3.105 \(\int \frac {x^4 (a+b \cosh ^{-1}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=212 \[ -\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d}+\frac {3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d}-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1}}{16 c^3 \sqrt {d-c^2 d x^2}} \]

[Out]

-3/16*b*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)-1/16*b*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^
2*d*x^2+d)^(1/2)+3/16*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^5/(-c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*a
rccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/4*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2/d

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Rubi [A]  time = 0.65, antiderivative size = 228, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5798, 5759, 5676, 30} \[ -\frac {x^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1}}{16 c^3 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]

[Out]

(-3*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1
6*c*Sqrt[d - c^2*d*x^2]) - (3*x*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(8*c^4*Sqrt[d - c^2*d*x^2]) - (x^3*(
1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(4*c^2*Sqrt[d - c^2*d*x^2]) + (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*
ArcCosh[c*x])^2)/(16*b*c^5*Sqrt[d - c^2*d*x^2])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5798

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

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^3 \, dx}{4 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{8 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{16 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.93, size = 171, normalized size = 0.81 \[ \frac {-\frac {16 a c x \left (2 c^2 x^2+3\right ) \sqrt {d-c^2 d x^2}}{d}-\frac {48 a \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{\sqrt {d}}+\frac {b \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (-16 \cosh \left (2 \cosh ^{-1}(c x)\right )-\cosh \left (4 \cosh ^{-1}(c x)\right )+4 \cosh ^{-1}(c x) \left (6 \cosh ^{-1}(c x)+8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {d-c^2 d x^2}}}{128 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]

[Out]

((-16*a*c*x*(3 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/d - (48*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2
*x^2))])/Sqrt[d] + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-16*Cosh[2*ArcCosh[c*x]] - Cosh[4*ArcCosh[c*x]] +
4*ArcCosh[c*x]*(6*ArcCosh[c*x] + 8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(128*c^
5)

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-(b*x^4*arccosh(c*x) + a*x^4)*sqrt(-c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x^4/sqrt(-c^2*d*x^2 + d), x)

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maple [B]  time = 0.82, size = 408, normalized size = 1.92 \[ -\frac {a \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2}}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4}}{16 d c \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{5}}{4 d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{3}}{8 d \,c^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{8 d \,c^{4} \left (c^{2} x^{2}-1\right )}-\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}}{128 d \,c^{5} \left (c^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)

[Out]

-1/4*a*x^3/c^2/d*(-c^2*d*x^2+d)^(1/2)-3/8*a/c^4*x/d*(-c^2*d*x^2+d)^(1/2)+3/8*a/c^4/(c^2*d)^(1/2)*arctan((c^2*d
)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-3/16*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^5/(c^2*x^2-1)*ar
ccosh(c*x)^2+1/16*b*(-d*(c^2*x^2-1))^(1/2)/d/c/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4+3/16*b*(-d*(c^2*x^2
-1))^(1/2)/d/c^3/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*arccos
h(c*x)*x^5-1/8*b*(-d*(c^2*x^2-1))^(1/2)/d/c^2/(c^2*x^2-1)*arccosh(c*x)*x^3+3/8*b*(-d*(c^2*x^2-1))^(1/2)/d/c^4/
(c^2*x^2-1)*arccosh(c*x)*x-15/128*b*(-d*(c^2*x^2-1))^(1/2)/d/c^5/(c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, a {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} x^{3}}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} x}{c^{4} d} - \frac {3 \, \arcsin \left (c x\right )}{c^{5} \sqrt {d}}\right )} + b \int \frac {x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/8*a*(2*sqrt(-c^2*d*x^2 + d)*x^3/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*x/(c^4*d) - 3*arcsin(c*x)/(c^5*sqrt(d))) +
b*integrate(x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2),x)

[Out]

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**4*(a + b*acosh(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

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