3.7 \(\int (a+b \sec ^{-1}(c x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0203169, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5214, 266, 63, 208} \[ a x-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c}+b x \sec ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSec[c*x],x]

[Out]

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

Rule 5214

Int[ArcSec[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcSec[c*x], x] - Dist[1/c, Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x], x
] /; FreeQ[c, x]

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \left (a+b \sec ^{-1}(c x)\right ) \, dx &=a x+b \int \sec ^{-1}(c x) \, dx\\ &=a x+b x \sec ^{-1}(c x)-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \sec ^{-1}(c x)+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c}\\ &=a x+b x \sec ^{-1}(c x)-(b c) \operatorname{Subst}\left (\int \frac{1}{c^2-c^2 x^2} \, dx,x,\sqrt{1-\frac{1}{c^2 x^2}}\right )\\ &=a x+b x \sec ^{-1}(c x)-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c}\\ \end{align*}

Mathematica [A]  time = 0.0628324, size = 59, normalized size = 1.84 \[ a x-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{\sqrt{c^2 x^2-1}}+b x \sec ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSec[c*x],x]

[Out]

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

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Maple [A]  time = 0.155, size = 38, normalized size = 1.2 \begin{align*} ax+bx{\rm arcsec} \left (cx\right )-{\frac{b}{c}\ln \left ( cx+cx\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arcsec(c*x),x)

[Out]

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

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Maxima [A]  time = 0.961775, size = 72, normalized size = 2.25 \begin{align*} a x + \frac{{\left (2 \, c x \operatorname{arcsec}\left (c x\right ) - \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )\right )} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.64069, size = 154, normalized size = 4.81 \begin{align*} \frac{a c x + 2 \, b c \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (b c x - b c\right )} \operatorname{arcsec}\left (c x\right ) + b \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{asec}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*asec(c*x), x)

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Giac [A]  time = 1.13203, size = 62, normalized size = 1.94 \begin{align*}{\left (x \arccos \left (\frac{1}{c x}\right ) + \frac{c \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{{\left | c \right |}^{2} \mathrm{sgn}\left (x\right )}\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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