3.1.25 \(\int (a+b \text {sech}^{-1}(c x)) \, dx\) [25]

Optimal. Leaf size=40 \[ a x+b x \text {sech}^{-1}(c x)+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6412, 222} \begin {gather*} a x+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{c}+b x \text {sech}^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSech[c*x],x]

[Out]

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

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 6412

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 77, normalized size = 1.92 \begin {gather*} a x+b x \text {sech}^{-1}(c x)-\frac {2 b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c-c^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSech[c*x],x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 42, normalized size = 1.05

method	result	size
default	\(a x +b x \,\mathrm {arcsech}\left (c x \right )-\frac {b \arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{c}\)	\(42\)
derivativedivides	\(\frac {a c x +\mathrm {arcsech}\left (c x \right ) b c x -\arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right ) b}{c}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arcsech(c*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 31, normalized size = 0.78 \begin {gather*} a x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (20) = 40\).
time = 0.36, size = 119, normalized size = 2.98 \begin {gather*} \frac {a c x - b c \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, b \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + {\left (b c x - b c\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {asech}{\left (c x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*asech(c*x), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*arcsech(c*x) + a, x)

________________________________________________________________________________________

Mupad [B]
time = 1.34, size = 44, normalized size = 1.10 \begin {gather*} a\,x+b\,x\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )+\frac {b\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*acosh(1/(c*x)),x)

[Out]

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

________________________________________________________________________________________