3.80 \(\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=306 \[ -\frac{a+b \text{sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{2 d^2 \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{2 d^2 e} \]

[Out]

(b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*d*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSech[c*
x])/(2*e*(d + e*x)^2) + (b*c^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sq
rt[1 - c^2*x^2])])/(2*(c^2*d^2 - e^2)^(3/2)) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqr
t[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*d^2*Sqrt[c^2*d^2 - e^2]) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcT
anh[Sqrt[1 - c^2*x^2]])/(2*d^2*e)

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Rubi [A]  time = 0.193247, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6288, 961, 266, 63, 208, 731, 725, 204} \[ -\frac{a+b \text{sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{2 d^2 \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{2 d^2 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])/(d + e*x)^3,x]

[Out]

(b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*d*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSech[c*
x])/(2*e*(d + e*x)^2) + (b*c^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sq
rt[1 - c^2*x^2])])/(2*(c^2*d^2 - e^2)^(3/2)) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqr
t[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*d^2*Sqrt[c^2*d^2 - e^2]) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcT
anh[Sqrt[1 - c^2*x^2]])/(2*d^2*e)

Rule 6288

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a +
b*ArcSech[c*x]))/(e*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(e*(m + 1)), Int[(d + e*x)^(m + 1)
/(x*Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rule 961

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

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]

Rule 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.629015, size = 342, normalized size = 1.12 \[ \frac{1}{2} \left (-\frac{a}{e (d+e x)^2}-\frac{i b \left (2 c^2 d^2-e^2\right ) \log \left (\frac{4 d^2 e \sqrt{c^2 d^2-e^2} \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d^2-e^2}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d^2-e^2}+i c^2 d x+i e\right )}{b \left (2 c^2 d^2-e^2\right ) (d+e x)}\right )}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{b \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{d^2 e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c e x+e)}{d (c d-e) (c d+e) (d+e x)}-\frac{b \text{sech}^{-1}(c x)}{e (d+e x)^2}-\frac{b \log (x)}{d^2 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a/(e*(d + e*x)^2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(e + c*e*x))/(d*(c*d - e)*(c*d + e)*(d + e*x)) - (b*ArcSe
ch[c*x])/(e*(d + e*x)^2) - (b*Log[x])/(d^2*e) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 +
c*x)]])/(d^2*e) - (I*b*(2*c^2*d^2 - e^2)*Log[(4*d^2*e*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e
^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d^2 - e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(2*c^2*d^2 - e^2)*(d +
e*x))])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2

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Maple [B]  time = 0.274, size = 1090, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^2*a/(c*e*x+c*d)^2/e-1/2*c^2*b/(c*e*x+c*d)^2/e*arcsech(c*x)+1/2*c^4*b*(-(c*x-1)/c/x)^(1/2)*x^2*((c*x+1)/
c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/(c*d+e)/(c*d-e)/(c*e*x+c*d)*arctanh(1/(-c^2*x^2+1)^(1/2))+1/2*c^4*b/e*(-(c*x-1)/
c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)*d/(c*d+e)/(c*d-e)/(c*e*x+c*d)*arctanh(1/(-c^2*x^2+1)^(1/2)
)-c^4*b*(-(c*x-1)/c/x)^(1/2)*x^2*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/(c*d+e)/(c*d-e)/(-(c^2*d^2-e^2)/e^2)^(
1/2)/(c*e*x+c*d)*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1)^(1/2)*e+c^2*d*x+e)/(c*e*x+c*d))-c^4*b/e*(-(c*x-
1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)*d/(c*d+e)/(c*d-e)/(-(c^2*d^2-e^2)/e^2)^(1/2)/(c*e*x+c*d
)*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1)^(1/2)*e+c^2*d*x+e)/(c*e*x+c*d))-1/2*c^2*b*e^2*(-(c*x-1)/c/x)^(
1/2)*x^2*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d^2/(c*d+e)/(c*d-e)/(c*e*x+c*d)*arctanh(1/(-c^2*x^2+1)^(1/2))-
1/2*c^2*b*e*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d/(c*d+e)/(c*d-e)/(c*e*x+c*d)*arctan
h(1/(-c^2*x^2+1)^(1/2))+1/2*c^2*b*e*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/d/(c*d+e)/(c*d-e)/(c*e*x+c*d)+1
/2*c^2*b*e^2*(-(c*x-1)/c/x)^(1/2)*x^2*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d^2/(c*d+e)/(c*d-e)/(-(c^2*d^2-e^
2)/e^2)^(1/2)/(c*e*x+c*d)*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1)^(1/2)*e+c^2*d*x+e)/(c*e*x+c*d))+1/2*c^
2*b*e*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d/(c*d+e)/(c*d-e)/(-(c^2*d^2-e^2)/e^2)^(1/
2)/(c*e*x+c*d)*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1)^(1/2)*e+c^2*d*x+e)/(c*e*x+c*d))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.68898, size = 2423, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a + b)*d^2*e^4 - (b*c^2*d^2*e^4 - b*e^6)*x^2 + (2*b*c^2*d^4*e - b*
d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(-c^2*d^2 + e^2)*log((c^2*d*e*x
- (c^3*d^2 - c*e^2)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2 - sqrt(-c^2*d^2 + e^2)*(c^2*d*x + c*e*x*sqrt(-(c^2
*x^2 - 1)/(c^2*x^2)) + e))/(e*x + d)) - 2*(b*c^2*d^3*e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e
^4 + (b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*log((c*x*s
qrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*log((c*x*sqrt(-(c^2*x^2 - 1)
/(c^2*x^2)) + 1)/(c*x)) - ((b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 + (b*c^3*d^4*e^2 - b*c*d^2*e^4)*x)*sqrt(-(c^2*x^2 -
1)/(c^2*x^2)))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^
7*e^2 - 2*c^2*d^5*e^4 + d^3*e^6)*x), -1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a + b)*d^2*e^4 - (b*c^2*d^2*e^
4 - b*e^6)*x^2 - 2*(2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*
x)*sqrt(c^2*d^2 - e^2)*arctan(-(sqrt(c^2*d^2 - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - sqrt(c^2*d^2 - e^2)
*(e*x + d))/((c^2*d^2 - e^2)*x)) - 2*(b*c^2*d^3*e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 +
(b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*log((c*x*sqrt(-
(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2
*x^2)) + 1)/(c*x)) - ((b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 + (b*c^3*d^4*e^2 - b*c*d^2*e^4)*x)*sqrt(-(c^2*x^2 - 1)/(
c^2*x^2)))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2
- 2*c^2*d^5*e^4 + d^3*e^6)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/(d + e*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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