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

Optimal. Leaf size=106 \[ \frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac{b \left (\frac{e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)}{4 c}-\frac{3 b d \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

[Out]

(-3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x))/(4*c) - (b*(2*d^2 + e
^2/c^2)*ArcCosh[c*x])/(4*e) + ((d + e*x)^2*(a + b*ArcCosh[c*x]))/(2*e)

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Rubi [A]  time = 0.0444908, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5802, 90, 80, 52} \[ \frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac{b \left (\frac{e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)}{4 c}-\frac{3 b d \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*ArcCosh[c*x]),x]

[Out]

(-3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x))/(4*c) - (b*(2*d^2 + e
^2/c^2)*ArcCosh[c*x])/(4*e) + ((d + e*x)^2*(a + b*ArcCosh[c*x]))/(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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac{(b c) \int \frac{(d+e x)^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac{b \int \frac{2 c^2 d^2+e^2+3 c^2 d e x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c e}\\ &=-\frac{3 b d \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac{\left (b \left (2 c^2 d^2+e^2\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c e}\\ &=-\frac{3 b d \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}-\frac{b \left (2 d^2+\frac{e^2}{c^2}\right ) \cosh ^{-1}(c x)}{4 e}+\frac{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.0928296, size = 117, normalized size = 1.1 \[ a d x+\frac{1}{2} a e x^2-\frac{b e \tanh ^{-1}\left (\frac{\sqrt{c x-1}}{\sqrt{c x+1}}\right )}{2 c^2}-\frac{b d \sqrt{c x-1} \sqrt{c x+1}}{c}+b d x \cosh ^{-1}(c x)+\frac{1}{2} b e x^2 \cosh ^{-1}(c x)-\frac{b e x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 123, normalized size = 1.2 \begin{align*}{\frac{a{x}^{2}e}{2}}+adx+{\frac{b{\rm arccosh} \left (cx\right ){x}^{2}e}{2}}+b{\rm arccosh} \left (cx\right )xd-{\frac{bex}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bd}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{be}{4\,{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)),x)

[Out]

1/2*a*x^2*e+a*d*x+1/2*b*arccosh(c*x)*x^2*e+b*arccosh(c*x)*x*d-1/4*b*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-b*d*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/c-1/4/c^2*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 [A]  time = 1.15359, size = 146, normalized size = 1.38 \begin{align*} \frac{1}{2} \, a e x^{2} + \frac{1}{4} \,{\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 )} b e + a d x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} 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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.27267, size = 197, normalized size = 1.86 \begin{align*} \frac{2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x +{\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x - b e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c e x + 4 \, 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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.417238, size = 105, normalized size = 0.99 \begin{align*} \begin{cases} a d x + \frac{a e x^{2}}{2} + b d x \operatorname{acosh}{\left (c x \right )} + \frac{b e x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{b d \sqrt{c^{2} x^{2} - 1}}{c} - \frac{b e x \sqrt{c^{2} x^{2} - 1}}{4 c} - \frac{b e \operatorname{acosh}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \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)),x)

[Out]

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

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Giac [A]  time = 1.26328, size = 170, normalized size = 1.6 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d + a d x + \frac{1}{4} \,{\left (2 \, a x^{2} +{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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