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

Optimal result

Integrand size = 12, antiderivative size = 31 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5328, 267} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{x} \]

[In]

Int[(a + b*ArcSec[c*x])/x^2,x]

[Out]

b*c*Sqrt[1 - 1/(c^2*x^2)] - (a + b*ArcSec[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 5328

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSec[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 \sec ^{-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 \sec ^{-1}(c x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=-\frac {a}{x}+b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \sec ^{-1}(c x)}{x} \]

[In]

Integrate[(a + b*ArcSec[c*x])/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87

method	result	size
parts	\(-\frac {a}{x}+b c \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\)	\(58\)
derivativedivides	\(c \left (-\frac {a}{c x}+b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\)	\(62\)
default	\(c \left (-\frac {a}{c x}+b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\)	\(62\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=-\frac {b \operatorname {arcsec}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1} b + a}{x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=\begin {cases} - \frac {a}{x} + b c \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b \operatorname {asec}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a + \tilde {\infty } b}{x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx={\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx={\left (b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {b \arccos \left (\frac {1}{c x}\right )}{c x} - \frac {a}{c x}\right )} c \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2} \, dx=b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {a}{x}-\frac {b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x} \]

[In]

int((a + b*acos(1/(c*x)))/x^2,x)

[Out]

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