3.82 \(\int \frac {(d+e x^2) (a+b \csc ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=197 \[ -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {b c \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {c^2 x^2-1}}{49 x^6 \sqrt {c^2 x^2}}-\frac {8 b c^5 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt {c^2 x^2}}-\frac {4 b c^3 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt {c^2 x^2}} \]

[Out]

-1/7*d*(a+b*arccsc(c*x))/x^7-1/5*e*(a+b*arccsc(c*x))/x^5-8/3675*b*c^5*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/(c^2*x
^2)^(1/2)-1/49*b*c*d*(c^2*x^2-1)^(1/2)/x^6/(c^2*x^2)^(1/2)-1/1225*b*c*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/x^4/(c
^2*x^2)^(1/2)-4/3675*b*c^3*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/x^2/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 5239, 12, 453, 271, 264} \[ -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {8 b c^5 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt {c^2 x^2}}-\frac {4 b c^3 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {c^2 x^2-1}}{49 x^6 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8,x]

[Out]

(-8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt
[c^2*x^2]) - (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) - (4*b*c^3*(30*c^2*d + 49*e)*
Sqrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(7*x^7) - (e*(a + b*ArcCsc[c*x]))/(5*x^
5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 5239

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac {(b c x) \int \frac {-5 d-7 e x^2}{35 x^8 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac {(b c x) \int \frac {-5 d-7 e x^2}{x^8 \sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {c^2 x^2}}\\ &=-\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{245 \sqrt {c^2 x^2}}\\ &=-\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac {\left (4 b c^3 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{1225 \sqrt {c^2 x^2}}\\ &=-\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}-\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac {\left (8 b c^5 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{3675 \sqrt {c^2 x^2}}\\ &=-\frac {8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}-\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 110, normalized size = 0.56 \[ -\frac {105 a \left (5 d+7 e x^2\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (49 e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+15 d \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )\right )+105 b \csc ^{-1}(c x) \left (5 d+7 e x^2\right )}{3675 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8,x]

[Out]

-1/3675*(105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 +
6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(5*d + 7*e*x^2)*ArcCsc[c*x])/x^7

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fricas [A]  time = 0.60, size = 109, normalized size = 0.55 \[ -\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \operatorname {arccsc}\left (c x\right ) + {\left (8 \, {\left (30 \, b c^{6} d + 49 \, b c^{4} e\right )} x^{6} + 4 \, {\left (30 \, b c^{4} d + 49 \, b c^{2} e\right )} x^{4} + 3 \, {\left (30 \, b c^{2} d + 49 \, b e\right )} x^{2} + 75 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*arccsc(c*x) + (8*(30*b*c^6*d + 49*b*c^4*e)*x^6 + 4*(3
0*b*c^4*d + 49*b*c^2*e)*x^4 + 3*(30*b*c^2*d + 49*b*e)*x^2 + 75*b*d)*sqrt(c^2*x^2 - 1))/x^7

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giac [B]  time = 0.16, size = 374, normalized size = 1.90 \[ -\frac {1}{3675} \, {\left (75 \, b c^{6} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 315 \, b c^{6} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {525 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{x} - 525 \, b c^{6} d {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {1575 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 525 \, b c^{6} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 147 \, b c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} e + \frac {1575 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - 490 \, b c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} e + \frac {525 \, b c^{5} d \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {735 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) e}{x} + 735 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} e + \frac {1470 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) e}{x} + \frac {735 \, b c^{3} \arcsin \left (\frac {1}{c x}\right ) e}{x} + \frac {735 \, a e}{c x^{5}} + \frac {525 \, a d}{c x^{7}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="giac")

[Out]

-1/3675*(75*b*c^6*d*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1) + 315*b*c^6*d*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*
x^2) + 1) + 525*b*c^5*d*(1/(c^2*x^2) - 1)^3*arcsin(1/(c*x))/x - 525*b*c^6*d*(-1/(c^2*x^2) + 1)^(3/2) + 1575*b*
c^5*d*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))/x + 525*b*c^6*d*sqrt(-1/(c^2*x^2) + 1) + 147*b*c^4*(1/(c^2*x^2) - 1)
^2*sqrt(-1/(c^2*x^2) + 1)*e + 1575*b*c^5*d*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 490*b*c^4*(-1/(c^2*x^2) + 1)^
(3/2)*e + 525*b*c^5*d*arcsin(1/(c*x))/x + 735*b*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))*e/x + 735*b*c^4*sqrt(-
1/(c^2*x^2) + 1)*e + 1470*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))*e/x + 735*b*c^3*arcsin(1/(c*x))*e/x + 735*a*
e/(c*x^5) + 525*a*d/(c*x^7))*c

