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

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

[Out]

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

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Rubi [A]  time = 0.647655, 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 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]

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 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 30

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

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*}

Mathematica [A]  time = 0.843036, 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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Maple [B]  time = 0.355, size = 408, normalized size = 1.9 \begin{align*} -{\frac{{x}^{3}a}{4\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{3\,ax}{8\,d{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{3\,a}{8\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b{x}^{4}}{16\,cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}+{\frac{3\,b{x}^{2}}{16\,d{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{3\,b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{16\,{c}^{5}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{5}}{4\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}}{8\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,b{\rm arccosh} \left (cx\right )x}{8\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{15\,b}{128\,{c}^{5}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}} \end{align*}

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))+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^2-3/16*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)*arccosh(c*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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

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)