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

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

[Out]

-2/27*c*(a^2*x^2+1)^(3/2)/a+14/3*c*x*arcsinh(a*x)+2/9*a^2*c*x^3*arcsinh(a*x)-1/3*c*(a^2*x^2+1)^(3/2)*arcsinh(a
*x)^2/a+2/3*c*x*arcsinh(a*x)^3+1/3*c*x*(a^2*x^2+1)*arcsinh(a*x)^3-40/9*c*(a^2*x^2+1)^(1/2)/a-2*c*arcsinh(a*x)^
2*(a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.22, antiderivative size = 153, normalized size of antiderivative = 1.00, 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 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])

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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.08, 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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fricas [A]  time = 0.58, size = 140, normalized size = 0.92 \[ \frac {9 \, {\left (a^{3} c x^{3} + 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 9 \, {\left (a^{2} c x^{2} + 7 \, c\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (a^{3} c x^{3} + 21 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (a^{2} c x^{2} + 61 \, c\right )} \sqrt {a^{2} x^{2} + 1}}{27 \, a} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.06, size = 128, normalized size = 0.84 \[ \frac {c \left (9 \arcsinh \left (a x \right )^{3} a^{3} x^{3}-9 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+27 \arcsinh \left (a x \right )^{3} a x +6 \arcsinh \left (a x \right ) a^{3} x^{3}-63 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-2 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+126 a x \arcsinh \left (a x \right )-122 \sqrt {a^{2} x^{2}+1}\right )}{27 a} \]

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*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*a^2*x^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+1)^(1/2)*x^2*a^2+126*a*x*arcsinh(a*x)-122*(a^2*x^2
+1)^(1/2))

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maxima [A]  time = 0.56, size = 124, normalized size = 0.81 \[ -\frac {1}{3} \, {\left (\sqrt {a^{2} x^{2} + 1} c x^{2} + \frac {7 \, \sqrt {a^{2} x^{2} + 1} c}{a^{2}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{3} \, {\left (a^{2} c x^{3} + 3 \, c x\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {2}{27} \, {\left (\sqrt {a^{2} x^{2} + 1} c x^{2} - \frac {3 \, {\left (a^{2} c x^{3} + 21 \, c x\right )} \operatorname {arsinh}\left (a x\right )}{a} + \frac {61 \, \sqrt {a^{2} x^{2} + 1} c}{a^{2}}\right )} a \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {asinh}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^3*(c + a^2*c*x^2),x)

[Out]

int(asinh(a*x)^3*(c + a^2*c*x^2), x)

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sympy [A]  time = 2.08, size = 150, normalized size = 0.98 \[ \begin {cases} \frac {a^{2} c x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {asinh}{\left (a x \right )}}{9} - \frac {a c x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3} - \frac {2 a c x^{2} \sqrt {a^{2} x^{2} + 1}}{27} + c x \operatorname {asinh}^{3}{\left (a x \right )} + \frac {14 c x \operatorname {asinh}{\left (a x \right )}}{3} - \frac {7 c \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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