3.75 \(\int (d+e x)^2 (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=201 \[ \frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (6 c^2 d^2+e^2\right ) \sin ^{-1}(c x)}{6 c^3}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{3 e}-\frac{b d e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{6 c^2} \]

[Out]

-((b*d*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/c^2) - (b*e^2*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c
*x]*Sqrt[1 - c^2*x^2])/(6*c^2) + ((d + e*x)^3*(a + b*ArcSech[c*x]))/(3*e) + (b*(6*c^2*d^2 + e^2)*Sqrt[(1 + c*x
)^(-1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3) - (b*d^3*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^
2]])/(3*e)

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Rubi [A]  time = 0.217373, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6288, 1809, 844, 216, 266, 63, 208} \[ \frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (6 c^2 d^2+e^2\right ) \sin ^{-1}(c x)}{6 c^3}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{3 e}-\frac{b d e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{6 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*ArcSech[c*x]),x]

[Out]

-((b*d*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/c^2) - (b*e^2*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c
*x]*Sqrt[1 - c^2*x^2])/(6*c^2) + ((d + e*x)^3*(a + b*ArcSech[c*x]))/(3*e) + (b*(6*c^2*d^2 + e^2)*Sqrt[(1 + c*x
)^(-1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3) - (b*d^3*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^
2]])/(3*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 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 216

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 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]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(d+e x)^3}{x \sqrt{1-c^2 x^2}} \, dx}{3 e}\\ &=-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-2 c^2 d^3-e \left (6 c^2 d^2+e^2\right ) x-6 c^2 d e^2 x^2}{x \sqrt{1-c^2 x^2}} \, dx}{6 c^2 e}\\ &=-\frac{b d e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{2 c^4 d^3+c^2 e \left (6 c^2 d^2+e^2\right ) x}{x \sqrt{1-c^2 x^2}} \, dx}{6 c^4 e}\\ &=-\frac{b d e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b d^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{3 e}+\frac{\left (b \left (6 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{6 c^2}\\ &=-\frac{b d e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{6 c^3}+\frac{\left (b d^3 \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 )}{6 e}\\ &=-\frac{b d e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{6 c^3}-\frac{\left (b d^3 \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 )}{3 c^2 e}\\ &=-\frac{b d e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{(d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{6 c^3}-\frac{b d^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{3 e}\\ \end{align*}

Mathematica [C]  time = 0.214008, size = 147, normalized size = 0.73 \[ \frac{2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+2 b c^3 x \text{sech}^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+i b \left (6 c^2 d^2+e^2\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )-b c e \sqrt{\frac{1-c x}{c x+1}} (c x+1) (6 d+e x)}{6 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + b*ArcSech[c*x]),x]

[Out]

(-(b*c*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(6*d + e*x)) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 2*b*c^3*x*
(3*d^2 + 3*d*e*x + e^2*x^2)*ArcSech[c*x] + I*b*(6*c^2*d^2 + e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*
(1 + c*x)])/(6*c^3)

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Maple [A]  time = 0.213, size = 215, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( cxe+cd \right ) ^{3}a}{3\,e{c}^{2}}}+{\frac{b}{{c}^{2}} \left ({\frac{{e}^{2}{\rm arcsech} \left (cx\right ){c}^{3}{x}^{3}}{3}}+e{\rm arcsech} \left (cx\right ){c}^{3}{x}^{2}d+{\rm arcsech} \left (cx\right ){c}^{3}x{d}^{2}+{\frac{{\rm arcsech} \left (cx\right ){c}^{3}{d}^{3}}{3\,e}}+{\frac{cx}{6\,e}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -2\,{c}^{3}{d}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +6\,{c}^{2}{d}^{2}e\arcsin \left ( cx \right ) -{e}^{3}cx\sqrt{-{c}^{2}{x}^{2}+1}-6\,cd{e}^{2}\sqrt{-{c}^{2}{x}^{2}+1}+{e}^{3}\arcsin \left ( cx \right ) \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arcsech(c*x)),x)

[Out]

1/c*(1/3*(c*e*x+c*d)^3*a/c^2/e+b/c^2*(1/3*e^2*arcsech(c*x)*c^3*x^3+e*arcsech(c*x)*c^3*x^2*d+arcsech(c*x)*c^3*x
*d^2+1/3/e*arcsech(c*x)*c^3*d^3+1/6/e*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(-2*c^3*d^3*arctanh(1/(-c^2
*x^2+1)^(1/2))+6*c^2*d^2*e*arcsin(c*x)-e^3*c*x*(-c^2*x^2+1)^(1/2)-6*c*d*e^2*(-c^2*x^2+1)^(1/2)+e^3*arcsin(c*x)
)/(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.50683, size = 205, normalized size = 1.02 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} + a d e x^{2} +{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} b d e + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{arsech}\left (c x\right ) - \frac{\frac{\sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac{{\left (c x \operatorname{arsech}\left (c x\right ) - \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e^2*x^3 + a*d*e*x^2 + (x^2*arcsech(c*x) - x*sqrt(1/(c^2*x^2) - 1)/c)*b*d*e + 1/6*(2*x^3*arcsech(c*x) - (
sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + arctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*b*e^2 + a*d^2*x +
(c*x*arcsech(c*x) - arctan(sqrt(1/(c^2*x^2) - 1)))*b*d^2/c

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Fricas [B]  time = 2.12997, size = 601, normalized size = 2.99 \begin{align*} \frac{2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x - 2 \,{\left (6 \, b c^{2} d^{2} + b e^{2}\right )} \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 2 \,{\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \,{\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (b c^{2} e^{2} x^{2} + 6 \, b c^{2} d e x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{6 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a*c^3*e^2*x^3 + 6*a*c^3*d*e*x^2 + 6*a*c^3*d^2*x - 2*(6*b*c^2*d^2 + b*e^2)*arctan((c*x*sqrt(-(c^2*x^2 -
1)/(c^2*x^2)) - 1)/(c*x)) - 2*(3*b*c^3*d^2 + 3*b*c^3*d*e + b*c^3*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
- 1)/x) + 2*(b*c^3*e^2*x^3 + 3*b*c^3*d*e*x^2 + 3*b*c^3*d^2*x - 3*b*c^3*d^2 - 3*b*c^3*d*e - b*c^3*e^2)*log((c*x
*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (b*c^2*e^2*x^2 + 6*b*c^2*d*e*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/
c^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{asech}{\left (c x \right )}\right ) \left (d + e x\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a+b*asech(c*x)),x)

[Out]

Integral((a + b*asech(c*x))*(d + e*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(b*arcsech(c*x) + a), x)