3.1.39 \(\int \frac {(a+b \text {sech}^{-1}(c x))^2}{x^3} \, dx\) [39]

Optimal. Leaf size=118 \[ -\frac {b^2 (1-c x) (1+c x)}{4 x^2}-\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2} \]

[Out]

-1/4*b^2*(-c*x+1)*(c*x+1)/x^2-1/2*a*b*c^2*arcsech(c*x)-1/4*b^2*c^2*arcsech(c*x)^2-1/2*(-c*x+1)*(c*x+1)*(a+b*ar
csech(c*x))^2/x^2+1/2*b*(c*x+1)*(a+b*arcsech(c*x))*((-c*x+1)/(c*x+1))^(1/2)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6420, 5554, 3391} \begin {gather*} -\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2-\frac {b^2 (1-c x) (c x+1)}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^2/x^3,x]

[Out]

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

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 5554

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

Rule 6420

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rubi steps

\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx &=-\left (c^2 \text {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \text {Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {b^2 (1-c x) (1+c x)}{4 x^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} \left (b c^2\right ) \text {Subst}\left (\int (a+b x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {b^2 (1-c x) (1+c x)}{4 x^2}-\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 183, normalized size = 1.55 \begin {gather*} \frac {-2 a^2-b^2+2 a b \sqrt {\frac {1-c x}{1+c x}}+2 a b c x \sqrt {\frac {1-c x}{1+c x}}+2 b \left (-2 a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 \left (-2+c^2 x^2\right ) \text {sech}^{-1}(c x)^2-2 a b c^2 x^2 \log (x)+2 a b c^2 x^2 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])^2/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 192, normalized size = 1.63

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{2 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(192\)
default	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{2 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^2/x^3,x, algorithm="maxima")

[Out]

-1/4*a*b*((2*c^4*x*sqrt(1/(c^2*x^2) - 1)/(c^2*x^2*(1/(c^2*x^2) - 1) - 1) - c^3*log(c*x*sqrt(1/(c^2*x^2) - 1) +
1) + c^3*log(c*x*sqrt(1/(c^2*x^2) - 1) - 1))/c + 4*arcsech(c*x)/x^2) + b^2*integrate(log(sqrt(1/(c*x) + 1)*sq
rt(1/(c*x) - 1) + 1/(c*x))^2/x^3, x) - 1/2*a^2/x^2

________________________________________________________________________________________

Fricas [A]
time = 0.42, size = 165, normalized size = 1.40 \begin {gather*} \frac {2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, {\left (a b c^{2} x^{2} + b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, a b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^2/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**2/x**3,x)

[Out]

Integral((a + b*asech(c*x))**2/x**3, 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))^2/x^3,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)^2/x^3, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))^2/x^3,x)

[Out]

int((a + b*acosh(1/(c*x)))^2/x^3, x)

________________________________________________________________________________________