\(\int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx\) [11]

Optimal result

Integrand size = 12, antiderivative size = 60 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=-\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \csc ^{-1}(c x)}{3 x^3} \]

[Out]

1/9*b*c^3*(1-1/c^2/x^2)^(3/2)+1/3*(-a-b*arccsc(c*x))/x^3-1/3*b*c^3*(1-1/c^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5329, 272, 45} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=-\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \]

[In]

Int[(a + b*ArcCsc[c*x])/x^4,x]

[Out]

-1/3*(b*c^3*Sqrt[1 - 1/(c^2*x^2)]) + (b*c^3*(1 - 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsc[c*x])/(3*x^3)

Rule 45

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 272

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 5329

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*} \text {integral}& = -\frac {a+b \csc ^{-1}(c x)}{3 x^3}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^5} \, dx}{3 c} \\ & = -\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{6 c} \\ & = -\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {b \text {Subst}\left (\int \left (\frac {c^2}{\sqrt {1-\frac {x}{c^2}}}-c^2 \sqrt {1-\frac {x}{c^2}}\right ) \, dx,x,\frac {1}{x^2}\right )}{6 c} \\ & = -\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \csc ^{-1}(c x)}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=-\frac {a}{3 x^3}+b \left (-\frac {2 c^3}{9}-\frac {c}{9 x^2}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{3 x^3} \]

[In]

Integrate[(a + b*ArcCsc[c*x])/x^4,x]

[Out]

-1/3*a/x^3 + b*((-2*c^3)/9 - c/(9*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(3*x^3)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18

method	result	size
parts	\(-\frac {a}{3 x^{3}}+b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\)	\(71\)
derivativedivides	\(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(75\)
default	\(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(75\)

[In]

int((a+b*arccsc(c*x))/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a/x^3+b*c^3*(-1/3/c^3/x^3*arccsc(c*x)-1/9*(c^2*x^2-1)*(2*c^2*x^2+1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c^4/x^4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=-\frac {3 \, b \operatorname {arccsc}\left (c x\right ) + {\left (2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} x^{2} - 1} + 3 \, a}{9 \, x^{3}} \]

[In]

integrate((a+b*arccsc(c*x))/x^4,x, algorithm="fricas")

[Out]

-1/9*(3*b*arccsc(c*x) + (2*b*c^2*x^2 + b)*sqrt(c^2*x^2 - 1) + 3*a)/x^3

Sympy [A] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.87 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=- \frac {a}{3 x^{3}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]

[In]

integrate((a+b*acsc(c*x))/x**4,x)

[Out]

-a/(3*x**3) - b*acsc(c*x)/(3*x**3) - b*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*sqrt(-c**2*x**2 + 1)/(3*x**3), True))/(
3*c)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=\frac {1}{9} \, b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a}{3 \, x^{3}} \]

[In]

integrate((a+b*arccsc(c*x))/x^4,x, algorithm="maxima")

[Out]

1/9*b*((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) - 1/3*a/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.45 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=\frac {1}{9} \, {\left (b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {3 \, b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {3 \, b c \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {3 \, a}{c x^{3}}\right )} c \]

[In]

integrate((a+b*arccsc(c*x))/x^4,x, algorithm="giac")

[Out]

1/9*(b*c^2*(-1/(c^2*x^2) + 1)^(3/2) - 3*b*c^2*sqrt(-1/(c^2*x^2) + 1) - 3*b*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))
/x - 3*b*c*arcsin(1/(c*x))/x - 3*a/(c*x^3))*c

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^4} \,d x \]

[In]

int((a + b*asin(1/(c*x)))/x^4,x)

[Out]

int((a + b*asin(1/(c*x)))/x^4, x)