3.1.9 \(\int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx\) [9]

Optimal. Leaf size=32 \[ -b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5329, 267} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{x}-b c \sqrt {1-\frac {1}{c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/x^2,x]

[Out]

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

Rule 267

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

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*} \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{x}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^3} \, dx}{c}\\ &=-b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 41, normalized size = 1.28 \begin {gather*} -\frac {a}{x}-b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])/x^2,x]

[Out]

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

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Maple [A]
time = 0.15, size = 63, normalized size = 1.97

method	result	size
derivativedivides	\(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\)	\(63\)
default	\(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 33, normalized size = 1.03 \begin {gather*} -{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 26, normalized size = 0.81 \begin {gather*} -\frac {b \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b + a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.48, size = 37, normalized size = 1.16 \begin {gather*} \begin {cases} - \frac {a}{x} - b c \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a + \tilde {\infty } b}{x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x - b*c*sqrt(1 - 1/(c**2*x**2)) - b*acsc(c*x)/x, Ne(c, 0)), (-(a + zoo*b)/x, True))

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Giac [A]
time = 0.44, size = 42, normalized size = 1.31 \begin {gather*} -{\left (b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a}{c x}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.64, size = 37, normalized size = 1.16 \begin {gather*} -\frac {a}{x}-b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/x^2,x)

[Out]

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

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