3.63 \(\int \frac{x}{(a+b \text{sech}^{-1}(c x))^3} \, dx\)
Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{x}{\left (a+b \text{sech}^{-1}(c x)\right )^3},x\right ) \]
[Out]
Unintegrable[x/(a + b*ArcSech[c*x])^3, x]
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Rubi [A] time = 0.0158324, 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 = {}
\[ \int \frac{x}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[x/(a + b*ArcSech[c*x])^3,x]
[Out]
Defer[Int][x/(a + b*ArcSech[c*x])^3, x]
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx &=\int \frac{x}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx\\ \end{align*}
Mathematica [A] time = 5.79114, size = 0, normalized size = 0. \[ \int \frac{x}{\left (a+b \text{sech}^{-1}(c x)\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[x/(a + b*ArcSech[c*x])^3,x]
[Out]
Integrate[x/(a + b*ArcSech[c*x])^3, x]
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Maple [A] time = 0.67, size = 0, normalized size = 0. \begin{align*} \int{\frac{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(x/(a+b*arcsech(c*x))^3,x)
[Out]
int(x/(a+b*arcsech(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(x/(a+b*arcsech(c*x))^3,x, algorithm="maxima")
[Out]
-1/2*((2*(2*b*c^4*x^5 - 3*b*c^2*x^3 + b*x)*x*log(x) + (4*(b*c^4*log(c) - a*c^4)*x^5 - (b*c^2*(6*log(c) + 1) -
6*a*c^2)*x^3 + (b*(2*log(c) + 1) - 2*a)*x)*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + (3*(b*c^6*x^7 - 5*b*c^4*x^5 +
6*b*c^2*x^3 - 2*b*x)*x*log(x) + (3*(b*c^6*log(c) - a*c^6)*x^7 - (b*c^4*(15*log(c) + 2) - 15*a*c^4)*x^5 + (b*c
^2*(18*log(c) + 5) - 18*a*c^2)*x^3 - 3*(b*(2*log(c) + 1) - 2*a)*x)*x)*(c*x + 1)*(c*x - 1) - 2*(b*c^6*x^7 - 3*b
*c^4*x^5 + 3*b*c^2*x^3 - b*x)*x*log(x) - ((5*b*c^6*x^7 - 17*b*c^4*x^5 + 18*b*c^2*x^3 - 6*b*x)*x*log(x) + ((b*c
^6*(5*log(c) + 1) - 5*a*c^6)*x^7 - (b*c^4*(17*log(c) + 5) - 17*a*c^4)*x^5 + (b*c^2*(18*log(c) + 7) - 18*a*c^2)
*x^3 - 3*(b*(2*log(c) + 1) - 2*a)*x)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - ((b*c^6*(2*log(c) + 1) - 2*a*c^6)*x^7 -
3*(b*c^4*(2*log(c) + 1) - 2*a*c^4)*x^5 + 3*(b*c^2*(2*log(c) + 1) - 2*a*c^2)*x^3 - (b*(2*log(c) + 1) - 2*a)*x)
*x - (2*(2*b*c^4*x^5 - 3*b*c^2*x^3 + b*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2)*x + 3*(b*c^6*x^7 - 5*b*c^4*x^5 + 6*
b*c^2*x^3 - 2*b*x)*(c*x + 1)*(c*x - 1)*x - (5*b*c^6*x^7 - 17*b*c^4*x^5 + 18*b*c^2*x^3 - 6*b*x)*sqrt(c*x + 1)*s
qrt(-c*x + 1)*x - 2*(b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - b*x)*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 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*lo
g(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*lo
g(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*log(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*lo
g(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*(4*(6*c^4*x
^4 - 6*c^2*x^2 + 1)*(c*x + 1)^2*(c*x - 1)^2*x - (33*c^6*x^6 - 108*c^4*x^4 + 88*c^2*x^2 - 16)*(c*x + 1)^(3/2)*(
-c*x + 1)^(3/2)*x - 12*(c^8*x^8 - 7*c^6*x^6 + 14*c^4*x^4 - 10*c^2*x^2 + 2)*(c*x + 1)*(c*x - 1)*x + (15*c^8*x^8
- 67*c^6*x^6 + 108*c^4*x^4 - 72*c^2*x^2 + 16)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x + 4*(c^8*x^8 - 4*c^6*x^6 + 6*c^4
*x^4 - 4*c^2*x^2 + 1)*x)/((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*l
og(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)
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{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(x/(a+b*arcsech(c*x))^3,x, algorithm="fricas")
[Out]
integral(x/(b^3*arcsech(c*x)^3 + 3*a*b^2*arcsech(c*x)^2 + 3*a^2*b*arcsech(c*x) + a^3), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{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(x/(a+b*asech(c*x))**3,x)
[Out]
Integral(x/(a + b*asech(c*x))**3, x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\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(x/(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate(x/(b*arcsech(c*x) + a)^3, x)