3.1.16 \(\int (d+e x) (a+b \cosh ^{-1}(c x)) \, dx\) [16]

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

[Out]

-1/4*b*(2*d^2+e^2/c^2)*arccosh(c*x)/e+1/2*(e*x+d)^2*(a+b*arccosh(c*x))/e-3/4*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
-1/4*b*(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 = 106, normalized size of antiderivative = 1.00, 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 = {5963, 92, 81, 54} \begin {gather*} \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} \end {gather*}

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 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) \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*}

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Mathematica [A]
time = 0.07, size = 117, normalized size = 1.10 \begin {gather*} a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+b d x \cosh ^{-1}(c x)+\frac {1}{2} b e x^2 \cosh ^{-1}(c x)-\frac {b e \tanh ^{-1}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{2 c^2} \end {gather*}

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 = 2.10, size = 133, normalized size = 1.25

method	result	size
derivativedivides	\(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+b \,\mathrm {arccosh}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{4}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\)	\(133\)
default	\(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+b \,\mathrm {arccosh}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{4}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c*(c^2*d*x+1/2*c^2*e*x^2)+b*arccosh(c*x)*d*c*x+1/2*b*c*arccosh(c*x)*e*x^2-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)
*d-1/4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e*x-1/4*b/c*(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 = 0.27, size = 101, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, a x^{2} e + a d x + \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} c\right )}{c^{3}}\right )}\right )} b e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \end {gather*}

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

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Fricas [A]
time = 0.41, size = 125, normalized size = 1.18 \begin {gather*} \frac {2 \, a c^{2} x^{2} \cosh \left (1\right ) + 2 \, a c^{2} x^{2} \sinh \left (1\right ) + 4 \, a c^{2} d x + {\left (4 \, b c^{2} d x + {\left (2 \, b c^{2} x^{2} - b\right )} \cosh \left (1\right ) + {\left (2 \, b c^{2} x^{2} - b\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} {\left (b c x \cosh \left (1\right ) + b c x \sinh \left (1\right ) + 4 \, b c d\right )}}{4 \, c^{2}} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.11, size = 105, normalized size = 0.99 \begin {gather*} \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 {gather*}

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 = 0.46, size = 124, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{c}\right )} b d + \frac {1}{4} \, {\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 e + a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.03, size = 83, normalized size = 0.78 \begin {gather*} \frac {a\,x\,\left (2\,d+e\,x\right )}{2}+b\,d\,x\,\mathrm {acosh}\left (c\,x\right )+b\,e\,x\,\mathrm {acosh}\left (c\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,c^2\,x}\right )-\frac {b\,d\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c}-\frac {b\,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((a + b*acosh(c*x))*(d + e*x),x)

[Out]

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

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