Integrand size = 12, antiderivative size = 114 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {5 b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7} \]
1/7*x^7*(a+b*arcsec(c*x))-5/112*b*arctanh((1-1/c^2/x^2)^(1/2))/c^7-5/112*b *x^2*(1-1/c^2/x^2)^(1/2)/c^5-5/168*b*x^4*(1-1/c^2/x^2)^(1/2)/c^3-1/42*b*x^ 6*(1-1/c^2/x^2)^(1/2)/c
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a x^7}{7}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (-\frac {5 x^2}{112 c^5}-\frac {5 x^4}{168 c^3}-\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \sec ^{-1}(c x)-\frac {5 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{112 c^7} \]
(a*x^7)/7 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-5*x^2)/(112*c^5) - (5*x^4) /(168*c^3) - x^6/(42*c)) + (b*x^7*ArcSec[c*x])/7 - (5*b*Log[x*(1 + Sqrt[(- 1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)
Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5743, 798, 52, 52, 52, 73, 221}
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.
| \(\displaystyle \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5743 |
| \(\displaystyle \frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \int \frac {x^5}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{7 c}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {b \int \frac {x^8}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{14 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {b \left (\frac {5 \int \frac {x^6}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{6 c^2}-\frac {1}{3} x^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{14 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {b \left (\frac {5 \left (\frac {3 \int \frac {x^4}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^2}-\frac {1}{3} x^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{14 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {b \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{2 c^2}-x^2 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^2}-\frac {1}{3} x^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{14 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b \left (\frac {5 \left (\frac {3 \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\int \frac {1}{c^2-\frac {c^2}{x^4}}d\sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^2}-\frac {1}{3} x^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{14 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {b \left (\frac {5 \left (\frac {3 \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^2}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^2}-\frac {1}{3} x^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{14 c}\) |
(x^7*(a + b*ArcSec[c*x]))/7 + (b*(-1/3*(Sqrt[1 - 1/(c^2*x^2)]*x^6) + (5*(- 1/2*(Sqrt[1 - 1/(c^2*x^2)]*x^4) + (3*(-(Sqrt[1 - 1/(c^2*x^2)]*x^2) - ArcTa nh[Sqrt[1 - 1/(c^2*x^2)]]/c^2))/(4*c^2)))/(6*c^2)))/(14*c)
3.1.1.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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 = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55
| method | result | size |
| parts | \(\frac {a \,x^{7}}{7}+\frac {b \,x^{7} \operatorname {arcsec}\left (c x \right )}{7}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4}}{42 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2}}{168 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right )}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(177\) |
| derivativedivides | \(\frac {\frac {a \,c^{7} x^{7}}{7}+\frac {b \,c^{7} x^{7} \operatorname {arcsec}\left (c x \right )}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{4} x^{4}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{7}}\) | \(184\) |
| default | \(\frac {\frac {a \,c^{7} x^{7}}{7}+\frac {b \,c^{7} x^{7} \operatorname {arcsec}\left (c x \right )}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{4} x^{4}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{7}}\) | \(184\) |
1/7*a*x^7+1/7*b*x^7*arcsec(c*x)-1/42*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^ 2)^(1/2)*x^4-5/168*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2-5/112 *b/c^7*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)-5/112*b/c^8*(c^2*x^2-1)^(1/ 2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*ln(c*x+(c^2*x^2-1)^(1/2))
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {48 \, a c^{7} x^{7} + 96 \, b c^{7} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \operatorname {arcsec}\left (c x\right ) + 15 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{336 \, c^{7}} \]
1/336*(48*a*c^7*x^7 + 96*b*c^7*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 48*(b*c^ 7*x^7 - b*c^7)*arcsec(c*x) + 15*b*log(-c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5 *x^5 + 10*b*c^3*x^3 + 15*b*c*x)*sqrt(c^2*x^2 - 1))/c^7
Time = 8.73 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.94 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a x^{7}}{7} + \frac {b x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]
a*x**7/7 + b*x**7*asec(c*x)/7 - b*Piecewise((c*x**7/(6*sqrt(c**2*x**2 - 1) ) + x**5/(24*c*sqrt(c**2*x**2 - 1)) + 5*x**3/(48*c**3*sqrt(c**2*x**2 - 1)) - 5*x/(16*c**5*sqrt(c**2*x**2 - 1)) + 5*acosh(c*x)/(16*c**6), Abs(c**2*x* *2) > 1), (-I*c*x**7/(6*sqrt(-c**2*x**2 + 1)) - I*x**5/(24*c*sqrt(-c**2*x* *2 + 1)) - 5*I*x**3/(48*c**3*sqrt(-c**2*x**2 + 1)) + 5*I*x/(16*c**5*sqrt(- c**2*x**2 + 1)) - 5*I*asin(c*x)/(16*c**6), True))/(7*c)
Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a x^{7} + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b \]
1/7*a*x^7 + 1/672*(96*x^7*arcsec(c*x) - (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1/(c^2*x^2) + 1)^(3/2) + 33*sqrt(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) + 15* log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c ^6)/c)*b
Leaf count of result is larger than twice the leaf count of optimal. 8644 vs. \(2 (96) = 192\).
Time = 1.12 (sec) , antiderivative size = 8644, normalized size of antiderivative = 75.82 \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/336*c*(48*b*arccos(1/(c*x))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1) ^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1) ^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*( 1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) - 15*b*log(abs(sqrt(-1/ (c^2*x^2) + 1) + 1/(c*x) + 1))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1 )^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1 )^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8* (1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) + 15*b*log(abs(sqrt(-1 /(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8 *(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) + 48*a/(c^8 + 7*c^8*( 1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1 )^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1 )^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8* (1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) - 336*b*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^8 + 7*c^8*(1/(c^2*...
Timed out. \[ \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^6\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]