3.92 \(\int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx\)
Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{1}{x \sinh ^{-1}(a+b x)^3},x\right ) \]
[Out]
Unintegrable[1/(x*ArcSinh[a + b*x]^3), x]
________________________________________________________________________________________
Rubi [A] time = 0.0395817, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/(x*ArcSinh[a + b*x]^3),x]
[Out]
Defer[Subst][Defer[Int][1/((-(a/b) + x/b)*ArcSinh[x]^3), x], x, a + b*x]/b
Rubi steps
\begin{align*} \int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+\frac{x}{b}\right ) \sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ \end{align*}
Mathematica [A] time = 2.02481, size = 0, normalized size = 0. \[ \int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/(x*ArcSinh[a + b*x]^3),x]
[Out]
Integrate[1/(x*ArcSinh[a + b*x]^3), x]
________________________________________________________________________________________
Maple [A] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/arcsinh(b*x+a)^3,x)
[Out]
int(1/x/arcsinh(b*x+a)^3,x)
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/arcsinh(b*x+a)^3,x, algorithm="maxima")
[Out]
-1/2*(b^8*x^8 + 7*a*b^7*x^7 + 3*(7*a^2*b^6 + b^6)*x^6 + 5*(7*a^3*b^5 + 3*a*b^5)*x^5 + (35*a^4*b^4 + 30*a^2*b^4
+ 3*b^4)*x^4 + 3*(7*a^5*b^3 + 10*a^3*b^3 + 3*a*b^3)*x^3 + (7*a^6*b^2 + 15*a^4*b^2 + 9*a^2*b^2 + b^2)*x^2 + (b
^5*x^5 + 4*a*b^4*x^4 + (6*a^2*b^3 + b^3)*x^3 + 2*(2*a^3*b^2 + a*b^2)*x^2 + (a^4*b + a^2*b)*x)*(b^2*x^2 + 2*a*b
*x + a^2 + 1)^(3/2) + (3*b^6*x^6 + 15*a*b^5*x^5 + 5*(6*a^2*b^4 + b^4)*x^4 + 15*(2*a^3*b^3 + a*b^3)*x^3 + (15*a
^4*b^2 + 15*a^2*b^2 + 2*b^2)*x^2 + (3*a^5*b + 5*a^3*b + 2*a*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + (a^7*b + 3*a
^5*b + 3*a^3*b + a*b)*x - (a*b^7*x^7 + 7*a^2*b^6*x^6 + a^8 + 3*a^6 + 3*(7*a^3*b^5 + a*b^5)*x^5 + 5*(7*a^4*b^4
+ 3*a^2*b^4)*x^4 + 3*a^4 + (35*a^5*b^3 + 30*a^3*b^3 + 3*a*b^3)*x^3 + 3*(7*a^6*b^2 + 10*a^4*b^2 + 3*a^2*b^2)*x^
2 + (a*b^4*x^4 + a^5 + 2*(2*a^2*b^3 + b^3)*x^3 + 2*a^3 + 6*(a^3*b^2 + a*b^2)*x^2 + 2*(2*a^4*b + 3*a^2*b + b)*x
+ a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*a*b^5*x^5 + 3*a^6 + (15*a^2*b^4 + 4*b^4)*x^4 + 7*a^4 + (30*a^3*
b^3 + 19*a*b^3)*x^3 + (30*a^4*b^2 + 33*a^2*b^2 + 5*b^2)*x^2 + 5*a^2 + 5*(3*a^5*b + 5*a^3*b + 2*a*b)*x + 1)*(b^
2*x^2 + 2*a*b*x + a^2 + 1) + a^2 + (7*a^7*b + 15*a^5*b + 9*a^3*b + a*b)*x + (3*a*b^6*x^6 + 3*a^7 + 2*(9*a^2*b^
