3.1086 \(\int e^{2 \tanh ^{-1}(a x)} x^3 (c-a^2 c x^2)^{3/2} \, dx\)

Optimal. Leaf size=161 \[ -\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}+\frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}-\frac {(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3} \]

[Out]

-11/35*x^2*(-a^2*c*x^2+c)^(3/2)/a^2-1/3*x^3*(-a^2*c*x^2+c)^(3/2)/a-1/7*x^4*(-a^2*c*x^2+c)^(3/2)-1/420*(105*a*x
+88)*(-a^2*c*x^2+c)^(3/2)/a^4+1/8*c^(3/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/a^4+1/8*c*x*(-a^2*c*x^2+c)^
(1/2)/a^3

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Rubi [A]  time = 0.34, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1809, 833, 780, 195, 217, 203} \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}+\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^(3/2),x]

[Out]

(c*x*Sqrt[c - a^2*c*x^2])/(8*a^3) - (11*x^2*(c - a^2*c*x^2)^(3/2))/(35*a^2) - (x^3*(c - a^2*c*x^2)^(3/2))/(3*a
) - (x^4*(c - a^2*c*x^2)^(3/2))/7 - ((88 + 105*a*x)*(c - a^2*c*x^2)^(3/2))/(420*a^4) + (c^(3/2)*ArcTan[(a*Sqrt
[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^4)

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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6151

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

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x^3 (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x^3 \left (-11 a^2 c-14 a^3 c x\right ) \sqrt {c-a^2 c x^2} \, dx}{7 a^2}\\ &=-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}+\frac {\int x^2 \left (42 a^3 c^2+66 a^4 c^2 x\right ) \sqrt {c-a^2 c x^2} \, dx}{42 a^4 c}\\ &=-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x \left (-132 a^4 c^3-210 a^5 c^3 x\right ) \sqrt {c-a^2 c x^2} \, dx}{210 a^6 c^2}\\ &=-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c \int \sqrt {c-a^2 c x^2} \, dx}{4 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^2 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 113, normalized size = 0.70 \[ \frac {c \left (120 a^6 x^6+280 a^5 x^5+144 a^4 x^4-70 a^3 x^3-88 a^2 x^2-105 a x-176\right ) \sqrt {c-a^2 c x^2}-105 c^{3/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{840 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^(3/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-176 - 105*a*x - 88*a^2*x^2 - 70*a^3*x^3 + 144*a^4*x^4 + 280*a^5*x^5 + 120*a^6*x^6) -
105*c^(3/2)*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(840*a^4)

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fricas [A]  time = 0.74, size = 234, normalized size = 1.45 \[ \left [\frac {105 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, a^{4}}, -\frac {105 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[1/1680*(105*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(120*a^6*c*x^6 + 280*a^
5*c*x^5 + 144*a^4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4, -1/840*(
105*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (120*a^6*c*x^6 + 280*a^5*c*x^5 + 144*a^
4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4]

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giac [A]  time = 0.35, size = 117, normalized size = 0.73 \[ \frac {1}{840} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (3 \, a^{2} c x + 7 \, a c\right )} x + 18 \, c\right )} x - \frac {35 \, c}{a}\right )} x - \frac {44 \, c}{a^{2}}\right )} x - \frac {105 \, c}{a^{3}}\right )} x - \frac {176 \, c}{a^{4}}\right )} - \frac {c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{3} \sqrt {-c} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

1/840*sqrt(-a^2*c*x^2 + c)*((2*((4*(5*(3*a^2*c*x + 7*a*c)*x + 18*c)*x - 35*c/a)*x - 44*c/a^2)*x - 105*c/a^3)*x
- 176*c/a^4) - 1/8*c^2*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(a^3*sqrt(-c)*abs(a))

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maple [B]  time = 0.05, size = 268, normalized size = 1.66 \[ \frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{7 a^{2} c}+\frac {16 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{35 c \,a^{4}}+\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 a^{3} c}-\frac {7 x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 a^{3}}-\frac {7 c x \sqrt {-a^{2} c \,x^{2}+c}}{8 a^{3}}-\frac {7 c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 a^{3} \sqrt {a^{2} c}}-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{4}}+\frac {c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{a^{3}}+\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a^{3} \sqrt {a^{2} c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x)

[Out]

1/7*x^2*(-a^2*c*x^2+c)^(5/2)/a^2/c+16/35/c/a^4*(-a^2*c*x^2+c)^(5/2)+1/3/a^3*x*(-a^2*c*x^2+c)^(5/2)/c-7/12/a^3*
x*(-a^2*c*x^2+c)^(3/2)-7/8*c*x*(-a^2*c*x^2+c)^(1/2)/a^3-7/8/a^3*c^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2
*c*x^2+c)^(1/2))-2/3/a^4*(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(3/2)+1/a^3*c*(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2)
*x+1/a^3*c^2/(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 = 0.75, size = 213, normalized size = 1.32 \[ \frac {1}{840} \, a {\left (\frac {120 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}}{a^{3} c} - \frac {490 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {280 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{4} c} + \frac {840 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{4}} - \frac {735 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{4}} - \frac {735 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{5}} - \frac {560 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{5}} + \frac {384 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{5} c} - \frac {1680 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{5}} + \frac {840 \, c^{3} \arcsin \left (a x - 2\right )}{a^{8} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

1/840*a*(120*(-a^2*c*x^2 + c)^(5/2)*x^2/(a^3*c) - 490*(-a^2*c*x^2 + c)^(3/2)*x/a^4 + 280*(-a^2*c*x^2 + c)^(5/2
)*x/(a^4*c) + 840*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c*x/a^4 - 735*sqrt(-a^2*c*x^2 + c)*c*x/a^4 - 735*c^(3/2)*arc
sin(a*x)/a^5 - 560*(-a^2*c*x^2 + c)^(3/2)/a^5 + 384*(-a^2*c*x^2 + c)^(5/2)/(a^5*c) - 1680*sqrt(a^2*c*x^2 - 4*a
*c*x + 3*c)*c/a^5 + 840*c^3*arcsin(a*x - 2)/(a^8*(-c/a^2)^(3/2)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3*(c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

int(-(x^3*(c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)

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sympy [A]  time = 18.31, size = 420, normalized size = 2.61 \[ a^{2} c \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**(3/2),x)

[Out]

a**2*c*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)) + 2*a*c*Piecewi
se((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)) + sq
rt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a
**5), True)) + c*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, True))

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