3.64 \(\int \frac{1}{(a+b \text{sech}^{-1}(c x))^3} \, dx\)
Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^3},x\right ) \]
[Out]
Unintegrable[(a + b*ArcSech[c*x])^(-3), x]
________________________________________________________________________________________
Rubi [A] time = 0.0059659, 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 = {}
\[ \int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*ArcSech[c*x])^(-3),x]
[Out]
Defer[Int][(a + b*ArcSech[c*x])^(-3), x]
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx &=\int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx\\ \end{align*}
Mathematica [A] time = 3.58711, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(a + b*ArcSech[c*x])^(-3),x]
[Out]
Integrate[(a + b*ArcSech[c*x])^(-3), x]
________________________________________________________________________________________
Maple [A] time = 0.589, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arcsech} \left (cx\right ) \right ) ^{-3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*arcsech(c*x))^3,x)
[Out]
int(1/(a+b*arcsech(c*x))^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/(a+b*arcsech(c*x))^3,x, algorithm="maxima")
[Out]
1/2*((b*c^6*(log(c) + 1) - a*c^6)*x^7 - 3*(b*c^4*(log(c) + 1) - a*c^4)*x^5 - (3*(b*c^4*log(c) - a*c^4)*x^5 - (
b*c^2*(4*log(c) + 1) - 4*a*c^2)*x^3 + (b*(log(c) + 1) - a)*x + (3*b*c^4*x^5 - 4*b*c^2*x^3 + b*x)*log(x))*(c*x
+ 1)^(3/2)*(-c*x + 1)^(3/2) + 3*(b*c^2*(log(c) + 1) - a*c^2)*x^3 - (2*(b*c^6*log(c) - a*c^6)*x^7 - 2*(b*c^4*(5
*log(c) + 1) - 5*a*c^4)*x^5 + (b*c^2*(11*log(c) + 5) - 11*a*c^2)*x^3 - 3*(b*(log(c) + 1) - a)*x + (2*b*c^6*x^7
- 10*b*c^4*x^5 + 11*b*c^2*x^3 - 3*b*x)*log(x))*(c*x + 1)*(c*x - 1) + ((b*c^6*(3*log(c) + 1) - 3*a*c^6)*x^7 -
5*(b*c^4*(2*log(c) + 1) - 2*a*c^4)*x^5 + (b*c^2*(10*log(c) + 7) - 10*a*c^2)*x^3 - 3*(b*(log(c) + 1) - a)*x + (
3*b*c^6*x^7 - 10*b*c^4*x^5 + 10*b*c^2*x^3 - 3*b*x)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b*(log(c) + 1) - a)
*x - (b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - (3*b*c^4*x^5 - 4*b*c^2*x^3 + b*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/
2) - (2*b*c^6*x^7 - 10*b*c^4*x^5 + 11*b*c^2*x^3 - 3*b*x)*(c*x + 1)*(c*x - 1) + (3*b*c^6*x^7 - 10*b*c^4*x^5 + 1
0*b*c^2*x^3 - 3*b*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - b*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b*c^6*x^7 -
3*b*c^4*x^5 + 3*b*c^2*x^3 - b*x)*log(x))/((b^4*c^6*log(c)^2 - 2*a*b^3*c^6*log(c) + a^2*b^2*c^6)*x^6 - b^4*log(
c)^2 - 3*(b^4*c^4*log(c)^2 - 2*a*b^3*c^4*log(c) + a^2*b^2*c^4)*x^4 + 2*a*b^3*log(c) - (b^4*log(c)^2 + b^4*log(
x)^2 - 2*a*b^3*log(c) + a^2*b^2 + 2*(b^4*log(c) - a*b^3)*log(x))*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - a^2*b^2 +
3*(b^4*log(c)^2 - 2*a*b^3*log(c) + a^2*b^2 - (b^4*c^2*log(c)^2 - 2*a*b^3*c^2*log(c) + a^2*b^2*c^2)*x^2 - (b^4*
c^2*x^2 - b^4)*log(x)^2 + 2*(b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*c^2)*x^2)*log(x))*(c*x + 1)*(c*x - 1
) + 3*(b^4*c^2*log(c)^2 - 2*a*b^3*c^2*log(c) + a^2*b^2*c^2)*x^2 + (b^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2
- (c*x + 1)^(3/2)*(-c*x + 1)^(3/2)*b^4 - b^4 - 3*(b^4*c^2*x^2 - b^4)*(c*x + 1)*(c*x - 1) - 3*(b^4*c^4*x^4 - 2
*b^4*c^2*x^2 + b^4)*sqrt(c*x + 1)*sqrt(-c*x + 1))*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 + (b^4*c^6*x^6 - 3*b
