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

Optimal. Leaf size=153 \[ -\frac{2 c \left (a^2 x^2+1\right )^{3/2}}{27 a}-\frac{40 c \sqrt{a^2 x^2+1}}{9 a}+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)+\frac{1}{3} c x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{c \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{14}{3} c x \sinh ^{-1}(a x) \]

[Out]

(-40*c*Sqrt[1 + a^2*x^2])/(9*a) - (2*c*(1 + a^2*x^2)^(3/2))/(27*a) + (14*c*x*ArcSinh[a*x])/3 + (2*a^2*c*x^3*Ar
cSinh[a*x])/9 - (2*c*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a - (c*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(3*a) + (2*c
*x*ArcSinh[a*x]^3)/3 + (c*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/3

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Rubi [A]  time = 0.217183, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {5684, 5653, 5717, 261, 5679, 444, 43} \[ -\frac{2 c \left (a^2 x^2+1\right )^{3/2}}{27 a}-\frac{40 c \sqrt{a^2 x^2+1}}{9 a}+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)+\frac{1}{3} c x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{c \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{14}{3} c x \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)*ArcSinh[a*x]^3,x]

[Out]

(-40*c*Sqrt[1 + a^2*x^2])/(9*a) - (2*c*(1 + a^2*x^2)^(3/2))/(27*a) + (14*c*x*ArcSinh[a*x])/3 + (2*a^2*c*x^3*Ar
cSinh[a*x])/9 - (2*c*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a - (c*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(3*a) + (2*c
*x*ArcSinh[a*x]^3)/3 + (c*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/3

Rule 5684

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

Rule 5653

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

Rule 5717

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5679

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \sinh ^{-1}(a x)^3 \, dx-(a c) \int x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \, dx\\ &=-\frac{c \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \left (1+a^2 x^2\right ) \sinh ^{-1}(a x) \, dx-(2 a c) \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{2}{3} c x \sinh ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)-\frac{2 c \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}-\frac{c \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+(4 c) \int \sinh ^{-1}(a x) \, dx-\frac{1}{3} (2 a c) \int \frac{x \left (1+\frac{a^2 x^2}{3}\right )}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{14}{3} c x \sinh ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)-\frac{2 c \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}-\frac{c \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3-\frac{1}{3} (a c) \operatorname{Subst}\left (\int \frac{1+\frac{a^2 x}{3}}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-(4 a c) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{4 c \sqrt{1+a^2 x^2}}{a}+\frac{14}{3} c x \sinh ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)-\frac{2 c \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}-\frac{c \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3-\frac{1}{3} (a c) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+a^2 x}}+\frac{1}{3} \sqrt{1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{40 c \sqrt{1+a^2 x^2}}{9 a}-\frac{2 c \left (1+a^2 x^2\right )^{3/2}}{27 a}+\frac{14}{3} c x \sinh ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)-\frac{2 c \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}-\frac{c \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{1}{3} c x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0819733, size = 99, normalized size = 0.65 \[ \frac{c \left (-2 \sqrt{a^2 x^2+1} \left (a^2 x^2+61\right )+9 a x \left (a^2 x^2+3\right ) \sinh ^{-1}(a x)^3-9 \sqrt{a^2 x^2+1} \left (a^2 x^2+7\right ) \sinh ^{-1}(a x)^2+6 a x \left (a^2 x^2+21\right ) \sinh ^{-1}(a x)\right )}{27 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-2*Sqrt[1 + a^2*x^2]*(61 + a^2*x^2) + 6*a*x*(21 + a^2*x^2)*ArcSinh[a*x] - 9*Sqrt[1 + a^2*x^2]*(7 + a^2*x^2
)*ArcSinh[a*x]^2 + 9*a*x*(3 + a^2*x^2)*ArcSinh[a*x]^3))/(27*a)

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Maple [A]  time = 0.036, size = 128, normalized size = 0.8 \begin{align*}{\frac{c}{27\,a} \left ( 9\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}-9\,{a}^{2}{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}+27\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax+6\,{\it Arcsinh} \left ( ax \right ){a}^{3}{x}^{3}-63\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-2\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+126\,ax{\it Arcsinh} \left ( ax \right ) -122\,\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27/a*c*(9*arcsinh(a*x)^3*a^3*x^3-9*a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+27*arcsinh(a*x)^3*a*x+6*arcsinh(
a*x)*a^3*x^3-63*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-2*a^2*x^2*(a^2*x^2+1)^(1/2)+126*a*x*arcsinh(a*x)-122*(a^2*x^2
+1)^(1/2))

