3.2 \(\int x^5 (a+b \sec ^{-1}(c x)) \, dx\)
Optimal. Leaf size=89 \[ \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^5 \sqrt {1-\frac {1}{c^2 x^2}}}{30 c}-\frac {4 b x \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^5}-\frac {2 b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^3} \]
[Out]
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
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Rubi [A] time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{5220, 271, 191} \[ \frac {1}{6} x^6 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^5 \sqrt {1-\frac {1}{c^2 x^2}}}{30 c}-\frac {2 b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^3}-\frac {4 b x \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^5} \]
Antiderivative was successfully verified.
[In]
Int[x^5*(a + b*ArcSec[c*x]),x]
[Out]
(-4*b*Sqrt[1 - 1/(c^2*x^2)]*x)/(45*c^5) - (2*b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(45*c^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*
x^5)/(30*c) + (x^6*(a + b*ArcSec[c*x]))/6
Rule 191
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]
Rule 271
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(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]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Rule 5220
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSec[c*x]
))/(d*(m + 1)), x] - Dist[(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]
Rubi steps
\begin {align*} \int x^5 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\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}\\ &=-\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 )-\frac {(2 b) \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=-\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 )-\frac {(4 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=-\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 )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 72, normalized size = 0.81 \[ \frac {a x^6}{6}+b \sqrt {\frac {c^2 x^2-1}{c^2 x^2}} \left (-\frac {4 x}{45 c^5}-\frac {2 x^3}{45 c^3}-\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
Integrate[x^5*(a + b*ArcSec[c*x]),x]
[Out]
(a*x^6)/6 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-4*x)/(45*c^5) - (2*x^3)/(45*c^3) - x^5/(30*c)) + (b*x^6*ArcSec
[c*x])/6
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fricas [A] time = 0.71, size = 63, normalized size = 0.71 \[ \frac {15 \, b c^{6} x^{6} \operatorname {arcsec}\left (c x\right ) + 15 \, a c^{6} x^{6} - {\left (3 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(a+b*arcsec(c*x)),x, algorithm="fricas")
[Out]
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
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giac [B] time = 0.23, size = 3862, normalized size = 43.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(a+b*arcsec(c*x)),x, algorithm="giac")
[Out]
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)^12)*(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 + c^7*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12)*(1/(c*x) + 1)^2) +
225*b*(1/(c^2*x^2) - 1)^2*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)*(1/(c*x) + 1)^4
) + 70*b*(-1/(c^2*x^2) + 1)^(3/2)/((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)^3) + 225*a
*(1/(c^2*x^2) - 1)^2/((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)^4) - 300*b*(1/(c^2*x^2)
- 1)^3*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)*(1/(c*x) + 1)^6) - 156*b*(1/(c^2*x
^2) - 1)^2*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)^5) - 300*a
*(1/(c^2*x^2) - 1)^3/((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)^6) + 225*b*(1/(c^2*x^2)
- 1)^4*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)*(1/(c*x) + 1)^8) - 156*b*(1/(c^2*x
^2) - 1)^3*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)^7) + 225*a
*(1/(c^2*x^2) - 1)^4/((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)^8) - 90*b*(1/(c^2*x^2)
- 1)^5*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)*(1/(c*x) + 1)^10) - 70*b*(1/(c^2*x^
2) - 1)^4*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)^9) - 90*a*(
1/(c^2*x^2) - 1)^5/((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)^10) + 15*b*(1/(c^2*x^2) -
1)^6*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)*(1/(c*x) + 1)^12) - 30*b*(1/(c^2*x^2
) - 1)^5*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)^11) + 15*a*(
1/(c^2*x^2) - 1)^6/((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)^12))
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maple [A] time = 0.05, size = 83, normalized size = 0.93 \[ \frac {\frac {c^{6} x^{6} a}{6}+b \left (\frac {c^{6} x^{6} \mathrm {arcsec}\left (c x \right )}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(a+b*arcsec(c*x)),x)
[Out]
1/c^6*(1/6*c^6*x^6*a+b*(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))
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maxima [A] time = 0.33, size = 81, normalized size = 0.91 \[ \frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsec}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(a+b*arcsec(c*x)),x, algorithm="maxima")
[Out]
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
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(a + b*acos(1/(c*x))),x)
[Out]
int(x^5*(a + b*acos(1/(c*x))), x)
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sympy [A] time = 4.19, size = 153, normalized size = 1.72 \[ \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {asec}{\left (c x \right )}}{6} - \frac {b \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**5*(a+b*asec(c*x)),x)
[Out]
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*sqr
t(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), True))/(6*c)
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