3.1 \(\int x^6 (a+b \sec ^{-1}(c x)) \, dx\)
Optimal. Leaf size=114 \[ \frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^6 \sqrt {1-\frac {1}{c^2 x^2}}}{42 c}-\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}-\frac {5 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{112 c^5}-\frac {5 b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{168 c^3} \]
[Out]
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
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Rubi [A] time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used =
{5220, 266, 51, 63, 208} \[ \frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^6 \sqrt {1-\frac {1}{c^2 x^2}}}{42 c}-\frac {5 b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{168 c^3}-\frac {5 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{112 c^5}-\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7} \]
Antiderivative was successfully verified.
[In]
Int[x^6*(a + b*ArcSec[c*x]),x]
[Out]
(-5*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(112*c^5) - (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(168*c^3) - (b*Sqrt[1 - 1/(c^2*x^
2)]*x^6)/(42*c) + (x^7*(a + b*ArcSec[c*x]))/7 - (5*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(112*c^7)
Rule 51
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] - Dist[(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] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]
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^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\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}\\ &=\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{14 c}\\ &=-\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) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{84 c^3}\\ &=-\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) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{112 c^5}\\ &=-\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) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{224 c^7}\\ &=-\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) \operatorname {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^5}\\ &=-\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 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 107, normalized size = 0.94 \[ \frac {a x^7}{7}-\frac {5 b \log \left (x \left (\sqrt {\frac {c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{112 c^7}+b \sqrt {\frac {c^2 x^2-1}{c^2 x^2}} \left (-\frac {5 x^2}{112 c^5}-\frac {5 x^4}{168 c^3}-\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
Integrate[x^6*(a + b*ArcSec[c*x]),x]
[Out]
(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*Ar
cSec[c*x])/7 - (5*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)
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fricas [A] time = 0.94, size = 116, normalized size = 1.02 \[ \frac {48 \, a c^{7} x^{7} + 96 \, b c^{7} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \operatorname {arcsec}\left (c x\right ) + 15 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{336 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsec(c*x)),x, algorithm="fricas")
[Out]
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*lo
g(-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
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giac [B] time = 1.75, size = 8644, normalized size = 75.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsec(c*x)),x, algorithm="giac")
[Out]
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*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)*(1/(c*x) + 1)^2) - 105*b*(1/(c^2*x^2) - 1)*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)^1
0 + 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)*(1/(c*x) + 1)^2) +
105*b*(1/(c^2*x^2) - 1)*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)*(1/(c*x) + 1)^2) - 66*b*sqrt(-1/(c^2*x^2) + 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)*(1/(c*x) + 1)) -
336*a*(1/(c^2*x^2) - 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)*(1/(c*x) + 1)^2) + 1008*b*(1/(c^2*x^2) - 1)^2*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)*(1/(c*x) + 1)^4) - 315*b*(1/(c^2*x^2) - 1)^2*lo
g(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)*(1/(c*x) + 1)^4) + 315*b*(1/(c^2*x^2) - 1)^2*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)^1
4)*(1/(c*x) + 1)^4) + 56*b*(-1/(c^2*x^2) + 1)^(3/2)/((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)*(1/(c*x) + 1)^3) + 1008*a*(1/(c^2*x^2) - 1)^2/((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)*(1/(c*x) + 1)^4) - 1680*b*(1/(c^2*x^
2) - 1)^3*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)*(1/(c*x) + 1)^6) - 525*b*(1/(c^2*x^2) - 1)^3*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)*(1/(c*x) + 1)^6)
+ 525*b*(1/(c^2*x^2) - 1)^3*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)*(1/(c*x) + 1)^6) - 170*b*(1/(c^2*x^2) - 1)^2*s
qrt(-1/(c^2*x^2) + 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)*(1/(c*x) + 1)^5) - 1680*a*(1/(c^2*x^2) - 1)^3/((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)*(1/(c*x) + 1)^6) + 1680*b*(1/(c^2*x^2) - 1)^4*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)*(1/(c*x) + 1)^8)
- 525*b*(1/(c^2*x^2) - 1)^4*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)*(1/(c*x) + 1)^8) + 525*b*(1/(c^2*x^2) - 1)^4*l
og(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)*(1/(c*x) + 1)^8) + 1680*a*(1/(c^2*x^2) - 1)^4/((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)*(1/(c*x) + 1)^8) - 1008*b*(1/(c^2*x^2) - 1
)^5*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)*(1/(c*x) + 1)^10) - 315*b*(1/(c^2*x^2) - 1)^5*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)*(1/(c*x) + 1)^10) + 3
15*b*(1/(c^2*x^2) - 1)^5*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)*(1/(c*x) + 1)^10) + 170*b*(1/(c^2*x^2) - 1)^4*sqrt
(-1/(c^2*x^2) + 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)*(1/(c*x) + 1)^9) - 1008*a*(1/(c^2*x^2) - 1)^5/((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)*(1/(c*x) + 1)^10) + 336*b*(1/(c^2*x^2) - 1)^6*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)^1
0 + 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)*(1/(c*x) + 1)^12) -
105*b*(1/(c^2*x^2) - 1)^6*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)*(1/(c*x) + 1)^12) + 105*b*(1/(c^2*x^2) - 1)^6*lo
g(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)*(1/(c*x) + 1)^12) + 56*b*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) + 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)*(1/(c*x) + 1)^11) + 3
36*a*(1/(c^2*x^2) - 1)^6/((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)*(1/(c*x) + 1)^12) - 48*b*(1/(c^2*x^2) - 1)^7*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)*(1/(c*x) + 1)^14) - 15*b*(1/(c^2*x^2) - 1)^7*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)*(1/(c*x) + 1)^14) + 15*b*(1/(c^2*x^2) - 1)^7*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 + 3
5*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
)*(1/(c*x) + 1)^14) + 66*b*(1/(c^2*x^2) - 1)^6*sqrt(-1/(c^2*x^2) + 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)*(1/(c*x) + 1)^13) - 48*a*(1/(c^2*x^2) - 1)^7/((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)^1
0 + 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)*(1/(c*x) + 1)^14))
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maple [A] time = 0.08, size = 177, normalized size = 1.55 \[ \frac {x^{7} a}{7}+\frac {b \,x^{7} \mathrm {arcsec}\left (c x \right )}{7}-\frac {b \,x^{6}}{42 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{4}}{168 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \,x^{2}}{336 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^6*(a+b*arcsec(c*x)),x)
[Out]
1/7*x^7*a+1/7*b*x^7*arcsec(c*x)-1/42/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^6-1/168/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/
2)*x^4-5/336/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2+5/112/c^7*b/((c^2*x^2-1)/c^2/x^2)^(1/2)-5/112/c^8*b*(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))
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maxima [A] time = 0.42, size = 162, normalized size = 1.42 \[ \frac {1}{7} \, a x^{7} + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsec(c*x)),x, algorithm="maxima")
[Out]
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*sqr
t(-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) + 1
5*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
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^6*(a + b*acos(1/(c*x))),x)
[Out]
int(x^6*(a + b*acos(1/(c*x))), x)
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sympy [A] time = 8.46, size = 221, normalized size = 1.94 \[ \frac {a x^{7}}{7} + \frac {b x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**6*(a+b*asec(c*x)),x)
[Out]
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)
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