3.2.17 \(\int (a+b \cosh ^{-1}(c+d x))^3 \, dx\) [117]

Optimal. Leaf size=114 \[ 6 a b^2 x-\frac {6 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{d}+\frac {6 b^3 (c+d x) \cosh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5995, 5879, 5915, 75} \begin {gather*} 6 a b^2 x-\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {6 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1}}{d}+\frac {6 b^3 (c+d x) \cosh ^{-1}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^3,x]

[Out]

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

Rule 75

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] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5879

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 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 5995

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x+\frac {6 b^3 (c+d x) \cosh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x-\frac {6 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{d}+\frac {6 b^3 (c+d x) \cosh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 168, normalized size = 1.47 \begin {gather*} \frac {a \left (a^2+6 b^2\right ) (c+d x)-3 b \left (a^2+2 b^2\right ) \sqrt {-1+c+d x} \sqrt {1+c+d x}-3 b \left (-a^2 (c+d x)-2 b^2 (c+d x)+2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)-3 b^2 \left (-a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)^2+b^3 (c+d x) \cosh ^{-1}(c+d x)^3}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3,x]

[Out]

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

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Maple [A]
time = 49.31, size = 180, normalized size = 1.58

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\mathrm {arccosh}\left (d x +c \right )^{3} \left (d x +c \right )-3 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}+6 \left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+3 a \,b^{2} \left (\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\)	\(180\)
default	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\mathrm {arccosh}\left (d x +c \right )^{3} \left (d x +c \right )-3 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}+6 \left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+3 a \,b^{2} \left (\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((d*x+c)*a^3+b^3*(arccosh(d*x+c)^3*(d*x+c)-3*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)+6*(d*x+c)*ar
ccosh(d*x+c)-6*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))+3*a*b^2*(arccosh(d*x+c)^2*(d*x+c)-2*arccosh(d*x+c)*(d*x+c-1)^(
1/2)*(d*x+c+1)^(1/2)+2*d*x+2*c)+3*a^2*b*((d*x+c)*arccosh(d*x+c)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))

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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((a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (106) = 212\).
time = 0.37, size = 239, normalized size = 2.10 \begin {gather*} \frac {{\left (b^{3} d x + b^{3} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + {\left (a^{3} + 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - 3 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b^{2} - {\left (a^{2} b + 2 \, b^{3}\right )} d x - {\left (a^{2} b + 2 \, b^{3}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (a^{2} b + 2 \, b^{3}\right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (109) = 218\).
time = 0.19, size = 282, normalized size = 2.47 \begin {gather*} \begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {acosh}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {acosh}{\left (c + d x \right )} - \frac {3 a^{2} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {3 a b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 6 a b^{2} x - \frac {6 a b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {acosh}^{3}{\left (c + d x \right )}}{d} + \frac {6 b^{3} c \operatorname {acosh}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {acosh}^{3}{\left (c + d x \right )} + 6 b^{3} x \operatorname {acosh}{\left (c + d x \right )} - \frac {3 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**3,x)

[Out]

Piecewise((a**3*x + 3*a**2*b*c*acosh(c + d*x)/d + 3*a**2*b*x*acosh(c + d*x) - 3*a**2*b*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)/d + 3*a*b**2*c*acosh(c + d*x)**2/d + 3*a*b**2*x*acosh(c + d*x)**2 + 6*a*b**2*x - 6*a*b**2*sqrt(c
**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/d + b**3*c*acosh(c + d*x)**3/d + 6*b**3*c*acosh(c + d*x)/d + b**
3*x*acosh(c + d*x)**3 + 6*b**3*x*acosh(c + d*x) - 3*b**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**
2/d - 6*b**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/d, Ne(d, 0)), (x*(a + b*acosh(c))**3, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^3,x)

[Out]

int((a + b*acosh(c + d*x))^3, x)

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