5 + b^5)*x^5 + 8*a^5 + (45*a^3*b^4 + 16*a*b^4)*x^4 + (60*a^4*b^3 + 44*a^2*b^3 + 3*b^3)*x^3 + 7*a^3 + (45*a^5*b
^2 + 56*a^3*b^2 + 13*a*b^2)*x^2 + (18*a^6*b + 34*a^4*b + 17*a^2*b + b)*x + 2*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (3*b^7*x^7 + 18*a*b^6*x^6 + (45*a^2*b^5 + 7*b^5)*x^5 +
4*(15*a^3*b^4 + 7*a*b^4)*x^4 + (45*a^4*b^3 + 42*a^2*b^3 + 5*b^3)*x^3 + 2*(9*a^5*b^2 + 14*a^3*b^2 + 5*a*b^2)*x
^2 + (3*a^6*b + 7*a^4*b + 5*a^2*b + b)*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((b^8*x^8 + 6*a*b^7*x^7 + 3*(5*a^
2*b^6 + b^6)*x^6 + 4*(5*a^3*b^5 + 3*a*b^5)*x^5 + 3*(5*a^4*b^4 + 6*a^2*b^4 + b^4)*x^4 + 6*(a^5*b^3 + 2*a^3*b^3
+ a*b^3)*x^3 + (a^6*b^2 + 3*a^4*b^2 + 3*a^2*b^2 + b^2)*x^2 + (b^5*x^5 + 3*a*b^4*x^4 + 3*a^2*b^3*x^3 + a^3*b^2*
x^2)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 3*(b^6*x^6 + 4*a*b^5*x^5 + (6*a^2*b^4 + b^4)*x^4 + 2*(2*a^3*b^3 + a
*b^3)*x^3 + (a^4*b^2 + a^2*b^2)*x^2)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 3*(b^7*x^7 + 5*a*b^6*x^6 + 2*(5*a^2*b^5 +
b^5)*x^5 + 2*(5*a^3*b^4 + 3*a*b^4)*x^4 + (5*a^4*b^3 + 6*a^2*b^3 + b^3)*x^3 + (a^5*b^2 + 2*a^3*b^2 + a*b^2)*x^
2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2) + integrate(1/2*(a*b
^9*x^9 + 10*a^2*b^8*x^8 + 2*a^10 + 8*a^8 + 4*(11*a^3*b^7 + a*b^7)*x^7 + 16*(7*a^4*b^6 + 2*a^2*b^6)*x^6 + 12*a^
6 + 2*(91*a^5*b^5 + 54*a^3*b^5 + 3*a*b^5)*x^5 + 4*(49*a^6*b^4 + 50*a^4*b^4 + 9*a^2*b^4)*x^4 + 8*a^4 + 4*(35*a^
7*b^3 + 55*a^5*b^3 + 21*a^3*b^3 + a*b^3)*x^3 + (a*b^5*x^5 + 2*a^6 + 2*(3*a^2*b^4 + 2*b^4)*x^4 + 4*a^4 + 2*(7*a
^3*b^3 + 8*a*b^3)*x^3 + 8*(2*a^4*b^2 + 3*a^2*b^2 + b^2)*x^2 + 2*a^2 + (9*a^5*b + 16*a^3*b + 7*a*b)*x)*(b^2*x^2
+ 2*a*b*x + a^2 + 1)^2 + 16*(4*a^8*b^2 + 9*a^6*b^2 + 6*a^4*b^2 + a^2*b^2)*x^2 + (4*a*b^6*x^6 + 8*a^7 + 4*(7*a
^2*b^5 + 3*b^5)*x^5 + 20*a^5 + 16*(5*a^3*b^4 + 4*a*b^4)*x^4 + 2*(60*a^4*b^3 + 70*a^2*b^3 + 11*b^3)*x^3 + 16*a^
3 + (100*a^5*b^2 + 156*a^3*b^2 + 57*a*b^2)*x^2 + (44*a^6*b + 88*a^4*b + 51*a^2*b + 7*b)*x + 4*a)*(b^2*x^2 + 2*
a*b*x + a^2 + 1)^(3/2) + (6*a*b^7*x^7 + 12*a^8 + 12*(4*a^2*b^6 + b^6)*x^6 + 36*a^6 + 6*(27*a^3*b^5 + 14*a*b^5)
*x^5 + 4*(75*a^4*b^4 + 63*a^2*b^4 + 5*b^4)*x^4 + 38*a^4 + (330*a^5*b^3 + 408*a^3*b^3 + 95*a*b^3)*x^3 + 2*(108*
a^6*b^2 + 186*a^4*b^2 + 84*a^2*b^2 + 5*b^2)*x^2 + 16*a^2 + (78*a^7*b + 180*a^5*b + 131*a^3*b + 29*a*b)*x + 2)*
