\(\int \frac {1}{x (a+b \text {sech}^{-1}(c x))^3} \, dx\) [65]
Optimal result
Integrand size = 14, antiderivative size = 14 \[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3},x\right )
\]
[Out]
Unintegrable(1/x/(a+b*arcsech(c*x))^3,x)
Rubi [N/A]
Not integrable
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of
steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx
\]
[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*}
\text {integral}& = \int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 2.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx
\]
[In]
Integrate[1/(x*(a + b*ArcSech[c*x])^3),x]
[Out]
Integrate[1/(x*(a + b*ArcSech[c*x])^3), x]
Maple [N/A] (verified)
Not integrable
Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
\[\int \frac {1}{x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{3}}d x\]
[In]
int(1/x/(a+b*arcsech(c*x))^3,x)
[Out]
int(1/x/(a+b*arcsech(c*x))^3,x)
Fricas [N/A]
Not integrable
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 3.21
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x} \,d x }
\]
[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 [N/A]
Not integrable
Time = 1.96 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx
\]
[In]
integrate(1/x/(a+b*asech(c*x))**3,x)
[Out]
Integral(1/(x*(a + b*asech(c*x))**3), x)
Maxima [N/A]
Not integrable
Time = 1.90 (sec) , antiderivative size = 2638, normalized size of antiderivative = 188.43
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x} \,d x }
\]
[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)
Giac [N/A]
Not integrable
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x} \,d x }
\]
[In]
integrate(1/x/(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate(1/((b*arcsech(c*x) + a)^3*x), x)
Mupad [N/A]
Not integrable
Time = 4.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43
\[
\int \frac {1}{x \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x
\]
[In]
int(1/(x*(a + b*acosh(1/(c*x)))^3),x)
[Out]
int(1/(x*(a + b*acosh(1/(c*x)))^3), x)