3.1.2 \(\int x^3 (d-c^2 d x^2) (a+b \cosh ^{-1}(c x)) \, dx\) [2]

Optimal. Leaf size=135 \[ -\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \cosh ^{-1}(c x)}{24 c^4}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right ) \]

[Out]

-1/24*b*d*arccosh(c*x)/c^4+1/4*d*x^4*(a+b*arccosh(c*x))-1/6*c^2*d*x^6*(a+b*arccosh(c*x))-1/24*b*d*x*(c*x-1)^(1
/2)*(c*x+1)^(1/2)/c^3-1/36*b*d*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+1/36*b*c*d*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {14, 5921, 12, 471, 102, 92, 54} \begin {gather*} -\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d \cosh ^{-1}(c x)}{24 c^4}-\frac {b d x \sqrt {c x-1} \sqrt {c x+1}}{24 c^3}+\frac {1}{36} b c d x^5 \sqrt {c x-1} \sqrt {c x+1}-\frac {b d x^3 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

-1/24*(b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (b*c*d*x^5*Sq
rt[-1 + c*x]*Sqrt[1 + c*x])/36 - (b*d*ArcCosh[c*x])/(24*c^4) + (d*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d*x^6*(a
+ b*ArcCosh[c*x]))/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 5921

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^3 \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{12} (b c d) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} (b c d) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{36 c}\\ &=-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{12 c}\\ &=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{24 c^3}\\ &=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \cosh ^{-1}(c x)}{24 c^4}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 166, normalized size = 1.23 \begin {gather*} \frac {1}{4} a d x^4-\frac {1}{6} a c^2 d x^6-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} b d x^4 \cosh ^{-1}(c x)-\frac {1}{6} b c^2 d x^6 \cosh ^{-1}(c x)-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{12 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

(a*d*x^4)/4 - (a*c^2*d*x^6)/6 - (b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1
+ c*x])/(36*c) + (b*c*d*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/36 + (b*d*x^4*ArcCosh[c*x])/4 - (b*c^2*d*x^6*ArcCos
h[c*x])/6 - (b*d*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(12*c^4)

________________________________________________________________________________________

Maple [A]
time = 2.66, size = 186, normalized size = 1.38

method	result	size
derivativedivides	\(\frac {-d a \left (\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\left (c^{2} x^{2}-1\right )^{2}}{4}\right )-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{12}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{36}-\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{24}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\)	\(186\)
default	\(\frac {-d a \left (\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\left (c^{2} x^{2}-1\right )^{2}}{4}\right )-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{12}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{36}-\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{24}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^4*(-d*a*(1/6*(c^2*x^2-1)^3+1/4*(c^2*x^2-1)^2)-1/6*b*d*arccosh(c*x)*c^6*x^6+1/4*b*d*arccosh(c*x)*c^4*x^4-1/
12*b*d*arccosh(c*x)+1/36*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*x^5-1/36*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-
1/24*b*c*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/24*b*d*(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)))

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 202, normalized size = 1.50 \begin {gather*} -\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d 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^{2} d + \frac {1}{32} \, {\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 d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/6*a*c^2*d*x^6 + 1/4*a*d*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^2*d + 1/32*(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*d

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 108, normalized size = 0.80 \begin {gather*} -\frac {12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{72 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/72*(12*a*c^6*d*x^6 - 18*a*c^4*d*x^4 + 3*(4*b*c^6*d*x^6 - 6*b*c^4*d*x^4 + b*d)*log(c*x + sqrt(c^2*x^2 - 1))
- (2*b*c^5*d*x^5 - 2*b*c^3*d*x^3 - 3*b*c*d*x)*sqrt(c^2*x^2 - 1))/c^4

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 144, normalized size = 1.07 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{6}}{6} + \frac {a d x^{4}}{4} - \frac {b c^{2} d x^{6} \operatorname {acosh}{\left (c x \right )}}{6} + \frac {b c d x^{5} \sqrt {c^{2} x^{2} - 1}}{36} + \frac {b d x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {b d x^{3} \sqrt {c^{2} x^{2} - 1}}{36 c} - \frac {b d x \sqrt {c^{2} x^{2} - 1}}{24 c^{3}} - \frac {b d \operatorname {acosh}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {d x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-c**2*d*x**2+d)*(a+b*acosh(c*x)),x)

[Out]

Piecewise((-a*c**2*d*x**6/6 + a*d*x**4/4 - b*c**2*d*x**6*acosh(c*x)/6 + b*c*d*x**5*sqrt(c**2*x**2 - 1)/36 + b*
d*x**4*acosh(c*x)/4 - b*d*x**3*sqrt(c**2*x**2 - 1)/(36*c) - b*d*x*sqrt(c**2*x**2 - 1)/(24*c**3) - b*d*acosh(c*
x)/(24*c**4), Ne(c, 0)), (d*x**4*(a + I*pi*b/2)/4, True))

________________________________________________________________________________________

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^3*(-c^2*d*x^2+d)*(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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*acosh(c*x))*(d - c^2*d*x^2),x)

[Out]

int(x^3*(a + b*acosh(c*x))*(d - c^2*d*x^2), x)

________________________________________________________________________________________