3.7 \(\int \frac{\text{sech}^{-1}(a+b x)}{x^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{b^2 \text{sech}^{-1}(a+b x)}{2 a^2}-\frac{\left (1-2 a^2\right ) b^2 \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{b \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) x}-\frac{\text{sech}^{-1}(a+b x)}{2 x^2} \]

[Out]

(b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(2*a*(1 - a^2)*x) + (b^2*ArcSech[a + b*x])/(2*a^2) - ArcSe
ch[a + b*x]/(2*x^2) - ((1 - 2*a^2)*b^2*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/(a^2*(1 -
a^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.203781, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6321, 5468, 3785, 3919, 3831, 2659, 208} \[ \frac{b^2 \text{sech}^{-1}(a+b x)}{2 a^2}-\frac{\left (1-2 a^2\right ) b^2 \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{b \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) x}-\frac{\text{sech}^{-1}(a+b x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a + b*x]/x^3,x]

[Out]

(b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(2*a*(1 - a^2)*x) + (b^2*ArcSech[a + b*x])/(2*a^2) - ArcSe
ch[a + b*x]/(2*x^2) - ((1 - 2*a^2)*b^2*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/(a^2*(1 -
a^2)^(3/2))

Rule 6321

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Dist[(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ
[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 5468

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

Rule 3785

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n +
1))/(a*d*(n + 1)*(a^2 - b^2)), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

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

Mathematica [B]  time = 0.907066, size = 315, normalized size = 2.37 \[ \frac{1}{2} \left (-\frac{\left (2 a^2-1\right ) b^2 \log (x)}{a^2 \left (1-a^2\right )^{3/2}}-\frac{b^2 \log (a+b x)}{a^2}+\frac{b^2 \log \left (a \sqrt{-\frac{a+b x-1}{a+b x+1}}+b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{-\frac{a+b x-1}{a+b x+1}}+1\right )}{a^2}+\frac{\left (2 a^2-1\right ) b^2 \log \left (\sqrt{1-a^2} a \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} \sqrt{-\frac{a+b x-1}{a+b x+1}}-a^2-a b x+1\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{\text{sech}^{-1}(a+b x)}{x^2}-\frac{b \sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1)}{(a-1) a (a+1) x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[a + b*x]/x^3,x]

[Out]

(-((b*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x))/((-1 + a)*a*(1 + a)*x)) - ArcSech[a + b*x]/x^2 - ((
-1 + 2*a^2)*b^2*Log[x])/(a^2*(1 - a^2)^(3/2)) - (b^2*Log[a + b*x])/a^2 + (b^2*Log[1 + Sqrt[-((-1 + a + b*x)/(1
+ a + b*x))] + a*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/a^2 + ((
-1 + 2*a^2)*b^2*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*Sqrt[1 - a^2]*Sq
rt[-((-1 + a + b*x)/(1 + a + b*x))] + Sqrt[1 - a^2]*b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/(a^2*(1 - a^2)
^(3/2)))/2

________________________________________________________________________________________

Maple [B]  time = 0.259, size = 1233, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(b*x+a)/x^3,x)

[Out]

