Integrand size = 21, antiderivative size = 183 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {b c \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {-1+c^2 x^2}}{225 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c d \left (6 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x} \]
-1/5*d^2*(a+b*arccsc(c*x))/x^5-2/3*d*e*(a+b*arccsc(c*x))/x^3-e^2*(a+b*arcc sc(c*x))/x-1/225*b*c*(24*c^4*d^2+100*c^2*d*e+225*e^2)*(c^2*x^2-1)^(1/2)/(c ^2*x^2)^(1/2)-1/25*b*c*d^2*(c^2*x^2-1)^(1/2)/x^4/(c^2*x^2)^(1/2)-2/225*b*c *d*(6*c^2*d+25*e)*(c^2*x^2-1)^(1/2)/x^2/(c^2*x^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (225 e^2 x^4+50 d e x^2 \left (1+2 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \csc ^{-1}(c x)}{225 x^5} \]
-1/225*(15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)] *x*(225*e^2*x^4 + 50*d*e*x^2*(1 + 2*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^ 4*x^4)) + 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcCsc[c*x])/x^5
Time = 0.40 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5762, 27, 1588, 359, 242}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int -\frac {15 e^2 x^4+10 d e x^2+3 d^2}{15 x^6 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {15 e^2 x^4+10 d e x^2+3 d^2}{x^6 \sqrt {c^2 x^2-1}}dx}{15 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \int \frac {75 e^2 x^2+2 d \left (6 d c^2+25 e\right )}{x^4 \sqrt {c^2 x^2-1}}dx+\frac {3 d^2 \sqrt {c^2 x^2-1}}{5 x^5}\right )}{15 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \int \frac {1}{x^2 \sqrt {c^2 x^2-1}}dx+\frac {2 d \sqrt {c^2 x^2-1} \left (6 c^2 d+25 e\right )}{3 x^3}\right )+\frac {3 d^2 \sqrt {c^2 x^2-1}}{5 x^5}\right )}{15 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-\frac {b c x \left (\frac {3 d^2 \sqrt {c^2 x^2-1}}{5 x^5}+\frac {1}{5} \left (\frac {2 d \sqrt {c^2 x^2-1} \left (6 c^2 d+25 e\right )}{3 x^3}+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+100 c^2 d e+225 e^2\right )}{3 x}\right )\right )}{15 \sqrt {c^2 x^2}}\) |
-1/15*(b*c*x*((3*d^2*Sqrt[-1 + c^2*x^2])/(5*x^5) + ((2*d*(6*c^2*d + 25*e)* Sqrt[-1 + c^2*x^2])/(3*x^3) + ((24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*Sqrt[- 1 + c^2*x^2])/(3*x))/5))/Sqrt[c^2*x^2] - (d^2*(a + b*ArcCsc[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcCsc[c*x]))/(3*x^3) - (e^2*(a + b*ArcCsc[c*x]))/x
3.1.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
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]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[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])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.96
| method | result | size |
| parts | \(a \left (-\frac {e^{2}}{x}-\frac {d^{2}}{5 x^{5}}-\frac {2 d e}{3 x^{3}}\right )+b \,c^{5} \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c^{5} x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 x^{5} c^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c^{5} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{10} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{6}}\right )\) | \(175\) |
| derivativedivides | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(191\) |
| default | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(191\) |
a*(-e^2/x-1/5*d^2/x^5-2/3*d*e/x^3)+b*c^5*(-1/c^5*arccsc(c*x)*e^2/x-1/5*arc csc(c*x)*d^2/x^5/c^5-2/3/c^5*arccsc(c*x)*d*e/x^3-1/225/c^10*(c^2*x^2-1)*(2 4*c^8*d^2*x^4+100*c^6*d*e*x^4+12*c^6*d^2*x^2+225*c^4*e^2*x^4+50*c^4*d*e*x^ 2+9*c^4*d^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^6)