(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*a^2 + (17*a^9*b + 52*a^7*b + 54*a^5*b + 20*a^3*b + a*b)*x + (4*a*b^8*x^8 + 8
*a^9 + 4*(9*a^2*b^7 + b^7)*x^7 + 28*a^7 + 20*(7*a^3*b^6 + 2*a*b^6)*x^6 + 2*(154*a^4*b^5 + 84*a^2*b^5 + 3*b^5)*
x^5 + 36*a^5 + (420*a^5*b^4 + 380*a^3*b^4 + 51*a*b^4)*x^4 + (364*a^6*b^3 + 500*a^4*b^3 + 153*a^2*b^3 + 3*b^3)*
x^3 + 20*a^3 + (196*a^7*b^2 + 384*a^5*b^2 + 213*a^3*b^2 + 25*a*b^2)*x^2 + (60*a^8*b + 160*a^6*b + 141*a^4*b +
42*a^2*b + b)*x + 4*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((b^10*x^11 + 8*a*b^9*x^10 + 4*(7*a^2*b^8 + b^8)*x^9
+ 8*(7*a^3*b^7 + 3*a*b^7)*x^8 + 2*(35*a^4*b^6 + 30*a^2*b^6 + 3*b^6)*x^7 + 8*(7*a^5*b^5 + 10*a^3*b^5 + 3*a*b^5
)*x^6 + 4*(7*a^6*b^4 + 15*a^4*b^4 + 9*a^2*b^4 + b^4)*x^5 + 8*(a^7*b^3 + 3*a^5*b^3 + 3*a^3*b^3 + a*b^3)*x^4 + (
a^8*b^2 + 4*a^6*b^2 + 6*a^4*b^2 + 4*a^2*b^2 + b^2)*x^3 + (b^6*x^7 + 4*a*b^5*x^6 + 6*a^2*b^4*x^5 + 4*a^3*b^3*x^
4 + a^4*b^2*x^3)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + 4*(b^7*x^8 + 5*a*b^6*x^7 + (10*a^2*b^5 + b^5)*x^6 + (10*a^3
*b^4 + 3*a*b^4)*x^5 + (5*a^4*b^3 + 3*a^2*b^3)*x^4 + (a^5*b^2 + a^3*b^2)*x^3)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/
2) + 6*(b^8*x^9 + 6*a*b^7*x^8 + (15*a^2*b^6 + 2*b^6)*x^7 + 4*(5*a^3*b^5 + 2*a*b^5)*x^6 + (15*a^4*b^4 + 12*a^2*
b^4 + b^4)*x^5 + 2*(3*a^5*b^3 + 4*a^3*b^3 + a*b^3)*x^4 + (a^6*b^2 + 2*a^4*b^2 + a^2*b^2)*x^3)*(b^2*x^2 + 2*a*b
*x + a^2 + 1) + 4*(b^9*x^10 + 7*a*b^8*x^9 + 3*(7*a^2*b^7 + b^7)*x^8 + 5*(7*a^3*b^6 + 3*a*b^6)*x^7 + (35*a^4*b^
5 + 30*a^2*b^5 + 3*b^5)*x^6 + 3*(7*a^5*b^4 + 10*a^3*b^4 + 3*a*b^4)*x^5 + (7*a^6*b^3 + 15*a^4*b^3 + 9*a^2*b^3 +
b^3)*x^4 + (a^7*b^2 + 3*a^5*b^2 + 3*a^3*b^2 + a*b^2)*x^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1))), x)
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \operatorname{arsinh}\left (b x + a\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/arcsinh(b*x+a)^3,x, algorithm="fricas")
[Out]
integral(1/(x*arcsinh(b*x + a)^3), x)
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{asinh}^{3}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/asinh(b*x+a)**3,x)
[Out]
Integral(1/(x*asinh(a + b*x)**3), x)
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{arsinh}\left (b x + a\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/arcsinh(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(1/(x*arcsinh(b*x + a)^3), x)