3.1247 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{7/2} \, dx\)

Optimal. Leaf size=154 \[ \frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}+\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

[Out]

15/64*c^2*x*(-a^2*c*x^2+c)^(3/2)+3/16*c*x*(-a^2*c*x^2+c)^(5/2)+9/56*(-a^2*c*x^2+c)^(7/2)/a+1/8*(-a*x+1)*(-a^2*
c*x^2+c)^(7/2)/a+45/128*c^(7/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/a+45/128*c^3*x*(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6142, 671, 641, 195, 217, 203} \[ \frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(7/2)/E^(2*ArcTanh[a*x]),x]

[Out]

(45*c^3*x*Sqrt[c - a^2*c*x^2])/128 + (15*c^2*x*(c - a^2*c*x^2)^(3/2))/64 + (3*c*x*(c - a^2*c*x^2)^(5/2))/16 +
(9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 - a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) + (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/
Sqrt[c - a^2*c*x^2]])/(128*a)

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 6142

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(c + d*x^2)^(p
+ n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && I
LtQ[n/2, 0]

Rubi steps

\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{8} (9 c) \int (1-a x) \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{8} (9 c) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{16} \left (15 c^2\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{64} \left (45 c^3\right ) \int \sqrt {c-a^2 c x^2} \, dx\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{128} \left (45 c^4\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{128} \left (45 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 151, normalized size = 0.98 \[ -\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {a x+1} \left (112 a^8 x^8-368 a^7 x^7+88 a^6 x^6+936 a^5 x^5-978 a^4 x^4-558 a^3 x^3+1349 a^2 x^2-325 a x-256\right )+630 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(7/2)/E^(2*ArcTanh[a*x]),x]

[Out]

-1/896*(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-256 - 325*a*x + 1349*a^2*x^2 - 558*a^3*x^3 - 978*a^4*x^4 + 93
6*a^5*x^5 + 88*a^6*x^6 - 368*a^7*x^7 + 112*a^8*x^8) + 630*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqr
t[1 - a*x]*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 1.35, size = 286, normalized size = 1.86 \[ \left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, -\frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")

[Out]

[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(112*a^7*c^3*x^7 - 25
6*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*c^3*x + 256*c^3)
*sqrt(-a^2*c*x^2 + c))/a, -1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (112*
a^7*c^3*x^7 - 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*
c^3*x + 256*c^3)*sqrt(-a^2*c*x^2 + c))/a]

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giac [B]  time = 0.44, size = 416, normalized size = 2.70 \[ -\frac {{\left (80640 \, a^{9} c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - \frac {{\left (315 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{7} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 2415 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{6} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{5} c^{6} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 17609 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{4} c^{7} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 15159 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{8} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{9} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 315 \, a^{9} c^{11} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 2415 \, a^{9} c^{10} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)\right )} {\left (a x + 1\right )}^{8}}{c^{8}}\right )} {\left | a \right |}}{114688 \, a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")

[Out]

-1/114688*(80640*a^9*c^(7/2)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a*x + 1))*sgn(a) - (315*a^9*(c -
2*c/(a*x + 1))^7*c^4*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) - 2415*a^9*(c - 2*c/(a*x + 1))^6*c^5*sqr
t(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 8043*a^9*(c - 2*c/(a*x + 1))^5*c^6*sqrt(-c + 2*c/(a*x + 1))*sg
n(1/(a*x + 1))*sgn(a) + 17609*a^9*(c - 2*c/(a*x + 1))^4*c^7*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) -
15159*a^9*(c - 2*c/(a*x + 1))^3*c^8*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 8043*a^9*(c - 2*c/(a*x
+ 1))^2*c^9*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 315*a^9*c^11*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a
*x + 1))*sgn(a) + 2415*a^9*c^10*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^8/c^8)*abs(a)/a^
11

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maple [B]  time = 0.04, size = 276, normalized size = 1.79 \[ -\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}-\frac {7 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{48}-\frac {35 c^{2} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{192}-\frac {35 c^{3} x \sqrt {-a^{2} c \,x^{2}+c}}{128}-\frac {35 c^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{128 \sqrt {a^{2} c}}+\frac {2 \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {7}{2}}}{7 a}+\frac {c \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}} x}{3}+\frac {5 c^{2} \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{12}+\frac {5 c^{3} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}\, x}{8}+\frac {5 c^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{8 \sqrt {a^{2} c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x)

[Out]

-1/8*x*(-a^2*c*x^2+c)^(7/2)-7/48*c*x*(-a^2*c*x^2+c)^(5/2)-35/192*c^2*x*(-a^2*c*x^2+c)^(3/2)-35/128*c^3*x*(-a^2
*c*x^2+c)^(1/2)-35/128*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/7/a*(-(x+1/a)^2*a^2*c+
2*a*c*(x+1/a))^(7/2)+1/3*c*(-(x+1/a)^2*a^2*c+2*a*c*(x+1/a))^(5/2)*x+5/12*c^2*(-(x+1/a)^2*a^2*c+2*a*c*(x+1/a))^
(3/2)*x+5/8*c^3*(-(x+1/a)^2*a^2*c+2*a*c*(x+1/a))^(1/2)*x+5/8*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-(x+1/a
)^2*a^2*c+2*a*c*(x+1/a))^(1/2))

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maxima [A]  time = 0.42, size = 173, normalized size = 1.12 \[ -\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x + \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x + \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x + \frac {5}{8} \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3} x - \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x + 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} - \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-1/8*(-a^2*c*x^2 + c)^(7/2)*x + 3/16*(-a^2*c*x^2 + c)^(5/2)*c*x + 15/64*(-a^2*c*x^2 + c)^(3/2)*c^2*x + 5/8*sqr
t(a^2*c*x^2 + 4*a*c*x + 3*c)*c^3*x - 35/128*sqrt(-a^2*c*x^2 + c)*c^3*x - 5/8*c^5*arcsin(a*x + 2)/(a*(-c)^(3/2)
) - 35/128*c^(7/2)*arcsin(a*x)/a + 2/7*(-a^2*c*x^2 + c)^(7/2)/a + 5/4*sqrt(a^2*c*x^2 + 4*a*c*x + 3*c)*c^3/a

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a^2*c*x^2)^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)

[Out]

-int(((c - a^2*c*x^2)^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)

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sympy [C]  time = 20.11, size = 1091, normalized size = 7.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)

[Out]

a**6*c**3*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) -
I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)
*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**
9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2
+ 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt
(c)*asin(a*x)/(128*a**7), True)) - 2*a**5*c**3*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x*
*2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)),
(sqrt(c)*x**6/6, True)) - a**4*c**3*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5
/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2
- 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5
*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqr
t(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 4*a**3*c**3*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/
5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, Tr
ue)) - a**2*c**3*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 -
1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*s
qrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*
x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a*c**3*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2,
0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c**3*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 -
1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-
a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))

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