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

Optimal. Leaf size=123 \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{1}{6} d \left (\frac{3 e}{c^2}+\frac{2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac{\sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{9 c}+\frac{\cosh ^{-1}(c x) (d+e x)^3}{3 e} \]

[Out]

-(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5*c^2
*d*e*x))/(18*c^3) - (d*((2*d^2)/e + (3*e)/c^2)*ArcCosh[c*x])/6 + ((d + e*x)^3*ArcCosh[c*x])/(3*e)

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Rubi [A]  time = 0.103075, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5802, 100, 147, 52} \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{1}{6} d \left (\frac{3 e}{c^2}+\frac{2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac{\sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{9 c}+\frac{\cosh ^{-1}(c x) (d+e x)^3}{3 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*ArcCosh[c*x],x]

[Out]

-(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5*c^2
*d*e*x))/(18*c^3) - (d*((2*d^2)/e + (3*e)/c^2)*ArcCosh[c*x])/6 + ((d + e*x)^3*ArcCosh[c*x])/(3*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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int (d+e x)^2 \cosh ^{-1}(c x) \, dx &=\frac{(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac{c \int \frac{(d+e x)^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 e}\\ &=-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}+\frac{(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac{\int \frac{(d+e x) \left (3 c^2 d^2+2 e^2+5 c^2 d e x\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c e}\\ &=-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}-\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}+\frac{(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac{1}{6} \left (d \left (\frac{2 c d^2}{e}+\frac{3 e}{c}\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}-\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{1}{6} d \left (\frac{2 d^2}{e}+\frac{3 e}{c^2}\right ) \cosh ^{-1}(c x)+\frac{(d+e x)^3 \cosh ^{-1}(c x)}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.168389, size = 113, normalized size = 0.92 \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )-6 c^3 x \cosh ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+9 c d e \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{18 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*ArcCosh[c*x],x]

[Out]

-(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) - 6*c^3*x*(3*d^2 + 3*d*e*x + e^2*x
^2)*ArcCosh[c*x] + 9*c*d*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(18*c^3)

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Maple [B]  time = 0.013, size = 233, normalized size = 1.9 \begin{align*}{\frac{{e}^{2}{\rm arccosh} \left (cx\right ){x}^{3}}{3}}+e{\rm arccosh} \left (cx\right ){x}^{2}d+{\rm arccosh} \left (cx\right )x{d}^{2}+{\frac{{\rm arccosh} \left (cx\right ){d}^{3}}{3\,e}}-{\frac{{e}^{2}{x}^{2}}{9\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{d}^{3}}{3\,e}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{dex}{2\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{d}^{2}}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{de}{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}}}}-{\frac{2\,{e}^{2}}{9\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^2*arccosh(c*x)*x^3+e*arccosh(c*x)*x^2*d+arccosh(c*x)*x*d^2+1/3/e*arccosh(c*x)*d^3-1/9/c*e^2*(c*x-1)^(1/2
)*(c*x+1)^(1/2)*x^2-1/3/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^3*ln(c*x+(c^2*x^2-1)^(1/2))-1/2/c*e*
(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*x-1/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2-1/2/c^2*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2
*x^2-1)^(1/2)*d*ln(c*x+(c^2*x^2-1)^(1/2))-2/9/c^3*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)

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Maxima [A]  time = 1.08668, size = 201, normalized size = 1.63 \begin{align*} -\frac{1}{18} \,{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} e^{2} x^{2}}{c^{2}} + \frac{9 \, \sqrt{c^{2} x^{2} - 1} d e x}{c^{2}} + \frac{18 \, \sqrt{c^{2} x^{2} - 1} d^{2}}{c^{2}} + \frac{9 \, d e \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} e^{2}}{c^{4}}\right )} c + \frac{1}{3} \,{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \operatorname{arcosh}\left (c x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*sqrt(c^2*x^2 - 1)*e^2*x^2/c^2 + 9*sqrt(c^2*x^2 - 1)*d*e*x/c^2 + 18*sqrt(c^2*x^2 - 1)*d^2/c^2 + 9*d*e*
log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^2) + 4*sqrt(c^2*x^2 - 1)*e^2/c^4)*c + 1/3*(e^2*x^3 +
3*d*e*x^2 + 3*d^2*x)*arccosh(c*x)

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Fricas [A]  time = 2.00196, size = 230, normalized size = 1.87 \begin{align*} \frac{3 \,{\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(3*(2*c^3*e^2*x^3 + 6*c^3*d*e*x^2 + 6*c^3*d^2*x - 3*c*d*e)*log(c*x + sqrt(c^2*x^2 - 1)) - (2*c^2*e^2*x^2
+ 9*c^2*d*e*x + 18*c^2*d^2 + 4*e^2)*sqrt(c^2*x^2 - 1))/c^3

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Sympy [A]  time = 0.832006, size = 155, normalized size = 1.26 \begin{align*} \begin{cases} d^{2} x \operatorname{acosh}{\left (c x \right )} + d e x^{2} \operatorname{acosh}{\left (c x \right )} + \frac{e^{2} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{d^{2} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{d e x \sqrt{c^{2} x^{2} - 1}}{2 c} - \frac{e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{d e \operatorname{acosh}{\left (c x \right )}}{2 c^{2}} - \frac{2 e^{2} \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\frac{i \pi \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**2*x*acosh(c*x) + d*e*x**2*acosh(c*x) + e**2*x**3*acosh(c*x)/3 - d**2*sqrt(c**2*x**2 - 1)/c - d*e
*x*sqrt(c**2*x**2 - 1)/(2*c) - e**2*x**2*sqrt(c**2*x**2 - 1)/(9*c) - d*e*acosh(c*x)/(2*c**2) - 2*e**2*sqrt(c**
2*x**2 - 1)/(9*c**3), Ne(c, 0)), (I*pi*(d**2*x + d*e*x**2 + e**2*x**3/3)/2, True))

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Giac [A]  time = 1.16729, size = 174, normalized size = 1.41 \begin{align*} \frac{1}{3} \,{\left (x e + d\right )}^{3} e^{\left (-1\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{1}{18} \,{\left (\sqrt{c^{2} x^{2} - 1}{\left (x{\left (\frac{2 \, x e^{3}}{c} + \frac{9 \, d e^{2}}{c}\right )} + \frac{2 \,{\left (9 \, c^{3} d^{2} e + 2 \, c e^{3}\right )}}{c^{4}}\right )} - \frac{3 \,{\left (2 \, c^{2} d^{3} + 3 \, d e^{2}\right )} \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c{\left | c \right |}}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x*e + d)^3*e^(-1)*log(c*x + sqrt(c^2*x^2 - 1)) - 1/18*(sqrt(c^2*x^2 - 1)*(x*(2*x*e^3/c + 9*d*e^2/c) + 2*(
9*c^3*d^2*e + 2*c*e^3)/c^4) - 3*(2*c^2*d^3 + 3*d*e^2)*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c*abs(c)))*e^(-
1)