3.13 \(\int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx\)

Optimal. Leaf size=82 \[ -\frac {a+b \csc ^{-1}(c x)}{5 x^5}-\frac {1}{25} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{5/2}+\frac {2}{15} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{5} b c^5 \sqrt {1-\frac {1}{c^2 x^2}} \]

[Out]

2/15*b*c^5*(1-1/c^2/x^2)^(3/2)-1/25*b*c^5*(1-1/c^2/x^2)^(5/2)+1/5*(-a-b*arccsc(c*x))/x^5-1/5*b*c^5*(1-1/c^2/x^
2)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5221, 266, 43} \[ -\frac {a+b \csc ^{-1}(c x)}{5 x^5}-\frac {1}{25} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{5/2}+\frac {2}{15} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{5} b c^5 \sqrt {1-\frac {1}{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/x^6,x]

[Out]

-(b*c^5*Sqrt[1 - 1/(c^2*x^2)])/5 + (2*b*c^5*(1 - 1/(c^2*x^2))^(3/2))/15 - (b*c^5*(1 - 1/(c^2*x^2))^(5/2))/25 -
(a + b*ArcCsc[c*x])/(5*x^5)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 5221

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsc[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 \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{5 x^5}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^7} \, dx}{5 c}\\ &=-\frac {a+b \csc ^{-1}(c x)}{5 x^5}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{10 c}\\ &=-\frac {a+b \csc ^{-1}(c x)}{5 x^5}+\frac {b \operatorname {Subst}\left (\int \left (\frac {c^4}{\sqrt {1-\frac {x}{c^2}}}-2 c^4 \sqrt {1-\frac {x}{c^2}}+c^4 \left (1-\frac {x}{c^2}\right )^{3/2}\right ) \, dx,x,\frac {1}{x^2}\right )}{10 c}\\ &=-\frac {1}{5} b c^5 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{15} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{25} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{5/2}-\frac {a+b \csc ^{-1}(c x)}{5 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 69, normalized size = 0.84 \[ -\frac {a}{5 x^5}+b \left (-\frac {8 c^5}{75}-\frac {4 c^3}{75 x^2}-\frac {c}{25 x^4}\right ) \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])/x^6,x]

[Out]

-1/5*a/x^5 + b*((-8*c^5)/75 - c/(25*x^4) - (4*c^3)/(75*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/
(5*x^5)

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fricas [A]  time = 0.90, size = 50, normalized size = 0.61 \[ -\frac {15 \, b \operatorname {arccsc}\left (c x\right ) + {\left (8 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 3 \, b\right )} \sqrt {c^{2} x^{2} - 1} + 15 \, a}{75 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="fricas")

[Out]

-1/75*(15*b*arccsc(c*x) + (8*b*c^4*x^4 + 4*b*c^2*x^2 + 3*b)*sqrt(c^2*x^2 - 1) + 15*a)/x^5

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giac [B]  time = 0.14, size = 149, normalized size = 1.82 \[ -\frac {1}{75} \, {\left (3 \, b c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 10 \, b c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {15 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 15 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {30 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {15 \, b c^{3} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {15 \, a}{c x^{5}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="giac")

[Out]

-1/75*(3*b*c^4*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1) - 10*b*c^4*(-1/(c^2*x^2) + 1)^(3/2) + 15*b*c^3*(1/(c
^2*x^2) - 1)^2*arcsin(1/(c*x))/x + 15*b*c^4*sqrt(-1/(c^2*x^2) + 1) + 30*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)
)/x + 15*b*c^3*arcsin(1/(c*x))/x + 15*a/(c*x^5))*c

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maple [A]  time = 0.05, size = 83, normalized size = 1.01 \[ c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/x^6,x)

[Out]

c^5*(-1/5*a/c^5/x^5+b*(-1/5/c^5/x^5*arccsc(c*x)-1/75*(c^2*x^2-1)*(8*c^4*x^4+4*c^2*x^2+3)/((c^2*x^2-1)/c^2/x^2)
^(1/2)/c^6/x^6))

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maxima [A]  time = 0.36, size = 76, normalized size = 0.93 \[ -\frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="maxima")

[Out]

-1/75*b*((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/x^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/x^6,x)

[Out]

int((a + b*asin(1/(c*x)))/x^6, x)

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sympy [A]  time = 7.23, size = 158, normalized size = 1.93 \[ - \frac {a}{5 x^{5}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {b \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))/x**6,x)

[Out]

-a/(5*x**5) - b*acsc(c*x)/(5*x**5) - b*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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