3.65 \(\int \frac{1}{x (a+b \text{sech}^{-1}(c x))^3} \, dx\)
Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^3},x\right ) \]
[Out]
Unintegrable[1/(x*(a + b*ArcSech[c*x])^3), x]
________________________________________________________________________________________
Rubi [A] time = 0.0248769, 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 = {}
\[ \int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/(x*(a + b*ArcSech[c*x])^3),x]
[Out]
Defer[Int][1/(x*(a + b*ArcSech[c*x])^3), x]
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx &=\int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx\\ \end{align*}
Mathematica [A] time = 2.49891, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/(x*(a + b*ArcSech[c*x])^3),x]
[Out]
Integrate[1/(x*(a + b*ArcSech[c*x])^3), x]
________________________________________________________________________________________
Maple [A] time = 0.276, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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/x/(a+b*arcsech(c*x))^3,x)
[Out]
int(1/x/(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/x/(a+b*arcsech(c*x))^3,x, algorithm="maxima")
[Out]
-1/2*(b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - (2*(b*c^4*log(c) - a*c^4)*x^5 - (b*c^2*(2*log(c) + 1) - 2*a*c^2)
*x^3 + b*x + 2*(b*c^4*x^5 - b*c^2*x^3)*log(x))*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - ((b*c^6*log(c) - a*c^6)*x^7
- (b*c^4*(5*log(c) + 2) - 5*a*c^4)*x^5 + (b*c^2*(4*log(c) + 5) - 4*a*c^2)*x^3 - 3*b*x + (b*c^6*x^7 - 5*b*c^4*x
^5 + 4*b*c^2*x^3)*log(x))*(c*x + 1)*(c*x - 1) + ((b*c^6*(log(c) + 1) - a*c^6)*x^7 - (b*c^4*(3*log(c) + 5) - 3*
a*c^4)*x^5 + (b*c^2*(2*log(c) + 7) - 2*a*c^2)*x^3 - 3*b*x + (b*c^6*x^7 - 3*b*c^4*x^5 + 2*b*c^2*x^3)*log(x))*sq
rt(c*x + 1)*sqrt(-c*x + 1) - b*x + (2*(b*c^4*x^5 - b*c^2*x^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + (b*c^6*x^7 -
5*b*c^4*x^5 + 4*b*c^2*x^3)*(c*x + 1)*(c*x - 1) - (b*c^6*x^7 - 3*b*c^4*x^5 + 2*b*c^2*x^3)*sqrt(c*x + 1)*sqrt(-c
*x + 1))*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1))/((b^4*x*log(x)^2 + 2*(b^4*log(c) - a*b^3)*x*log(x) + (b^4*log(
c)^2 - 2*a*b^3*log(c) + a^2*b^2)*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - (b^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^
2*x^2 - b^4)*x*log(x)^2 + 3*((b^4*c^2*x^2 - b^4)*x*log(x)^2 - 2*(b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*
c^2)*x^2)*x*log(x) - (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)*x)*(c*x + 1)*(c*x - 1) + ((c*x + 1)^(3/2)*(-c*x + 1)^(3/2)*b^4*x + 3*(b^4*c^2*x^2 - b^4)*(c*x + 1
)*(c*x - 1)*x + 3*(b^4*c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x - (b^4*c^6*x^6 - 3*b^4*c^
4*x^4 + 3*b^4*c^2*x^2 - b^4)*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 - 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)*x*log(x) + 3*(
(b^4*c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*x*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*log(c) - a*b^3*c^2)*x^2)*x*log(x) + (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)*x)*sqrt(c*x
+ 1)*sqrt(-c*x + 1) - ((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) - a^2*b^2 + 3*(b^4*c^2*log(c)^2 - 2*a*b^3*c^
2*log(c) + a^2*b^2*c^2)*x^2)*x - 2*((b^4*x*log(x) + (b^4*log(c) - a*b^3)*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) +
3*((b^4*c^2*x^2 - b^4)*x*log(x) - (b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x)*(c*x + 1)*(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)*x*log(x) + 3*((b^4*c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*x*
log(x) + ((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)*x)*sqrt(
c*x + 1)*sqrt(-c*x + 1) - ((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)*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)) + integrate(-1/2*(4*(2
*c^4*x^4 - c^2*x^2)*(c*x + 1)^2*(c*x - 1)^2 - (7*c^6*x^6 - 22*c^4*x^4 + 12*c^2*x^2)*(c*x + 1)^(3/2)*(-c*x + 1)
^(3/2) - 2*(c^8*x^8 - 5*c^6*x^6 + 10*c^4*x^4 - 6*c^2*x^2)*(c*x + 1)*(c*x - 1) + (c^8*x^8 - 3*c^6*x^6 + 6*c^4*x
^4 - 4*c^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1))/((b^3*x*log(x) + (b^3*log(c) - a*b^2)*x)*(c*x + 1)^2*(c*x - 1)^2
- 4*((b^3*c^2*x^2 - b^3)*x*log(x) - (b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*x)*(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)*x*log(x) + ((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)*x)*(c*x + 1)*(c*x - 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)*x*log(x) - 4*((b^3*c^6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*x*log(x
) + ((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*l
og(c) - a*b^2*c^2)*x^2)*x)*sqrt(c*x + 1)*sqrt(-c*x + 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 + 6*(b^3*c^4*log(c) - a*b^2*c^4)*x^4 + b^3*log(c) - a*b^2 - 4*(b^3*c^2*log(c) - a*b^2*c^2)*
x^2)*x - ((c*x + 1)^2*(c*x - 1)^2*b^3*x - 4*(b^3*c^2*x^2 - b^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2)*x - 6*(b^3*c^
4*x^4 - 2*b^3*c^2*x^2 + b^3)*(c*x + 1)*(c*x - 1)*x - 4*(b^3*c^6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*sqr
t(c*x + 1)*sqrt(-c*x + 1)*x + (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)*x)*log(sqrt(
c*x + 1)*sqrt(-c*x + 1) + 1)), x)
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} x \operatorname{arsech}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname{arsech}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname{arsech}\left (c x\right ) + a^{3} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*arcsech(c*x))^3,x, algorithm="fricas")
[Out]
integral(1/(b^3*x*arcsech(c*x)^3 + 3*a*b^2*x*arcsech(c*x)^2 + 3*a^2*b*x*arcsech(c*x) + a^3*x), x)
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \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/x/(a+b*asech(c*x))**3,x)
[Out]
Integral(1/(x*(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} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate(1/((b*arcsech(c*x) + a)^3*x), x)