3.10 \(\int (d+e x) \cosh ^{-1}(c x)^2 \, dx\)

Optimal. Leaf size=122 \[ -\frac{e \cosh ^{-1}(c x)^2}{4 c^2}-\frac{d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^2}{2 e}-\frac{2 d \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac{e x^2}{4} \]

[Out]

2*d*x + (e*x^2)/4 - (2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcC
osh[c*x])/(2*c) - (d^2*ArcCosh[c*x]^2)/(2*e) - (e*ArcCosh[c*x]^2)/(4*c^2) + ((d + e*x)^2*ArcCosh[c*x]^2)/(2*e)

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Rubi [A]  time = 0.650873, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{e \cosh ^{-1}(c x)^2}{4 c^2}-\frac{d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^2}{2 e}-\frac{2 d \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac{e x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*ArcCosh[c*x]^2,x]

[Out]

2*d*x + (e*x^2)/4 - (2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcC
osh[c*x])/(2*c) - (d^2*ArcCosh[c*x]^2)/(2*e) - (e*ArcCosh[c*x]^2)/(4*c^2) + ((d + e*x)^2*ArcCosh[c*x]^2)/(2*e)

Rule 5802

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

Rule 5822

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (d+e x) \cosh ^{-1}(c x)^2 \, dx &=\frac{(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-\frac{c \int \frac{(d+e x)^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e}\\ &=\frac{(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-\frac{c \int \left (\frac{d^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 d e x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^2 x^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{e}\\ &=\frac{(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-(2 c d) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{\left (c d^2\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e}-(c e) \int \frac{x^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c}-\frac{e x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{2 c}-\frac{d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac{(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}+(2 d) \int 1 \, dx+\frac{1}{2} e \int x \, dx-\frac{e \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c}\\ &=2 d x+\frac{e x^2}{4}-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c}-\frac{e x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{2 c}-\frac{d^2 \cosh ^{-1}(c x)^2}{2 e}-\frac{e \cosh ^{-1}(c x)^2}{4 c^2}+\frac{(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.0767002, size = 105, normalized size = 0.86 \[ \frac{e \left (2 c^2 x^2-1\right ) \cosh ^{-1}(c x)^2}{4 c^2}+d x \cosh ^{-1}(c x)^2-\frac{2 d \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac{e x^2}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*ArcCosh[c*x]^2,x]

[Out]

2*d*x + (e*x^2)/4 - (2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcC
osh[c*x])/(2*c) + d*x*ArcCosh[c*x]^2 + (e*(-1 + 2*c^2*x^2)*ArcCosh[c*x]^2)/(4*c^2)

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Maple [A]  time = 0.042, size = 100, normalized size = 0.8 \begin{align*}{\frac{1}{c} \left ({\frac{e}{4\,c} \left ( 2\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{2}{x}^{2}-2\,{\rm arccosh} \left (cx\right )cx\sqrt{cx-1}\sqrt{cx+1}- \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}+{c}^{2}{x}^{2} \right ) }+d \left ( \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cx-2\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+2\,cx \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*arccosh(c*x)^2,x)

[Out]

1/c*(1/4*e*(2*arccosh(c*x)^2*c^2*x^2-2*arccosh(c*x)*c*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)-arccosh(c*x)^2+c^2*x^2)/c+
d*(arccosh(c*x)^2*c*x-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e*x^2 + 2*d*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate((c^3*e*x^4 + 2*c^3*d*x^3 - c*e*x^2 -
2*c*d*x + (c^2*e*x^3 + 2*c^2*d*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x
^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)

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Fricas [A]  time = 1.94098, size = 220, normalized size = 1.8 \begin{align*} \frac{c^{2} e x^{2} + 8 \, c^{2} d x +{\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 2 \, \sqrt{c^{2} x^{2} - 1}{\left (c e x + 4 \, c d\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^2*e*x^2 + 8*c^2*d*x + (2*c^2*e*x^2 + 4*c^2*d*x - e)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 2*sqrt(c^2*x^2 - 1
)*(c*e*x + 4*c*d)*log(c*x + sqrt(c^2*x^2 - 1)))/c^2

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Sympy [A]  time = 0.717199, size = 110, normalized size = 0.9 \begin{align*} \begin{cases} d x \operatorname{acosh}^{2}{\left (c x \right )} + 2 d x + \frac{e x^{2} \operatorname{acosh}^{2}{\left (c x \right )}}{2} + \frac{e x^{2}}{4} - \frac{2 d \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{e x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{2 c} - \frac{e \operatorname{acosh}^{2}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\- \frac{\pi ^{2} \left (d x + \frac{e x^{2}}{2}\right )}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*acosh(c*x)**2,x)

[Out]

Piecewise((d*x*acosh(c*x)**2 + 2*d*x + e*x**2*acosh(c*x)**2/2 + e*x**2/4 - 2*d*sqrt(c**2*x**2 - 1)*acosh(c*x)/
c - e*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(2*c) - e*acosh(c*x)**2/(4*c**2), Ne(c, 0)), (-pi**2*(d*x + e*x**2/2)/4
, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)*arccosh(c*x)^2, x)