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

Optimal. Leaf size=150 \[ -\frac{e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac{2 b d \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}+2 b^2 d x+\frac{1}{4} b^2 e x^2 \]

[Out]

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

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Rubi [A]  time = 0.756242, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac{2 b d \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}+2 b^2 d x+\frac{1}{4} b^2 e x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d*x + (b^2*e*x^2)/4 - (2*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (b*e*x*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c) - (d^2*(a + b*ArcCosh[c*x])^2)/(2*e) - (e*(a + b*ArcCosh[c*x])^2)/
(4*c^2) + ((d + e*x)^2*(a + b*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) \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac{(b c) \int \frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e}\\ &=\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac{(b c) \int \left (\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 d e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{e}\\ &=\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{\left (b c d^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e}-(b c e) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 b d \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\left (2 b^2 d\right ) \int 1 \, dx+\frac{1}{2} \left (b^2 e\right ) \int x \, dx-\frac{(b e) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c}\\ &=2 b^2 d x+\frac{1}{4} b^2 e x^2-\frac{2 b d \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.389524, size = 174, normalized size = 1.16 \[ \frac{c \left (2 a^2 c x (2 d+e x)-2 a b \sqrt{c x-1} \sqrt{c x+1} (4 d+e x)+b^2 c x (8 d+e x)\right )-2 b c \cosh ^{-1}(c x) \left (b \sqrt{c x-1} \sqrt{c x+1} (4 d+e x)-2 a c x (2 d+e x)\right )-2 a b e \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )+b^2 \cosh ^{-1}(c x)^2 \left (4 c^2 d x+e \left (2 c^2 x^2-1\right )\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(2*a^2*c*x*(2*d + e*x) - 2*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x) + b^2*c*x*(8*d + e*x)) - 2*b*c*(-2*
a*c*x*(2*d + e*x) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x))*ArcCosh[c*x] + b^2*(4*c^2*d*x + e*(-1 + 2*c^2*
x^2))*ArcCosh[c*x]^2 - 2*a*b*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(4*c^2)

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Maple [A]  time = 0.043, size = 245, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}{x}^{2}e}{2}}+{a}^{2}dx+{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{x}^{2}e}{2}}-{\frac{{b}^{2}{\rm arccosh} \left (cx\right )xe}{2\,c}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{{b}^{2}e{x}^{2}}{4}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}e}{4\,{c}^{2}}}+{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}xd-2\,{\frac{{b}^{2}{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}d}{c}}+2\,{b}^{2}dx+ab{\rm arccosh} \left (cx\right ){x}^{2}e+2\,ab{\rm arccosh} \left (cx\right )xd-{\frac{abex}{2\,c}\sqrt{cx-1}\sqrt{cx+1}}-2\,{\frac{ab\sqrt{cx-1}\sqrt{cx+1}d}{c}}-{\frac{abe}{2\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.30077, size = 406, normalized size = 2.71 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x +{\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \,{\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e -{\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 2 \,{\left (a b c e x + 4 \, a b c d\right )} \sqrt{c^{2} x^{2} - 1}}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((2*a^2 + b^2)*c^2*e*x^2 + 4*(a^2 + 2*b^2)*c^2*d*x + (2*b^2*c^2*e*x^2 + 4*b^2*c^2*d*x - b^2*e)*log(c*x + s
qrt(c^2*x^2 - 1))^2 + 2*(2*a*b*c^2*e*x^2 + 4*a*b*c^2*d*x - a*b*e - (b^2*c*e*x + 4*b^2*c*d)*sqrt(c^2*x^2 - 1))*
log(c*x + sqrt(c^2*x^2 - 1)) - 2*(a*b*c*e*x + 4*a*b*c*d)*sqrt(c^2*x^2 - 1))/c^2

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Sympy [A]  time = 1.02839, size = 240, normalized size = 1.6 \begin{align*} \begin{cases} a^{2} d x + \frac{a^{2} e x^{2}}{2} + 2 a b d x \operatorname{acosh}{\left (c x \right )} + a b e x^{2} \operatorname{acosh}{\left (c x \right )} - \frac{2 a b d \sqrt{c^{2} x^{2} - 1}}{c} - \frac{a b e x \sqrt{c^{2} x^{2} - 1}}{2 c} - \frac{a b e \operatorname{acosh}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname{acosh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac{b^{2} e x^{2} \operatorname{acosh}^{2}{\left (c x \right )}}{2} + \frac{b^{2} e x^{2}}{4} - \frac{2 b^{2} d \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{b^{2} e x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{2 c} - \frac{b^{2} e \operatorname{acosh}^{2}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right )^{2} \left (d x + \frac{e x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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