^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*log(x)^2 - 3*(b^4*log(c)^2 + (b^4*c^4*log(c)^2 - 2*a*b^3*c^4*log(c) + a^2*b^
2*c^4)*x^4 - 2*a*b^3*log(c) + a^2*b^2 - 2*(b^4*c^2*log(c)^2 - 2*a*b^3*c^2*log(c) + a^2*b^2*c^2)*x^2 + (b^4*c^4
*x^4 - 2*b^4*c^2*x^2 + b^4)*log(x)^2 + 2*((b^4*c^4*log(c) - a*b^3*c^4)*x^4 + b^4*log(c) - a*b^3 - 2*(b^4*c^2*l
og(c) - a*b^3*c^2)*x^2)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - 2*((b^4*c^6*log(c) - a*b^3*c^6)*x^6 - 3*(b^4*c^
4*log(c) - a*b^3*c^4)*x^4 - b^4*log(c) - (b^4*log(c) + b^4*log(x) - a*b^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) +
a*b^3 + 3*(b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*c^2)*x^2 - (b^4*c^2*x^2 - b^4)*log(x))*(c*x + 1)*(c*x
- 1) + 3*(b^4*c^2*log(c) - a*b^3*c^2)*x^2 - 3*((b^4*c^4*log(c) - a*b^3*c^4)*x^4 + b^4*log(c) - a*b^3 - 2*(b^4*
c^2*log(c) - a*b^3*c^2)*x^2 + (b^4*c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) + (b^4*
c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*log(x))*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + 2*((b^4*c^6*log
(c) - a*b^3*c^6)*x^6 - 3*(b^4*c^4*log(c) - a*b^3*c^4)*x^4 - b^4*log(c) + a*b^3 + 3*(b^4*c^2*log(c) - a*b^3*c^2
)*x^2)*log(x)) + integrate(-1/2*(c^8*x^8 - 4*c^6*x^6 + 6*c^4*x^4 + (15*c^4*x^4 - 12*c^2*x^2 + 1)*(c*x + 1)^2*(
c*x - 1)^2 - (18*c^6*x^6 - 57*c^4*x^4 + 40*c^2*x^2 - 4)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - 4*c^2*x^2 - 3*(2*c^
8*x^8 - 13*c^6*x^6 + 25*c^4*x^4 - 16*c^2*x^2 + 2)*(c*x + 1)*(c*x - 1) + (6*c^8*x^8 - 25*c^6*x^6 + 39*c^4*x^4 -
24*c^2*x^2 + 4)*sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/((b^3*c^8*log(c) - a*b^2*c^8)*x^8 - 4*(b^3*c^6*log(c) - a*b
^2*c^6)*x^6 + (b^3*log(c) + b^3*log(x) - a*b^2)*(c*x + 1)^2*(c*x - 1)^2 + 6*(b^3*c^4*log(c) - a*b^2*c^4)*x^4 +
4*(b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2)*x^2 - (b^3*c^2*x^2 - b^3)*log(x))*(c*x + 1)^(3/2)*(-c*x
+ 1)^(3/2) + b^3*log(c) - 6*((b^3*c^4*log(c) - a*b^2*c^4)*x^4 + b^3*log(c) - a*b^2 - 2*(b^3*c^2*log(c) - a*b^2
*c^2)*x^2 + (b^3*c^4*x^4 - 2*b^3*c^2*x^2 + b^3)*log(x))*(c*x + 1)*(c*x - 1) - a*b^2 - 4*(b^3*c^2*log(c) - a*b^
2*c^2)*x^2 - 4*((b^3*c^6*log(c) - a*b^2*c^6)*x^6 - 3*(b^3*c^4*log(c) - a*b^2*c^4)*x^4 - b^3*log(c) + a*b^2 + 3
*(b^3*c^2*log(c) - a*b^2*c^2)*x^2 + (b^3*c^6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*log(x))*sqrt(c*x + 1)*
sqrt(-c*x + 1) - (b^3*c^8*x^8 - 4*b^3*c^6*x^6 + 6*b^3*c^4*x^4 + (c*x + 1)^2*(c*x - 1)^2*b^3 - 4*b^3*c^2*x^2 -
4*(b^3*c^2*x^2 - b^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - 6*(b^3*c^4*x^4 - 2*b^3*c^2*x^2 + b^3)*(c*x + 1)*(c*x
- 1) + b^3 - 4*(b^3*c^6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*sqrt(c*x + 1)*sqrt(-c*x + 1))*log(sqrt(c*x
+ 1)*sqrt(-c*x + 1) + 1) + (b^3*c^8*x^8 - 4*b^3*c^6*x^6 + 6*b^3*c^4*x^4 - 4*b^3*c^2*x^2 + b^3)*log(x)), x)
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} \operatorname{arsech}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arsech}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arsech}\left (c x\right ) + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*arcsech(c*x))^3,x, algorithm="fricas")
[Out]
integral(1/(b^3*arcsech(c*x)^3 + 3*a*b^2*arcsech(c*x)^2 + 3*a^2*b*arcsech(c*x) + a^3), x)
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*asech(c*x))**3,x)
[Out]
Integral((a + b*asech(c*x))**(-3), x)
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate((b*arcsech(c*x) + a)^(-3), x)