3.336 \(\int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=205 \[ -\frac{3 a x^2 \sqrt{a^2 c x^2+c}}{8 \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{8 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{4 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{8 a \sqrt{a^2 x^2+1}}+\frac{3}{4} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x) \]

[Out]

(-3*a*x^2*Sqrt[c + a^2*c*x^2])/(8*Sqrt[1 + a^2*x^2]) + (3*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/4 - (3*Sqrt[c +
a^2*c*x^2]*ArcSinh[a*x]^2)/(8*a*Sqrt[1 + a^2*x^2]) - (3*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(4*Sqrt[1 +
a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(8*a*Sqrt[1 + a^2*
x^2])

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Rubi [A]  time = 0.176458, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5682, 5675, 5661, 5758, 30} \[ -\frac{3 a x^2 \sqrt{a^2 c x^2+c}}{8 \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{8 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{4 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{8 a \sqrt{a^2 x^2+1}}+\frac{3}{4} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3,x]

[Out]

(-3*a*x^2*Sqrt[c + a^2*c*x^2])/(8*Sqrt[1 + a^2*x^2]) + (3*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/4 - (3*Sqrt[c +
a^2*c*x^2]*ArcSinh[a*x]^2)/(8*a*Sqrt[1 + a^2*x^2]) - (3*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(4*Sqrt[1 +
a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(8*a*Sqrt[1 + a^2*
x^2])

Rule 5682

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

Rule 5675

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

Rule 5661

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

Rule 5758

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

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

Mathematica [A]  time = 0.140813, size = 86, normalized size = 0.42 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (2 \sinh ^{-1}(a x) \left (\sinh ^{-1}(a x)^3+\left (2 \sinh ^{-1}(a x)^2+3\right ) \sinh \left (2 \sinh ^{-1}(a x)\right )\right )-3 \left (2 \sinh ^{-1}(a x)^2+1\right ) \cosh \left (2 \sinh ^{-1}(a x)\right )\right )}{16 a \sqrt{a^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3,x]

[Out]

(Sqrt[c*(1 + a^2*x^2)]*(-3*(1 + 2*ArcSinh[a*x]^2)*Cosh[2*ArcSinh[a*x]] + 2*ArcSinh[a*x]*(ArcSinh[a*x]^3 + (3 +
2*ArcSinh[a*x]^2)*Sinh[2*ArcSinh[a*x]])))/(16*a*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 0.118, size = 231, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}}{8\,a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}-6\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+6\,{\it Arcsinh} \left ( ax \right ) -3}{ \left ( 32\,{a}^{2}{x}^{2}+32 \right ) a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ( 2\,{x}^{3}{a}^{3}+2\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+2\,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) }+{\frac{4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}+6\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+6\,{\it Arcsinh} \left ( ax \right ) +3}{ \left ( 32\,{a}^{2}{x}^{2}+32 \right ) a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+2\,ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3*(a^2*c*x^2+c)^(1/2),x)

[Out]

1/8*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a*arcsinh(a*x)^4+1/32*(c*(a^2*x^2+1))^(1/2)*(2*x^3*a^3+2*a^2*x^2*(
a^2*x^2+1)^(1/2)+2*a*x+(a^2*x^2+1)^(1/2))*(4*arcsinh(a*x)^3-6*arcsinh(a*x)^2+6*arcsinh(a*x)-3)/(a^2*x^2+1)/a+1
/32*(c*(a^2*x^2+1))^(1/2)*(2*x^3*a^3-2*a^2*x^2*(a^2*x^2+1)^(1/2)+2*a*x-(a^2*x^2+1)^(1/2))*(4*arcsinh(a*x)^3+6*
arcsinh(a*x)^2+6*arcsinh(a*x)+3)/(a^2*x^2+1)/a

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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(arcsinh(a*x)^3*(a^2*c*x^2+c)^(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 (\sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3*(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{asinh}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3*(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(c*(a**2*x**2 + 1))*asinh(a*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3*(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^3, x)