3.2.8 \(\int (a+b \cosh ^{-1}(c+d x))^2 \, dx\) [108]

Optimal. Leaf size=64 \[ 2 b^2 x-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5995, 5879, 5915, 8} \begin {gather*} -\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+2 b^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^2,x]

[Out]

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

Rule 8

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

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

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Mathematica [A]
time = 0.06, size = 105, normalized size = 1.64 \begin {gather*} \frac {a^2 (c+d x)+2 b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}-2 b \left (-a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)+b^2 (c+d x) \cosh ^{-1}(c+d x)^2}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 16.32, size = 100, normalized size = 1.56

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*((d*x+c)*a^2+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)+2*a
*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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (60) = 120\).
time = 0.40, size = 141, normalized size = 2.20 \begin {gather*} \frac {{\left (a^{2} + 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b + 2 \, {\left (a b d x + a b c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (60) = 120\).
time = 0.12, size = 143, normalized size = 2.23 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b c \operatorname {acosh}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + 2*a*b*c*acosh(c + d*x)/d + 2*a*b*x*acosh(c + d*x) - 2*a*b*sqrt(c**2 + 2*c*d*x + d**2*x**2
- 1)/d + b**2*c*acosh(c + d*x)**2/d + b**2*x*acosh(c + d*x)**2 + 2*b**2*x - 2*b**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)*acosh(c + d*x)/d, Ne(d, 0)), (x*(a + b*acosh(c))**2, 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))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^2,x)

[Out]

int((a + b*acosh(c + d*x))^2, x)

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