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Maxima [A]  time = 1.20982, size = 167, normalized size = 1.09 \begin{align*} -\frac{1}{3} \,{\left (\sqrt{a^{2} x^{2} + 1} c x^{2} + \frac{7 \, \sqrt{a^{2} x^{2} + 1} c}{a^{2}}\right )} a \operatorname{arsinh}\left (a x\right )^{2} + \frac{1}{3} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \operatorname{arsinh}\left (a x\right )^{3} - \frac{2}{27} \,{\left (\sqrt{a^{2} x^{2} + 1} c x^{2} - \frac{3 \,{\left (a^{2} c x^{3} + 21 \, c x\right )} \operatorname{arsinh}\left (a x\right )}{a} + \frac{61 \, \sqrt{a^{2} x^{2} + 1} c}{a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(a^2*x^2 + 1)*c*x^2 + 7*sqrt(a^2*x^2 + 1)*c/a^2)*a*arcsinh(a*x)^2 + 1/3*(a^2*c*x^3 + 3*c*x)*arcsinh(
a*x)^3 - 2/27*(sqrt(a^2*x^2 + 1)*c*x^2 - 3*(a^2*c*x^3 + 21*c*x)*arcsinh(a*x)/a + 61*sqrt(a^2*x^2 + 1)*c/a^2)*a

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Fricas [A]  time = 2.09969, size = 315, normalized size = 2.06 \begin{align*} \frac{9 \,{\left (a^{3} c x^{3} + 3 \, a c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 9 \,{\left (a^{2} c x^{2} + 7 \, c\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (a^{3} c x^{3} + 21 \, a c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 2 \,{\left (a^{2} c x^{2} + 61 \, c\right )} \sqrt{a^{2} x^{2} + 1}}{27 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*(a^3*c*x^3 + 3*a*c*x)*log(a*x + sqrt(a^2*x^2 + 1))^3 - 9*(a^2*c*x^2 + 7*c)*sqrt(a^2*x^2 + 1)*log(a*x +
sqrt(a^2*x^2 + 1))^2 + 6*(a^3*c*x^3 + 21*a*c*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 2*(a^2*c*x^2 + 61*c)*sqrt(a^2*
x^2 + 1))/a

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Sympy [A]  time = 2.46731, size = 150, normalized size = 0.98 \begin{align*} \begin{cases} \frac{a^{2} c x^{3} \operatorname{asinh}^{3}{\left (a x \right )}}{3} + \frac{2 a^{2} c x^{3} \operatorname{asinh}{\left (a x \right )}}{9} - \frac{a c x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3} - \frac{2 a c x^{2} \sqrt{a^{2} x^{2} + 1}}{27} + c x \operatorname{asinh}^{3}{\left (a x \right )} + \frac{14 c x \operatorname{asinh}{\left (a x \right )}}{3} - \frac{7 c \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{3 a} - \frac{122 c \sqrt{a^{2} x^{2} + 1}}{27 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c*x**3*asinh(a*x)**3/3 + 2*a**2*c*x**3*asinh(a*x)/9 - a*c*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)*
*2/3 - 2*a*c*x**2*sqrt(a**2*x**2 + 1)/27 + c*x*asinh(a*x)**3 + 14*c*x*asinh(a*x)/3 - 7*c*sqrt(a**2*x**2 + 1)*a
sinh(a*x)**2/(3*a) - 122*c*sqrt(a**2*x**2 + 1)/(27*a), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.67821, size = 196, normalized size = 1.28 \begin{align*} \frac{1}{3} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + \frac{1}{27} \,{\left (6 \,{\left (a^{2} x^{3} + 21 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{9 \,{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 6 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a} - \frac{2 \,{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 60 \, \sqrt{a^{2} x^{2} + 1}\right )}}{a}\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^2*c*x^3 + 3*c*x)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 1/27*(6*(a^2*x^3 + 21*x)*log(a*x + sqrt(a^2*x^2 + 1))
- 9*((a^2*x^2 + 1)^(3/2) + 6*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2/a - 2*((a^2*x^2 + 1)^(3/2) + 6
0*sqrt(a^2*x^2 + 1))/a)*c