3.8 \(\int \frac{\text{sech}^{-1}(a+b x)}{x^4} \, dx\)

Optimal. Leaf size=197 \[ -\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}+\frac{\left (6 a^4-5 a^2+2\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}+\frac{b \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a \left (1-a^2\right ) x^2}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3} \]

[Out]

(b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*a*(1 - a^2)*x^2) - ((2 - 5*a^2)*b^2*Sqrt[(1 - a - b*x)/
(1 + a + b*x)]*(1 + a + b*x))/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcSech[a + b*x])/(3*a^3) - ArcSech[a + b*x]/(3*x^3
) + ((2 - 5*a^2 + 6*a^4)*b^3*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/(3*a^3*(1 - a^2)^(5/
2))

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Rubi [A]  time = 0.315907, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {6321, 5468, 3785, 4060, 3919, 3831, 2659, 208} \[ -\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}+\frac{\left (6 a^4-5 a^2+2\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}+\frac{b \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a \left (1-a^2\right ) x^2}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a + b*x]/x^4,x]

[Out]

(b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*a*(1 - a^2)*x^2) - ((2 - 5*a^2)*b^2*Sqrt[(1 - a - b*x)/
(1 + a + b*x)]*(1 + a + b*x))/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcSech[a + b*x])/(3*a^3) - ArcSech[a + b*x]/(3*x^3
) + ((2 - 5*a^2 + 6*a^4)*b^3*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/(3*a^3*(1 - a^2)^(5/
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 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

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^4} \, dx &=-\left (b^3 \operatorname{Subst}\left (\int \frac{x \text{sech}(x) \tanh (x)}{(-a+\text{sech}(x))^4} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{1}{3} b^3 \operatorname{Subst}\left (\int \frac{1}{(-a+\text{sech}(x))^3} \, 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)}{6 a \left (1-a^2\right ) x^2}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{2 \left (1-a^2\right )-2 a \text{sech}(x)-\text{sech}^2(x)}{(-a+\text{sech}(x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{6 a \left (1-a^2\right )}\\ &=\frac{b \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) \text{sech}(x)}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{6 a^2 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{\text{sech}(x)}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{\left (\left (2-5 a^2+6 a^4\right ) b^3\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 )}{3 a^3 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac{\left (2-5 a^2\right ) b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac{b^3 \text{sech}^{-1}(a+b x)}{3 a^3}-\frac{\text{sech}^{-1}(a+b x)}{3 x^3}+\frac{\left (2-5 a^2+6 a^4\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{1+a} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.402399, size = 368, normalized size = 1.87 \[ \frac{1}{6} \left (\frac{b \sqrt{-\frac{a+b x-1}{a+b x+1}} \left (a^2 \left (5 b^2 x^2+5 b x+1\right )+a^3 (4 b x-1)-a^4-a b x+a-2 b x (b x+1)\right )}{(a-1)^2 a^2 (a+1)^2 x^2}-\frac{\left (6 a^4-5 a^2+2\right ) b^3 \log (x)}{a^3 \left (1-a^2\right )^{5/2}}+\frac{2 b^3 \log (a+b x)}{a^3}-\frac{2 b^3 \log \left (a \sqrt{-\frac{a+b x-1}{a+b x+1}}+b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{-\frac{a+b x-1}{a+b x+1}}+1\right )}{a^3}+\frac{\left (6 a^4-5 a^2+2\right ) b^3 \log \left (\sqrt{1-a^2} a \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} \sqrt{-\frac{a+b x-1}{a+b x+1}}-a^2-a b x+1\right )}{a^3 \left (1-a^2\right )^{5/2}}-\frac{2 \text{sech}^{-1}(a+b x)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[a + b*x]/x^4,x]

[Out]

((b*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(a - a^4 - a*b*x - 2*b*x*(1 + b*x) + a^3*(-1 + 4*b*x) + a^2*(1 + 5*b
*x + 5*b^2*x^2)))/((-1 + a)^2*a^2*(1 + a)^2*x^2) - (2*ArcSech[a + b*x])/x^3 - ((2 - 5*a^2 + 6*a^4)*b^3*Log[x])
/(a^3*(1 - a^2)^(5/2)) + (2*b^3*Log[a + b*x])/a^3 - (2*b^3*Log[1 + Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*S
qrt[-((-1 + a + b*x)/(1 + a + b*x))] + b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/a^3 + ((2 - 5*a^2 + 6*a^4)*
b^3*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]*Sqrt[-((-1 + a
+ b*x)/(1 + a + b*x))] + Sqrt[1 - a^2]*b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/(a^3*(1 - a^2)^(5/2)))/6

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Maple [B]  time = 0.256, size = 1946, normalized size = 9.9 \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^4,x)

[Out]

