3.1.3 \(\int (d+e x) \cosh ^{-1}(c x) \, dx\) [3]

Optimal. Leaf size=97 \[ -\frac {3 d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}-\frac {1}{4} \left (\frac {2 d^2}{e}+\frac {e}{c^2}\right ) \cosh ^{-1}(c x)+\frac {(d+e x)^2 \cosh ^{-1}(c x)}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5963, 92, 81, 54} \begin {gather*} -\frac {1}{4} \left (\frac {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)}{4 c}+\frac {\cosh ^{-1}(c x) (d+e x)^2}{2 e}-\frac {3 d \sqrt {c x-1} \sqrt {c x+1}}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*ArcCosh[c*x],x]

[Out]

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

Rule 54

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]

Rule 81

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 92

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 5963

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]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 73, normalized size = 0.75 \begin {gather*} -\frac {c \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)-2 c^2 x (2 d+e x) \cosh ^{-1}(c x)+2 e \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{4 c^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)*ArcCosh[c*x],x]

[Out]

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

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Maple [A]
time = 1.93, size = 104, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\mathrm {arccosh}\left (c x \right ) d c x +\frac {c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\)	\(104\)
default	\(\frac {\mathrm {arccosh}\left (c x \right ) d c x +\frac {c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*arccosh(c*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 85, normalized size = 0.88 \begin {gather*} -\frac {1}{4} \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x e}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} d}{c^{2}} + \frac {e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \operatorname {arcosh}\left (c x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 85, normalized size = 0.88 \begin {gather*} \frac {{\left (4 \, c^{2} d x + {\left (2 \, c^{2} x^{2} - 1\right )} \cosh \left (1\right ) + {\left (2 \, c^{2} x^{2} - 1\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} {\left (c x \cosh \left (1\right ) + c x \sinh \left (1\right ) + 4 \, c d\right )}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((4*c^2*d*x + (2*c^2*x^2 - 1)*cosh(1) + (2*c^2*x^2 - 1)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x
^2 - 1)*(c*x*cosh(1) + c*x*sinh(1) + 4*c*d))/c^2

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Sympy [C] Result contains complex when optimal does not.
time = 0.10, size = 80, normalized size = 0.82 \begin {gather*} \begin {cases} d x \operatorname {acosh}{\left (c x \right )} + \frac {e x^{2} \operatorname {acosh}{\left (c x \right )}}{2} - \frac {d \sqrt {c^{2} x^{2} - 1}}{c} - \frac {e x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {e \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {i \pi \left (d x + \frac {e x^{2}}{2}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 84, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {1}{4} \, \sqrt {c^{2} x^{2} - 1} {\left (\frac {e x}{c} + \frac {4 \, d}{c}\right )} + \frac {e \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{4 \, c {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.71, size = 68, normalized size = 0.70 \begin {gather*} d\,x\,\mathrm {acosh}\left (c\,x\right )+e\,x\,\mathrm {acosh}\left (c\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,c^2\,x}\right )-\frac {d\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c}-\frac {e\,x\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{4\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)*(d + e*x),x)

[Out]

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

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