\(\int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx\) [10]

Optimal result

Integrand size = 12, antiderivative size = 51 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x)-\frac {a+b \csc ^{-1}(c x)}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5329, 342, 327, 222} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=-\frac {a+b \csc ^{-1}(c x)}{2 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x) \]

[In]

Int[(a + b*ArcCsc[c*x])/x^3,x]

[Out]

-1/4*(b*c*Sqrt[1 - 1/(c^2*x^2)])/x + (b*c^2*ArcCsc[c*x])/4 - (a + b*ArcCsc[c*x])/(2*x^2)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

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

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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)}{2 x^2}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4} \, dx}{2 c} \\ & = -\frac {a+b \csc ^{-1}(c x)}{2 x^2}+\frac {b \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c} \\ & = -\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {a+b \csc ^{-1}(c x)}{2 x^2}+\frac {1}{4} (b c) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x)-\frac {a+b \csc ^{-1}(c x)}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{4 x}-\frac {b \csc ^{-1}(c x)}{2 x^2}+\frac {1}{4} b c^2 \arcsin \left (\frac {1}{c x}\right ) \]

[In]

Integrate[(a + b*ArcCsc[c*x])/x^3,x]

[Out]

-1/2*a/x^2 - (b*c*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*ArcCsc[c*x])/(2*x^2) + (b*c^2*ArcSin[1/(c*x)])/4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(46)=92\).

Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.88

method	result	size
parts	\(-\frac {a}{2 x^{2}}+b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\)	\(96\)
derivativedivides	\(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\)	\(99\)
default	\(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\)	\(99\)

[In]

int((a+b*arccsc(c*x))/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=\frac {{\left (b c^{2} x^{2} - 2 \, b\right )} \operatorname {arccsc}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1} b - 2 \, a}{4 \, x^{2}} \]

[In]

integrate((a+b*arccsc(c*x))/x^3,x, algorithm="fricas")

[Out]

1/4*((b*c^2*x^2 - 2*b)*arccsc(c*x) - sqrt(c^2*x^2 - 1)*b - 2*a)/x^2

Sympy [A] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.37 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=- \frac {a}{2 x^{2}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{2 x^{2}} - \frac {b \left (\begin {cases} \frac {i c^{3} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c^{2}}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{2 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {c^{3} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} + \frac {c^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{2 c} \]

[In]

integrate((a+b*acsc(c*x))/x**3,x)

[Out]

-a/(2*x**2) - b*acsc(c*x)/(2*x**2) - b*Piecewise((I*c**3*acosh(1/(c*x))/2 - I*c**2/(2*x*sqrt(-1 + 1/(c**2*x**2
))) + I/(2*x**3*sqrt(-1 + 1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (-c**3*asin(1/(c*x))/2 + c**2*sqrt(1 - 1/(c*
*2*x**2))/(2*x), True))/(2*c)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=\frac {1}{4} \, b {\left (\frac {\frac {c^{4} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}{c} - \frac {2 \, \operatorname {arccsc}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]

[In]

integrate((a+b*arccsc(c*x))/x^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=-\frac {1}{4} \, {\left (2 \, b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 2 \, a c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + b c \arcsin \left (\frac {1}{c x}\right ) + \frac {b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]

[In]

integrate((a+b*arccsc(c*x))/x^3,x, algorithm="giac")

[Out]

-1/4*(2*b*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 2*a*c*(1/(c^2*x^2) - 1) + b*c*arcsin(1/(c*x)) + b*sqrt(-1/(c^2
*x^2) + 1)/x)*c

Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx=-\frac {a}{2\,x^2}-\frac {b\,c^2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\left (\frac {2}{c^2\,x^2}-1\right )}{4}-\frac {b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}}{4\,x} \]

[In]

int((a + b*asin(1/(c*x)))/x^3,x)

[Out]

- a/(2*x^2) - (b*c^2*asin(1/(c*x))*(2/(c^2*x^2) - 1))/4 - (b*c*(1 - 1/(c^2*x^2))^(1/2))/(4*x)