-1/3*arcsech(b*x+a)/x^3-1/3*b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)^2/(1+a)/(a-1)/(
1-(b*x+a)^2)^(1/2)*a^3*arctanh(1/(1-(b*x+a)^2)^(1/2))-b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)
*x/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)*a*(-a^2+1)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+
a)+1)/b/x)-1/3*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1
/2)*a^4*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)/(a^2-1)^2/(1+a
)/(a-1)/(1-(b*x+a)^2)^(1/2)*a^2*(-a^2+1)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)+b^4*
(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)*a*arctanh(1/(
1-(b*x+a)^2)^(1/2))+5/6*b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a^2-1)^2/(1+a)/(a-1)/(1-(b
*x+a)^2)^(1/2)/a*(-a^2+1)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)+b^3*(-(b*x+a-1)/(b*
x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)*a^2*arctanh(1/(1-(b*x+a)^2)^(1
/2))+5/6*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)*a^2+5/6*b^3*(-(b*x+a-1
)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)*(-a^2+1)^(1/2)*ln(2*((-a^
2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)-b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)*x/(a
^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)/a*arctanh(1/(1-(b*x+a)^2)^(1/2))+2/3*b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((
b*x+a+1)/(b*x+a))^(1/2)/x/(a^2-1)^2/(1+a)/(a-1)*a^3-1/3*b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/
2)*x/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)/a^3*(-a^2+1)^(1/2)*ln(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(
b*x+a)+1)/b/x)-b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1
/2)*arctanh(1/(1-(b*x+a)^2)^(1/2))-7/6*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a
)/(a-1)-1/6*b*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/x^2/(a^2-1)^2/(1+a)/(a-1)*a^4-1/3*b^3*(-(b*
x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)/a^2*(-a^2+1)^(1/2)*l
n(2*((-a^2+1)^(1/2)*(1-(b*x+a)^2)^(1/2)-a*(b*x+a)+1)/b/x)+1/3*b^4*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a
))^(1/2)*x/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)/a^3*arctanh(1/(1-(b*x+a)^2)^(1/2))-5/6*b^2*(-(b*x+a-1)/(b
*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/x/(a^2-1)^2/(1+a)/(a-1)*a+1/3*b^3*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)
/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/(1-(b*x+a)^2)^(1/2)/a^2*arctanh(1/(1-(b*x+a)^2)^(1/2))+1/3*b^3*(-(b*x+a-
1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/(a^2-1)^2/(1+a)/(a-1)/a^2+1/3*b*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a
+1)/(b*x+a))^(1/2)/x^2/(a^2-1)^2/(1+a)/(a-1)*a^2+1/6*b^2*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/
x/(a^2-1)^2/(1+a)/(a-1)/a-1/6*b*(-(b*x+a-1)/(b*x+a))^(1/2)*((b*x+a+1)/(b*x+a))^(1/2)/x^2/(a^2-1)^2/(1+a)/(a-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (6 \, a^{4} b^{3} - 3 \, a^{2} b^{3} + b^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )}} - \frac{{\left (a^{6} b^{3} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{3} - a^{3} b^{3}\right )} x^{3} \log \left (b x + a + 1\right ) +{\left (a^{6} b^{3} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{3} + a^{3} b^{3}\right )} x^{3} \log \left (-b x - a + 1\right ) - 2 \,{\left (3 \, a^{5} b^{2} - 4 \, a^{3} b^{2} + a b^{2}\right )} x^{2} +{\left (a^{6} b - 2 \, a^{4} b + a^{2} b\right )} x + 2 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\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^{9} - 3 \, a^{7} + 3 \, a^{5} +{\left (a^{6} b^{3} - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{3} - b^{3}\right )} x^{3} - a^{3}\right )} \log \left (b x + a\right ) - 2 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}} - \int \frac{b^{2} x + a b}{3 \,{\left (b^{2} x^{5} + 2 \, a b x^{4} +{\left (a^{2} - 1\right )} x^{3} +{\left (b^{2} x^{5} + 2 \, a b x^{4} +{\left (a^{2} - 1\right )} x^{3}\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^4,x, algorithm="maxima")

[Out]

1/3*(6*a^4*b^3 - 3*a^2*b^3 + b^3)*log(x)/(a^9 - 3*a^7 + 3*a^5 - a^3) - 1/6*((a^6*b^3 - 3*a^5*b^3 + 3*a^4*b^3 -
a^3*b^3)*x^3*log(b*x + a + 1) + (a^6*b^3 + 3*a^5*b^3 + 3*a^4*b^3 + a^3*b^3)*x^3*log(-b*x - a + 1) - 2*(3*a^5*
b^2 - 4*a^3*b^2 + a*b^2)*x^2 + (a^6*b - 2*a^4*b + a^2*b)*x + 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*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^9 - 3*a^7 + 3*a^5 + (a^6*
b^3 - 3*a^4*b^3 + 3*a^2*b^3 - b^3)*x^3 - a^3)*log(b*x + a) - 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*log(b*x + a))/((a^9
- 3*a^7 + 3*a^5 - a^3)*x^3) - integrate(1/3*(b^2*x + a*b)/(b^2*x^5 + 2*a*b*x^4 + (a^2 - 1)*x^3 + (b^2*x^5 + 2
*a*b*x^4 + (a^2 - 1)*x^3)*e^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1))), x)

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Fricas [B]  time = 2.66797, size = 2163, normalized size = 10.98 \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^4,x, algorithm="fricas")

[Out]

[-1/12*((6*a^4 - 5*a^2 + 2)*sqrt(-a^2 + 1)*b^3*x^3*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) + 2*(a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*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 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*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) + 4*(a^9 - 3*a^7 + 3*a^5 - a^3)*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*((5*a^5 - 7*a^3 + 2*a)*b^3*x^3 + (4*a^6
- 5*a^4 + a^2)*b^2*x^2 - (a^7 - 2*a^5 + a^3)*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^9 - 3*a^7 + 3*a^5 - a^3)*x^3), -1/6*((6*a^4 - 5*a^2 + 2)*sqrt(a^2 - 1)*b^3*x^3*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^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*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^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*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^9 - 3*a^7 + 3*a^5 - a^3)*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)) - ((5*a^5 - 7*a^3 + 2*
a)*b^3*x^3 + (4*a^6 - 5*a^4 + a^2)*b^2*x^2 - (a^7 - 2*a^5 + a^3)*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^9 - 3*a^7 + 3*a^5 - a^3)*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}{\left (a + b x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(b*x+a)/x**4,x)

[Out]

Integral(asech(a + b*x)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (b x + a\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)/x^4,x, algorithm="giac")

[Out]

integrate(arcsech(b*x + a)/x^4, x)