3.527 \(\int (d+e x^2) (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=168 \[ -\frac {4 b e \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b e x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3 \]

[Out]

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

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Rubi [A]  time = 0.57, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5707, 5654, 5718, 8, 5662, 5759, 30} \[ -\frac {4 b e \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b e x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

2*b^2*d*x + (4*b^2*e*x)/(9*c^2) + (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))
/c - (4*b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (2*b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]*(a + b*ArcCosh[c*x]))/(9*c) + d*x*(a + b*ArcCosh[c*x])^2 + (e*x^3*(a + b*ArcCosh[c*x])^2)/3

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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5718

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

Rule 5759

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

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\int \left (d \left (a+b \cosh ^{-1}(c x)\right )^2+e x^2 \left (a+b \cosh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+e \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx\\ &=d x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-(2 b c d) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{3} (2 b c e) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{9} \left (2 b^2 e\right ) \int x^2 \, dx-\frac {(4 b e) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c}\\ &=2 b^2 d x+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (4 b^2 e\right ) \int 1 \, dx}{9 c^2}\\ &=2 b^2 d x+\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 174, normalized size = 1.04 \[ \frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (9 d+e x^2\right )+2 e\right )-6 b \cosh ^{-1}(c x) \left (b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (9 d+e x^2\right )+2 e\right )-3 a c^3 x \left (3 d+e x^2\right )\right )+9 b^2 c^3 x \cosh ^{-1}(c x)^2 \left (3 d+e x^2\right )+2 b^2 c x \left (c^2 \left (27 d+e x^2\right )+6 e\right )}{27 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*a^2*c^3*x*(3*d + e*x^2) - 6*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)) + 2*b^2*c*x*(6*e + c
^2*(27*d + e*x^2)) - 6*b*(-3*a*c^3*x*(3*d + e*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)))
*ArcCosh[c*x] + 9*b^2*c^3*x*(3*d + e*x^2)*ArcCosh[c*x]^2)/(27*c^3)

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fricas [A]  time = 0.67, size = 209, normalized size = 1.24 \[ \frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d + 4 \, b^{2} c e\right )} x + 6 \, {\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x - {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*((9*a^2 + 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 3*(9*(a^2
+ 2*b^2)*c^3*d + 4*b^2*c*e)*x + 6*(3*a*b*c^3*e*x^3 + 9*a*b*c^3*d*x - (b^2*c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*
sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(a*b*c^2*e*x^2 + 9*a*b*c^2*d + 2*a*b*e)*sqrt(c^2*x^2 - 1))
/c^3

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.10, size = 217, normalized size = 1.29 \[ \frac {\frac {a^{2} \left (\frac {1}{3} c^{3} x^{3} e +c^{3} d x \right )}{c^{2}}+\frac {b^{2} \left (\frac {e \left (9 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+c^{2} d \left (\mathrm {arccosh}\left (c x \right )^{2} c x -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{3} x^{3} e}{3}+\mathrm {arccosh}\left (c x \right ) c^{3} d x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} e +9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^2*(1/3*c^3*x^3*e+c^3*d*x)+b^2/c^2*(1/27*e*(9*arccosh(c*x)^2*c^3*x^3-6*arccosh(c*x)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)*c^2*x^2-12*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c^3*x^3+12*c*x)+c^2*d*(arccosh(c*x)^2*c*x-2*
arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c*x))+2*a*b/c^2*(1/3*arccosh(c*x)*c^3*x^3*e+arccosh(c*x)*c^3*d*x-1/
9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c^2*e*x^2+9*c^2*d+2*e)))

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maxima [A]  time = 0.51, size = 218, normalized size = 1.30 \[ \frac {1}{3} \, b^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*e*x^3*arccosh(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arccosh(c*x)^2 + 2/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*
x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*e - 2/27*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1
)/c^4)*arccosh(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*e + 2*b^2*d*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d*x +
2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^2*(d + e*x^2),x)

[Out]

int((a + b*acosh(c*x))^2*(d + e*x^2), x)

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sympy [A]  time = 1.36, size = 286, normalized size = 1.70 \[ \begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {acosh}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {2 a b d \sqrt {c^{2} x^{2} - 1}}{c} - \frac {2 a b e x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {4 a b e \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} + b^{2} d x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e x^{3}}{27} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {2 b^{2} e x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c} + \frac {4 b^{2} e x}{9 c^{2}} - \frac {4 b^{2} e \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right )^{2} \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*acosh(c*x) + 2*a*b*e*x**3*acosh(c*x)/3 - 2*a*b*d*sqrt(c**2*x**
2 - 1)/c - 2*a*b*e*x**2*sqrt(c**2*x**2 - 1)/(9*c) - 4*a*b*e*sqrt(c**2*x**2 - 1)/(9*c**3) + b**2*d*x*acosh(c*x)
**2 + 2*b**2*d*x + b**2*e*x**3*acosh(c*x)**2/3 + 2*b**2*e*x**3/27 - 2*b**2*d*sqrt(c**2*x**2 - 1)*acosh(c*x)/c
- 2*b**2*e*x**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c) + 4*b**2*e*x/(9*c**2) - 4*b**2*e*sqrt(c**2*x**2 - 1)*acos
h(c*x)/(9*c**3), Ne(c, 0)), ((a + I*pi*b/2)**2*(d*x + e*x**3/3), True))

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