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maple [A]  time = 0.06, size = 158, normalized size = 0.80 \[ c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arccsc}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\mathrm {arccsc}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 e \,c^{6} x^{6}+120 x^{4} c^{6} d +196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x)

[Out]

c^7*(a/c^2*(-1/7/c^5*d/x^7-1/5*e/c^5/x^5)+b/c^2*(-1/7*arccsc(c*x)/c^5*d/x^7-1/5*arccsc(c*x)*e/c^5/x^5-1/3675*(
c^2*x^2-1)*(240*c^8*d*x^6+392*c^6*e*x^6+120*c^6*d*x^4+196*c^4*e*x^4+90*c^4*d*x^2+147*c^2*e*x^2+75*c^2*d)/((c^2
*x^2-1)/c^2/x^2)^(1/2)/c^8/x^8))

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maxima [A]  time = 0.34, size = 172, normalized size = 0.87 \[ \frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arccsc}\left (c x\right )}{x^{7}}\right )} - \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="maxima")

[Out]

1/245*b*d*((5*c^8*(-1/(c^2*x^2) + 1)^(7/2) - 21*c^8*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^8*(-1/(c^2*x^2) + 1)^(3/2)
- 35*c^8*sqrt(-1/(c^2*x^2) + 1))/c - 35*arccsc(c*x)/x^7) - 1/75*b*e*((3*c^6*(-1/(c^2*x^2) + 1)^(5/2) - 10*c^6
*(-1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(-1/(c^2*x^2) + 1))/c + 15*arccsc(c*x)/x^5) - 1/5*a*e/x^5 - 1/7*a*d/x^7

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)*(a + b*asin(1/(c*x))))/x^8,x)

[Out]

int(((d + e*x^2)*(a + b*asin(1/(c*x))))/x^8, x)

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sympy [A]  time = 56.15, size = 372, normalized size = 1.89 \[ - \frac {a d}{7 x^{7}} - \frac {a e}{5 x^{5}} - \frac {b d \operatorname {acsc}{\left (c x \right )}}{7 x^{7}} - \frac {b e \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {b d \left (\begin {cases} \frac {16 c^{7} \sqrt {c^{2} x^{2} - 1}}{35 x} + \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{35 x^{3}} + \frac {6 c^{3} \sqrt {c^{2} x^{2} - 1}}{35 x^{5}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{7 x^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {16 i c^{7} \sqrt {- c^{2} x^{2} + 1}}{35 x} + \frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{35 x^{3}} + \frac {6 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{35 x^{5}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{7 x^{7}} & \text {otherwise} \end {cases}\right )}{7 c} - \frac {b e \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*acsc(c*x))/x**8,x)

[Out]

-a*d/(7*x**7) - a*e/(5*x**5) - b*d*acsc(c*x)/(7*x**7) - b*e*acsc(c*x)/(5*x**5) - b*d*Piecewise((16*c**7*sqrt(c
**2*x**2 - 1)/(35*x) + 8*c**5*sqrt(c**2*x**2 - 1)/(35*x**3) + 6*c**3*sqrt(c**2*x**2 - 1)/(35*x**5) + c*sqrt(c*
*2*x**2 - 1)/(7*x**7), Abs(c**2*x**2) > 1), (16*I*c**7*sqrt(-c**2*x**2 + 1)/(35*x) + 8*I*c**5*sqrt(-c**2*x**2
+ 1)/(35*x**3) + 6*I*c**3*sqrt(-c**2*x**2 + 1)/(35*x**5) + I*c*sqrt(-c**2*x**2 + 1)/(7*x**7), True))/(7*c) - b
*e*Piecewise((8*c**5*sqrt(c**2*x**2 - 1)/(15*x) + 4*c**3*sqrt(c**2*x**2 - 1)/(15*x**3) + c*sqrt(c**2*x**2 - 1)
/(5*x**5), Abs(c**2*x**2) > 1), (8*I*c**5*sqrt(-c**2*x**2 + 1)/(15*x) + 4*I*c**3*sqrt(-c**2*x**2 + 1)/(15*x**3
) + I*c*sqrt(-c**2*x**2 + 1)/(5*x**5), True))/(5*c)

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