3.1.9 \(\int (d+e x)^2 \cosh ^{-1}(c x)^2 \, dx\) [9]

Optimal. Leaf size=215 \[ 2 d^2 x+\frac {4 e^2 x}{9 c^2}+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e} \]

[Out]

2*d^2*x+4/9*e^2*x/c^2+1/2*d*e*x^2+2/27*e^2*x^3-1/3*d^3*arccosh(c*x)^2/e-1/2*d*e*arccosh(c*x)^2/c^2+1/3*(e*x+d)
^3*arccosh(c*x)^2/e-2*d^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-4/9*e^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/c^3-d*e*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-2/9*e^2*x^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1
/2)/c

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Rubi [A]
time = 0.67, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5963, 5975, 5893, 5915, 8, 5939, 30} \begin {gather*} -\frac {4 e^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {4 e^2 x}{9 c^2}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {2 d^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {d e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^3}{3 e}-\frac {2 e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c}+2 d^2 x+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*ArcCosh[c*x]^2,x]

[Out]

2*d^2*x + (4*e^2*x)/(9*c^2) + (d*e*x^2)/2 + (2*e^2*x^3)/27 - (2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])
/c - (4*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c^3) - (d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c
*x])/c - (2*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c) - (d^3*ArcCosh[c*x]^2)/(3*e) - (d*e*ArcCo
sh[c*x]^2)/(2*c^2) + ((d + e*x)^3*ArcCosh[c*x]^2)/(3*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)^2 \cosh ^{-1}(c x)^2 \, dx &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {(2 c) \int \frac {(d+e x)^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {(2 c) \int \left (\frac {d^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 e x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d e^2 x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^3 x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{3 e}\\ &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\left (2 c d^2\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (2 c d^3\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}-(2 c d e) \int \frac {x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{3} \left (2 c e^2\right ) \int \frac {x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}+\left (2 d^2\right ) \int 1 \, dx+(d e) \int x \, dx-\frac {(d e) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c}+\frac {1}{9} \left (2 e^2\right ) \int x^2 \, dx-\frac {\left (4 e^2\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c}\\ &=2 d^2 x+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}+\frac {\left (4 e^2\right ) \int 1 \, dx}{9 c^2}\\ &=2 d^2 x+\frac {4 e^2 x}{9 c^2}+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 131, normalized size = 0.61 \begin {gather*} \frac {c x \left (24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 \sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right ) \cosh ^{-1}(c x)+9 \left (-3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \cosh ^{-1}(c x)^2}{54 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*ArcCosh[c*x]^2,x]

[Out]

(c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*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))*ArcCosh[c*x] + 9*(-3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/(54*c^3
)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{2} \mathrm {arccosh}\left (c x \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/3*(x^3*e^2 + 3*d*x^2*e + 3*d^2*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(2/3*(c^3*x^5*e^2 + 3*
c^3*d*x^4*e - 3*c*d*x^2*e - 3*c*d^2*x + (3*c^3*d^2 - c*e^2)*x^3 + (c^2*x^4*e^2 + 3*c^2*d*x^3*e + 3*c^2*d^2*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)*s
qrt(c*x - 1) - c*x), x)

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Fricas [A]
time = 0.35, size = 291, normalized size = 1.35 \begin {gather*} \frac {27 \, c^{3} d x^{2} \cosh \left (1\right ) + 108 \, c^{3} d^{2} x + 4 \, {\left (c^{3} x^{3} + 6 \, c x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (2 \, c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, c^{3} d^{2} x + 3 \, {\left (2 \, c^{3} d x^{2} - c d\right )} \cosh \left (1\right ) + {\left (4 \, c^{3} x^{3} \cosh \left (1\right ) + 6 \, c^{3} d x^{2} - 3 \, c d\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 4 \, {\left (c^{3} x^{3} + 6 \, c x\right )} \sinh \left (1\right )^{2} - 6 \, {\left (9 \, c^{2} d x \cosh \left (1\right ) + 18 \, c^{2} d^{2} + 2 \, {\left (c^{2} x^{2} + 2\right )} \cosh \left (1\right )^{2} + 2 \, {\left (c^{2} x^{2} + 2\right )} \sinh \left (1\right )^{2} + {\left (9 \, c^{2} d x + 4 \, {\left (c^{2} x^{2} + 2\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, c^{3} d x^{2} + 8 \, {\left (c^{3} x^{3} + 6 \, c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{54 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(27*c^3*d*x^2*cosh(1) + 108*c^3*d^2*x + 4*(c^3*x^3 + 6*c*x)*cosh(1)^2 + 9*(2*c^3*x^3*cosh(1)^2 + 2*c^3*x^
3*sinh(1)^2 + 6*c^3*d^2*x + 3*(2*c^3*d*x^2 - c*d)*cosh(1) + (4*c^3*x^3*cosh(1) + 6*c^3*d*x^2 - 3*c*d)*sinh(1))
*log(c*x + sqrt(c^2*x^2 - 1))^2 + 4*(c^3*x^3 + 6*c*x)*sinh(1)^2 - 6*(9*c^2*d*x*cosh(1) + 18*c^2*d^2 + 2*(c^2*x
^2 + 2)*cosh(1)^2 + 2*(c^2*x^2 + 2)*sinh(1)^2 + (9*c^2*d*x + 4*(c^2*x^2 + 2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 -
1)*log(c*x + sqrt(c^2*x^2 - 1)) + (27*c^3*d*x^2 + 8*(c^3*x^3 + 6*c*x)*cosh(1))*sinh(1))/c^3

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Sympy [A]
time = 0.25, size = 223, normalized size = 1.04 \begin {gather*} \begin {cases} d^{2} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 d^{2} x + d e x^{2} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {d e x^{2}}{2} + \frac {e^{2} x^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{3} + \frac {2 e^{2} x^{3}}{27} - \frac {2 d^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {d e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {2 e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c} - \frac {d e \operatorname {acosh}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {4 e^{2} x}{9 c^{2}} - \frac {4 e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\- \frac {\pi ^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right )}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**2*x*acosh(c*x)**2 + 2*d**2*x + d*e*x**2*acosh(c*x)**2 + d*e*x**2/2 + e**2*x**3*acosh(c*x)**2/3 +
2*e**2*x**3/27 - 2*d**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/c - d*e*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/c - 2*e**2*x*
*2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c) - d*e*acosh(c*x)**2/(2*c**2) + 4*e**2*x/(9*c**2) - 4*e**2*sqrt(c**2*x*
*2 - 1)*acosh(c*x)/(9*c**3), Ne(c, 0)), (-pi**2*(d**2*x + d*e*x**2 + e**2*x**3/3)/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)^2*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 = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (c\,x\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)^2*(d + e*x)^2,x)

[Out]

int(acosh(c*x)^2*(d + e*x)^2, x)

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