3.1.13 \(\int x (d-c^2 d x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\) [13]

Optimal. Leaf size=136 \[ -\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}+\frac {5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2} \]

[Out]

5/144*b*d^2*x*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-1/36*b*d^2*x*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+5/96*b*d^2*arccosh(c*x)
/c^2-1/6*d^2*(-c^2*x^2+1)^3*(a+b*arccosh(c*x))/c^2-5/96*b*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

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Rubi [A]
time = 0.05, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5914, 38, 54} \begin {gather*} -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac {5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac {b d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{36 c}+\frac {5 b d^2 x (c x-1)^{3/2} (c x+1)^{3/2}}{144 c}-\frac {5 b d^2 x \sqrt {c x-1} \sqrt {c x+1}}{96 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c) + (5*b*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(144*c) - (b*d
^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(36*c) + (5*b*d^2*ArcCosh[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*
ArcCosh[c*x]))/(6*c^2)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

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 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}-\frac {\left (b d^2\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{6 c}\\ &=-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac {\left (5 b d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{36 c}\\ &=\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}-\frac {\left (5 b d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{48 c}\\ &=-\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac {\left (5 b d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 c}\\ &=-\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}+\frac {5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 132, normalized size = 0.97 \begin {gather*} \frac {d^2 \left (c x \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-33+26 c^2 x^2-8 c^4 x^4\right )+48 a c x \left (3-3 c^2 x^2+c^4 x^4\right )\right )+48 b c^2 x^2 \left (3-3 c^2 x^2+c^4 x^4\right ) \cosh ^{-1}(c x)-33 b \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )\right )}{288 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-33 + 26*c^2*x^2 - 8*c^4*x^4) + 48*a*c*x*(3 - 3*c^2*x^2 + c^4*x^4))
+ 48*b*c^2*x^2*(3 - 3*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 33*b*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]]))/(288*c
^2)

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Maple [A]
time = 2.75, size = 202, normalized size = 1.49

method	result	size
derivativedivides	\(\frac {\frac {d^{2} \left (c^{2} x^{2}-1\right )^{3} a}{6}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{6}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}+\frac {13 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{144}-\frac {11 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96}+\frac {5 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 \sqrt {c^{2} x^{2}-1}}}{c^{2}}\)	\(202\)
default	\(\frac {\frac {d^{2} \left (c^{2} x^{2}-1\right )^{3} a}{6}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{6}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}+\frac {13 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{144}-\frac {11 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96}+\frac {5 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 \sqrt {c^{2} x^{2}-1}}}{c^{2}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(1/6*d^2*(c^2*x^2-1)^3*a+1/6*d^2*b*arccosh(c*x)*c^6*x^6-1/2*d^2*b*arccosh(c*x)*c^4*x^4+1/2*d^2*b*arccosh
(c*x)*c^2*x^2-1/6*b*d^2*arccosh(c*x)-1/36*d^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*x^5+13/144*d^2*b*(c*x-1)^(1/2)
*(c*x+1)^(1/2)*c^3*x^3-11/96*b*c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+5/96*d^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2
*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (113) = 226\).
time = 0.27, size = 287, normalized size = 2.11 \begin {gather*} \frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} - \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\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^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^
2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*c^4*d^2 -
1/16*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c
^2*x^2 - 1)*c)/c^5)*c)*b*c^2*d^2 + 1/2*a*d^2*x^2 + 1/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(
2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*b*d^2

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Fricas [A]
time = 0.39, size = 149, normalized size = 1.10 \begin {gather*} \frac {48 \, a c^{6} d^{2} x^{6} - 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} - 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} - 11 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} d^{2} x^{5} - 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(48*a*c^6*d^2*x^6 - 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b*c^6*d^2*x^6 - 48*b*c^4*d^2*x^4 + 48*
b*c^2*d^2*x^2 - 11*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*d^2*x^5 - 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*s
qrt(c^2*x^2 - 1))/c^2

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Sympy [C] Result contains complex when optimal does not.
time = 0.53, size = 197, normalized size = 1.45 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{6}}{6} - \frac {a c^{2} d^{2} x^{4}}{2} + \frac {a d^{2} x^{2}}{2} + \frac {b c^{4} d^{2} x^{6} \operatorname {acosh}{\left (c x \right )}}{6} - \frac {b c^{3} d^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{36} - \frac {b c^{2} d^{2} x^{4} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {13 b c d^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{144} + \frac {b d^{2} x^{2} \operatorname {acosh}{\left (c x \right )}}{2} - \frac {11 b d^{2} x \sqrt {c^{2} x^{2} - 1}}{96 c} - \frac {11 b d^{2} \operatorname {acosh}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{2} \left (a + \frac {i \pi b}{2}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**4*d**2*x**6/6 - a*c**2*d**2*x**4/2 + a*d**2*x**2/2 + b*c**4*d**2*x**6*acosh(c*x)/6 - b*c**3*d*
*2*x**5*sqrt(c**2*x**2 - 1)/36 - b*c**2*d**2*x**4*acosh(c*x)/2 + 13*b*c*d**2*x**3*sqrt(c**2*x**2 - 1)/144 + b*
d**2*x**2*acosh(c*x)/2 - 11*b*d**2*x*sqrt(c**2*x**2 - 1)/(96*c) - 11*b*d**2*acosh(c*x)/(96*c**2), Ne(c, 0)), (
d**2*x**2*(a + I*pi*b/2)/2, 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(x*(-c^2*d*x^2+d)^2*(a+b*arccosh(c*x)),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 = -0.01 \begin {gather*} \int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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