3.1.10 \(\int (d+e x) \cosh ^{-1}(c x)^2 \, dx\) [10]

Optimal. Leaf size=122 \[ 2 d x+\frac {e x^2}{4}-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}-\frac {e \cosh ^{-1}(c x)^2}{4 c^2}+\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.43, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5963, 5975, 5893, 5915, 8, 5939, 30} \begin {gather*} -\frac {e \cosh ^{-1}(c x)^2}{4 c^2}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^2}{2 e}-\frac {2 d \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac {e x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*ArcCosh[c*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 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]

Rule 5975

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]))

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 105, normalized size = 0.86 \begin {gather*} 2 d x+\frac {e x^2}{4}-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}+d x \cosh ^{-1}(c x)^2+\frac {e \left (-1+2 c^2 x^2\right ) \cosh ^{-1}(c x)^2}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*ArcCosh[c*x]^2,x]

[Out]

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

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Maple [A]
time = 3.34, size = 100, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {e \left (-2 \sqrt {c x -1}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, x c +2 \mathrm {arccosh}\left (c x \right )^{2} x^{2} c^{2}-\mathrm {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\mathrm {arccosh}\left (c x \right )^{2} x c -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )}{c}\)	\(100\)
default	\(\frac {\frac {e \left (-2 \sqrt {c x -1}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, x c +2 \mathrm {arccosh}\left (c x \right )^{2} x^{2} c^{2}-\mathrm {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\mathrm {arccosh}\left (c x \right )^{2} x c -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )}{c}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 128, normalized size = 1.05 \begin {gather*} \frac {c^{2} x^{2} \cosh \left (1\right ) + c^{2} x^{2} \sinh \left (1\right ) + 8 \, c^{2} d x + {\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 )^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (c x \cosh \left (1\right ) + c x \sinh \left (1\right ) + 4 \, c d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^2*x^2*cosh(1) + c^2*x^2*sinh(1) + 8*c^2*d*x + (4*c^2*d*x + (2*c^2*x^2 - 1)*cosh(1) + (2*c^2*x^2 - 1)*si
nh(1))*log(c*x + sqrt(c^2*x^2 - 1))^2 - 2*sqrt(c^2*x^2 - 1)*(c*x*cosh(1) + c*x*sinh(1) + 4*c*d)*log(c*x + sqrt
(c^2*x^2 - 1)))/c^2

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Sympy [A]
time = 0.14, size = 110, normalized size = 0.90 \begin {gather*} \begin {cases} d x \operatorname {acosh}^{2}{\left (c x \right )} + 2 d x + \frac {e x^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{2} + \frac {e x^{2}}{4} - \frac {2 d \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{2 c} - \frac {e \operatorname {acosh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\- \frac {\pi ^{2} \left (d x + \frac {e x^{2}}{2}\right )}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {acosh}\left (c\,x\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)^2*(d + e*x),x)

[Out]

int(acosh(c*x)^2*(d + e*x), x)

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