3.47 \(\int \frac{1}{x (a+b \sec ^{-1}(c x))^3} \, dx\)
Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \sec ^{-1}(c x)\right )^3},x\right ) \]
[Out]
Unintegrable[1/(x*(a + b*ArcSec[c*x])^3), x]
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Rubi [A] time = 0.0249043, 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 = {}
\[ \int \frac{1}{x \left (a+b \sec ^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/(x*(a + b*ArcSec[c*x])^3),x]
[Out]
Defer[Int][1/(x*(a + b*ArcSec[c*x])^3), x]
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \sec ^{-1}(c x)\right )^3} \, dx &=\int \frac{1}{x \left (a+b \sec ^{-1}(c x)\right )^3} \, dx\\ \end{align*}
Mathematica [A] time = 1.6879, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \sec ^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/(x*(a + b*ArcSec[c*x])^3),x]
[Out]
Integrate[1/(x*(a + b*ArcSec[c*x])^3), x]
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Maple [A] time = 0.799, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\rm arcsec} \left (cx\right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(a+b*arcsec(c*x))^3,x)
[Out]
int(1/x/(a+b*arcsec(c*x))^3,x)
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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*arcsec(c*x))^3,x, algorithm="maxima")
[Out]
-(8*b^3*c^2*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 24*a*b^2*c^2*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2
+ 2*a*b^2*c^2*x^2*log(c^2*x^2)^2 + 16*a*b^2*c^2*x^2*log(c)*log(x) + 8*a*b^2*c^2*x^2*log(x)^2 + 8*(a*b^2*c^2*l
og(c)^2 + a^3*c^2)*x^2 + 2*(4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 - b^3*log(c^2*x^2)^2 - 4*b^3*log(c)^2
- 8*b^3*log(c)*log(x) - 4*b^3*log(x)^2 + 8*a*b^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b + 4*(b^3*log(c)
+ b^3*log(x))*log(c^2*x^2))*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*(b^3*c^2*x^2*log(c^2*x^2)^2 + 8*b^3*c^2*x^2*log(c
)*log(x) + 4*b^3*c^2*x^2*log(x)^2 + 4*(b^3*c^2*log(c)^2 + 3*a^2*b*c^2)*x^2 - 4*(b^3*c^2*x^2*log(c) + b^3*c^2*x
^2*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - (16*b^6*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^4 +
b^6*log(c^2*x^2)^4 + 16*b^6*log(c)^4 + 64*b^6*log(c)*log(x)^3 + 16*b^6*log(x)^4 + 64*a*b^5*arctan(sqrt(c*x +
1)*sqrt(c*x - 1))^3 + 32*a^2*b^4*log(c)^2 + 16*a^4*b^2 - 8*(b^6*log(c) + b^6*log(x))*log(c^2*x^2)^3 + 8*(b^6*l
og(c^2*x^2)^2 + 4*b^6*log(c)^2 + 8*b^6*log(c)*log(x) + 4*b^6*log(x)^2 + 12*a^2*b^4 - 4*(b^6*log(c) + b^6*log(x
))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 8*(3*b^6*log(c)^2 + 6*b^6*log(c)*log(x) + 3*b^6*log(x
)^2 + a^2*b^4)*log(c^2*x^2)^2 + 32*(3*b^6*log(c)^2 + a^2*b^4)*log(x)^2 + 16*(a*b^5*log(c^2*x^2)^2 + 4*a*b^5*lo
g(c)^2 + 8*a*b^5*log(c)*log(x) + 4*a*b^5*log(x)^2 + 4*a^3*b^3 - 4*(a*b^5*log(c) + a*b^5*log(x))*log(c^2*x^2))*
arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32*(b^6*log(c)^3 + 3*b^6*log(c)*log(x)^2 + b^6*log(x)^3 + a^2*b^4*log(c)
+ (3*b^6*log(c)^2 + a^2*b^4)*log(x))*log(c^2*x^2) + 64*(b^6*log(c)^3 + a^2*b^4*log(c))*log(x))*integrate(4*(b
*c^2*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a*c^2*x)/(4*b^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^4*log(c
^2*x^2)^2 + 4*b^4*log(c)^2 + 8*b^4*log(c)*log(x) + 4*b^4*log(x)^2 + 8*a*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)
) + 4*a^2*b^2 - 4*(b^4*log(c) + b^4*log(x))*log(c^2*x^2)), x) - 8*(a*b^2*c^2*x^2*log(c) + a*b^2*c^2*x^2*log(x)
)*log(c^2*x^2))/(16*b^6*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^4 + b^6*log(c^2*x^2)^4 + 16*b^6*log(c)^4 + 64*b^6*
log(c)*log(x)^3 + 16*b^6*log(x)^4 + 64*a*b^5*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 32*a^2*b^4*log(c)^2 + 16*
a^4*b^2 - 8*(b^6*log(c) + b^6*log(x))*log(c^2*x^2)^3 + 8*(b^6*log(c^2*x^2)^2 + 4*b^6*log(c)^2 + 8*b^6*log(c)*l
og(x) + 4*b^6*log(x)^2 + 12*a^2*b^4 - 4*(b^6*log(c) + b^6*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x
- 1))^2 + 8*(3*b^6*log(c)^2 + 6*b^6*log(c)*log(x) + 3*b^6*log(x)^2 + a^2*b^4)*log(c^2*x^2)^2 + 32*(3*b^6*log(c
)^2 + a^2*b^4)*log(x)^2 + 16*(a*b^5*log(c^2*x^2)^2 + 4*a*b^5*log(c)^2 + 8*a*b^5*log(c)*log(x) + 4*a*b^5*log(x)
^2 + 4*a^3*b^3 - 4*(a*b^5*log(c) + a*b^5*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32*(b^6*l
og(c)^3 + 3*b^6*log(c)*log(x)^2 + b^6*log(x)^3 + a^2*b^4*log(c) + (3*b^6*log(c)^2 + a^2*b^4)*log(x))*log(c^2*x
^2) + 64*(b^6*log(c)^3 + a^2*b^4*log(c))*log(x))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} x \operatorname{arcsec}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname{arcsec}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname{arcsec}\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*arcsec(c*x))^3,x, algorithm="fricas")
[Out]
integral(1/(b^3*x*arcsec(c*x)^3 + 3*a*b^2*x*arcsec(c*x)^2 + 3*a^2*b*x*arcsec(c*x) + a^3*x), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b \operatorname{asec}{\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*asec(c*x))**3,x)
[Out]
Integral(1/(x*(a + b*asec(c*x))**3), x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcsec}\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*arcsec(c*x))^3,x, algorithm="giac")
[Out]
integrate(1/((b*arcsec(c*x) + a)^3*x), x)