-1/2*arcsech(b*x+a)/x^2+1/2*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)/(1+a)*a^2/(a-1)
/(1-(b*x+a)^2)^(1/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))+1/2*b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/
2)/(a^2-1)/(1+a)*a^3/(a-1)/(1-(b*x+a)^2)^(1/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))+b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*(
(b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*
(b*x+a)+1)/b/x)*(-a^2+1)^(1/2)-b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)/(1+a)/(a-1)/
(1-(b*x+a)^2)^(1/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))+b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a
^2-1)/(1+a)*a/(a-1)/(1-(b*x+a)^2)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)*(-a^2+1)^(1
/2)-b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)/(1+a)*a/(a-1)/(1-(b*x+a)^2)^(1/2)*arctanh
(1/(1-(b*x+a)^2)^(1/2))-1/2*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)/(1+a)/a^2/(a-1)
/(1-(b*x+a)^2)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)*(-a^2+1)^(1/2)-1/2*b^2*(-(b*x+
a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)/(1+a)*a/(a-1)+1/2*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a
+1)/(b*x+a))^(1/2)*x/(a^2-1)/(1+a)/a^2/(a-1)/(1-(b*x+a)^2)^(1/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))-1/2*b^2*(-(b*x
+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)/(1+a)/a/(a-1)/(1-(b*x+a)^2)^(1/2)*ln(2*((-a^2+1)^(1/2)*
(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)*(-a^2+1)^(1/2)-1/2*b*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2
)/x/(a^2-1)/(1+a)*a^2/(a-1)+1/2*b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)/(1+a)/a/(a-1)
/(1-(b*x+a)^2)^(1/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))+1/2*b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/
2)/(a^2-1)/(1+a)/a/(a-1)+1/2*b*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/x/(a^2-1)/(1+a)/(a-1)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, a^{2} b^{2} - b^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )}} + \frac{{\left (a^{4} b^{2} - 2 \, a^{3} b^{2} + a^{2} b^{2}\right )} x^{2} \log \left (b x + a + 1\right ) +{\left (a^{4} b^{2} + 2 \, a^{3} b^{2} + a^{2} b^{2}\right )} x^{2} \log \left (-b x - a + 1\right ) - 2 \,{\left (a^{3} b - a b\right )} x - 2 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\sqrt{b x + a + 1} \sqrt{-b x - a + 1} b x + \sqrt{b x + a + 1} \sqrt{-b x - a + 1} a + b x + a\right ) + 2 \,{\left (a^{6} - 2 \, a^{4} -{\left (a^{4} b^{2} - 2 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (b x + a\right )}{4 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}} - \int \frac{b^{2} x + a b}{2 \,{\left (b^{2} x^{4} + 2 \, a b x^{3} +{\left (a^{2} - 1\right )} x^{2} +{\left (b^{2} x^{4} + 2 \, a b x^{3} +{\left (a^{2} - 1\right )} x^{2}\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (-b x - a + 1\right )\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*a^2*b^2 - b^2)*log(x)/(a^6 - 2*a^4 + a^2) + 1/4*((a^4*b^2 - 2*a^3*b^2 + a^2*b^2)*x^2*log(b*x + a + 1)
+ (a^4*b^2 + 2*a^3*b^2 + a^2*b^2)*x^2*log(-b*x - a + 1) - 2*(a^3*b - a*b)*x - 2*(a^6 - 2*a^4 + a^2)*log(sqrt(b
*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*a + b*x + a) + 2*(a^6 - 2*a^4 - (a^4
*b^2 - 2*a^2*b^2 + b^2)*x^2 + a^2)*log(b*x + a) + 2*(a^6 - 2*a^4 + a^2)*log(b*x + a))/((a^6 - 2*a^4 + a^2)*x^2
) - integrate(1/2*(b^2*x + a*b)/(b^2*x^4 + 2*a*b*x^3 + (a^2 - 1)*x^2 + (b^2*x^4 + 2*a*b*x^3 + (a^2 - 1)*x^2)*e
^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1))), x)

________________________________________________________________________________________

Fricas [B]  time = 2.4814, size = 1912, normalized size = 14.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((2*a^2 - 1)*sqrt(-a^2 + 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 + 2*(a*b^
2*x^2 + a^3 + (2*a^2 - 1)*b*x - a)*sqrt(-a^2 + 1)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2
)) + 2)/x^2) - (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x
+ a^2)) + 1)/x) + (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a
*b*x + a^2)) - 1)/x) + 2*(a^6 - 2*a^4 + a^2)*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a
*b*x + a^2)) + 1)/(b*x + a)) + 2*((a^3 - a)*b^2*x^2 + (a^4 - a^2)*b*x)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^
2*x^2 + 2*a*b*x + a^2)))/((a^6 - 2*a^4 + a^2)*x^2), 1/4*(2*(2*a^2 - 1)*sqrt(a^2 - 1)*b^2*x^2*arctan((a*b^2*x^2
+ a^3 + (2*a^2 - 1)*b*x - a)*sqrt(a^2 - 1)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2))/((a
^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^2*x^
2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/x) - (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^
2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x) - 2*(a^6 - 2*a^4 + a^2)*log(((b*x + a)*sqrt(-(b^
2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - 2*((a^3 - a)*b^2*x^2 + (a^4 - a^2)*b*x
)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)))/((a^6 - 2*a^4 + a^2)*x^2)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asech(a + b*x)/x**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (b x + a\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsech(b*x + a)/x^3, x)