3.1.15 \(\int (d+e x)^2 (a+b \cosh ^{-1}(c x)) \, dx\) [15]

Optimal. Leaf size=132 \[ -\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {b d \left (2 d^2+\frac {3 e^2}{c^2}\right ) \cosh ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e} \]

[Out]

-1/6*b*d*(2*d^2+3*e^2/c^2)*arccosh(c*x)/e+1/3*(e*x+d)^3*(a+b*arccosh(c*x))/e-1/9*b*(e*x+d)^2*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/c-1/18*b*(5*c^2*d*e*x+16*c^2*d^2+4*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3

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Rubi [A]
time = 0.07, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5963, 102, 152, 54} \begin {gather*} \frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac {b d \left (\frac {3 e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{6 e}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {b \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/c - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5
*c^2*d*e*x))/(18*c^3) - (b*d*(2*d^2 + (3*e^2)/c^2)*ArcCosh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcCosh[c*x]))/(3
*e)

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}\\ &=-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac {b \int \frac {(d+e x) \left (3 c^2 d^2+2 e^2+5 c^2 d e x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c e}\\ &=-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac {1}{6} \left (b d \left (\frac {2 c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {b d \left (2 d^2+\frac {3 e^2}{c^2}\right ) \cosh ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 142, normalized size = 1.08 \begin {gather*} a d^2 x+a d e x^2+\frac {1}{3} a e^2 x^3-\frac {b \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 )}{18 c^3}+\frac {1}{3} b x \left (3 d^2+3 d e x+e^2 x^2\right ) \cosh ^{-1}(c x)-\frac {b d e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^2*x + a*d*e*x^2 + (a*e^2*x^3)/3 - (b*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)))/(18*c^3) + (b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCosh[c*x])/3 - (b*d*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c
*x]])/(2*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(115)=230\).
time = 2.45, size = 262, normalized size = 1.98

method	result	size
derivativedivides	\(\frac {\frac {\left (e c x +c d \right )^{3} a}{3 c^{2} e}+\frac {b c \,\mathrm {arccosh}\left (c x \right ) d^{3}}{3 e}+b \,\mathrm {arccosh}\left (c x \right ) d^{2} c x +b c e \,\mathrm {arccosh}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{3}-\frac {b c \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 e \sqrt {c^{2} x^{2}-1}}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}-\frac {b e \sqrt {c x -1}\, \sqrt {c x +1}\, d x}{2}-\frac {b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{9}-\frac {b e \sqrt {c x -1}\, \sqrt {c x +1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c \sqrt {c^{2} x^{2}-1}}-\frac {2 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{2}}}{c}\)	\(262\)
default	\(\frac {\frac {\left (e c x +c d \right )^{3} a}{3 c^{2} e}+\frac {b c \,\mathrm {arccosh}\left (c x \right ) d^{3}}{3 e}+b \,\mathrm {arccosh}\left (c x \right ) d^{2} c x +b c e \,\mathrm {arccosh}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{3}-\frac {b c \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 e \sqrt {c^{2} x^{2}-1}}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}-\frac {b e \sqrt {c x -1}\, \sqrt {c x +1}\, d x}{2}-\frac {b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{9}-\frac {b e \sqrt {c x -1}\, \sqrt {c x +1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c \sqrt {c^{2} x^{2}-1}}-\frac {2 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{2}}}{c}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 167, normalized size = 1.27 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} + \frac {1}{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 )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3*e^2 + a*d*x^2*e + a*d^2*x + 1/2*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*s
qrt(c^2*x^2 - 1)*c)/c^3))*b*d*e + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^2/c + 1/9*(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))*b*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (114) = 228\).
time = 0.37, size = 282, normalized size = 2.14 \begin {gather*} \frac {6 \, a c^{3} x^{3} \cosh \left (1\right )^{2} + 6 \, a c^{3} x^{3} \sinh \left (1\right )^{2} + 18 \, a c^{3} d x^{2} \cosh \left (1\right ) + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, b c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, b c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} d x^{2} - b c d\right )} \cosh \left (1\right ) + {\left (4 \, b c^{3} x^{3} \cosh \left (1\right ) + 6 \, b c^{3} d x^{2} - 3 \, b c d\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (2 \, a c^{3} x^{3} \cosh \left (1\right ) + 3 \, a c^{3} d x^{2}\right )} \sinh \left (1\right ) - {\left (9 \, b c^{2} d x \cosh \left (1\right ) + 18 \, b c^{2} d^{2} + 2 \, {\left (b c^{2} x^{2} + 2 \, b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b c^{2} x^{2} + 2 \, b\right )} \sinh \left (1\right )^{2} + {\left (9 \, b c^{2} d x + 4 \, {\left (b c^{2} x^{2} + 2 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(6*a*c^3*x^3*cosh(1)^2 + 6*a*c^3*x^3*sinh(1)^2 + 18*a*c^3*d*x^2*cosh(1) + 18*a*c^3*d^2*x + 3*(2*b*c^3*x^3
*cosh(1)^2 + 2*b*c^3*x^3*sinh(1)^2 + 6*b*c^3*d^2*x + 3*(2*b*c^3*d*x^2 - b*c*d)*cosh(1) + (4*b*c^3*x^3*cosh(1)
+ 6*b*c^3*d*x^2 - 3*b*c*d)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + 6*(2*a*c^3*x^3*cosh(1) + 3*a*c^3*d*x^2)*sin
h(1) - (9*b*c^2*d*x*cosh(1) + 18*b*c^2*d^2 + 2*(b*c^2*x^2 + 2*b)*cosh(1)^2 + 2*(b*c^2*x^2 + 2*b)*sinh(1)^2 + (
9*b*c^2*d*x + 4*(b*c^2*x^2 + 2*b)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/c^3

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Sympy [C] Result contains complex when optimal does not.
time = 0.19, size = 197, normalized size = 1.49 \begin {gather*} \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {acosh}{\left (c x \right )} + b d e x^{2} \operatorname {acosh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {b d^{2} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {b d e x \sqrt {c^{2} x^{2} - 1}}{2 c} - \frac {b e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {b d e \operatorname {acosh}{\left (c x \right )}}{2 c^{2}} - \frac {2 b e^{2} \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))*(d + e*x)^2,x)

[Out]

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

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