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {225 \, a e^{2} x^{4} + 150 \, a d e x^{2} + 45 \, a d^{2} + 15 \, {\left (15 \, b e^{2} x^{4} + 10 \, b d e x^{2} + 3 \, b d^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left ({\left (24 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 225 \, b e^{2}\right )} x^{4} + 9 \, b d^{2} + 2 \, {\left (6 \, b c^{2} d^{2} + 25 \, b d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, x^{5}} \]
-1/225*(225*a*e^2*x^4 + 150*a*d*e*x^2 + 45*a*d^2 + 15*(15*b*e^2*x^4 + 10*b *d*e*x^2 + 3*b*d^2)*arccsc(c*x) + ((24*b*c^4*d^2 + 100*b*c^2*d*e + 225*b*e ^2)*x^4 + 9*b*d^2 + 2*(6*b*c^2*d^2 + 25*b*d*e)*x^2)*sqrt(c^2*x^2 - 1))/x^5
Time = 5.19 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.83 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=- \frac {a d^{2}}{5 x^{5}} - \frac {2 a d e}{3 x^{3}} - \frac {a e^{2}}{x} - b c e^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {2 b d e \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b e^{2} \operatorname {acsc}{\left (c x \right )}}{x} - \frac {b d^{2} \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {2 b d e \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
-a*d**2/(5*x**5) - 2*a*d*e/(3*x**3) - a*e**2/x - b*c*e**2*sqrt(1 - 1/(c**2 *x**2)) - b*d**2*acsc(c*x)/(5*x**5) - 2*b*d*e*acsc(c*x)/(3*x**3) - b*e**2* acsc(c*x)/x - b*d**2*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) - 2*b* d*e*Piecewise((2*c**3*sqrt(c**2*x**2 - 1)/(3*x) + c*sqrt(c**2*x**2 - 1)/(3 *x**3), Abs(c**2*x**2) > 1), (2*I*c**3*sqrt(-c**2*x**2 + 1)/(3*x) + I*c*sq rt(-c**2*x**2 + 1)/(3*x**3), True))/(3*c)
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b e^{2} - \frac {1}{75} \, b d^{2} {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} + \frac {2}{9} \, b d e {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a e^{2}}{x} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {a d^{2}}{5 \, x^{5}} \]
-(c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*b*e^2 - 1/75*b*d^2*((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) + 2/9*b*d*e*((c^4*(-1/(c^2*x^2) + 1 )^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c - 3*arccsc(c*x)/x^3) - a*e^2/x - 2/3*a*d*e/x^3 - 1/5*a*d^2/x^5
Time = 0.30 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.72 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {1}{225} \, {\left (9 \, b c^{4} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 30 \, b c^{4} d^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 45 \, b c^{4} d^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {90 \, b c^{3} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - 50 \, b c^{2} d e {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 150 \, b c^{2} d e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {150 \, b c d e {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {150 \, b c d e \arcsin \left (\frac {1}{c x}\right )}{x} + 225 \, b e^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {225 \, b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {225 \, a e^{2}}{c x} + \frac {150 \, a d e}{c x^{3}} + \frac {45 \, a d^{2}}{c x^{5}}\right )} c \]
-1/225*(9*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1) - 30*b*c^4* d^2*(-1/(c^2*x^2) + 1)^(3/2) + 45*b*c^3*d^2*(1/(c^2*x^2) - 1)^2*arcsin(1/( c*x))/x + 45*b*c^4*d^2*sqrt(-1/(c^2*x^2) + 1) + 90*b*c^3*d^2*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 50*b*c^2*d*e*(-1/(c^2*x^2) + 1)^(3/2) + 45*b*c^3* d^2*arcsin(1/(c*x))/x + 150*b*c^2*d*e*sqrt(-1/(c^2*x^2) + 1) + 150*b*c*d*e *(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x + 150*b*c*d*e*arcsin(1/(c*x))/x + 225 *b*e^2*sqrt(-1/(c^2*x^2) + 1) + 225*b*e^2*arcsin(1/(c*x))/(c*x) + 225*a*e^ 2/(c*x) + 150*a*d*e/(c*x^3) + 45*a*d^2/(c*x^5))*c
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]