1/(c^2*x^2) - 1)^5/( 1/(c*x) + 1)^10 + c^7*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12) - 90*b*(1/(c^2 *x^2) - 1)*arccos(1/(c*x))/((c^7 + 6*c^7*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 15*c^7*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 20*c^7*(1/(c^2*x^2) - 1)^3 /(1/(c*x) + 1)^6 + 15*c^7*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 6*c^7*(1/( c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + c^7*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^1 2)*(1/(c*x) + 1)^2) - 30*b*sqrt(-1/(c^2*x^2) + 1)/((c^7 + 6*c^7*(1/(c^2*x^ 2) - 1)/(1/(c*x) + 1)^2 + 15*c^7*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 20* c^7*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 15*c^7*(1/(c^2*x^2) - 1)^4/(1/(c *x) + 1)^8 + 6*c^7*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + c^7*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12)*(1/(c*x) + 1)) - 90*a*(1/(c^2*x^2) - 1)/((c^7 + 6*c^7*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 15*c^7*(1/(c^2*x^2) - 1)^2/(1/(c *x) + 1)^4 + 20*c^7*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 15*c^7*(1/(c^2*x ^2) - 1)^4/(1/(c*x) + 1)^8 + 6*c^7*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10...
Timed out. \[ \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^5\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]