\(\int \frac {1}{(1-c^2 x^2) (a+b \text {arccosh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2} \, dx\) [273]
Optimal result
Integrand size = 40, antiderivative size = 40 \[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2},x\right )
\]
[Out]
Unintegrable(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)
Rubi [N/A]
Not integrable
Time = 0.03 (sec) , antiderivative size = 40, 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 = {} \[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx
\]
[In]
Int[1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]
[Out]
Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 4.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05
\[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx
\]
[In]
Integrate[1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]
[Out]
Integrate[1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]
Maple [N/A] (verified)
Not integrable
Time = 0.95 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90
\[\int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{2}}d x\]
[In]
int(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)
[Out]
int(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)
Fricas [N/A]
Not integrable
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.28
\[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="fricas")
[Out]
integral(-1/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2
- a*b)*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1))), x)
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate(1/(-c**2*x**2+1)/(a+b*acosh((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2,x)
[Out]
Timed out
Maxima [N/A]
Not integrable
Time = 2.92 (sec) , antiderivative size = 1177, normalized size of antiderivative = 29.42
\[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="maxima")
[Out]
2*(2*c*x*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + (c*x + 1)*sqrt(-c*x + 1)
- (-c*x + 1)^(3/2))/(2*(c*x + 1)*sqrt(-c*x + 1)*a*b*c - 2*(-c*x + 1)^(3/2)*a*b*c - ((c*x - 1)*b^2*c*log(c*x +
1) - 2*(c*x - 1)*a*b*c)*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) - ((c*x +
1)*sqrt(-c*x + 1)*b^2*c - (-c*x + 1)^(3/2)*b^2*c)*log(c*x + 1) + 2*((c*x - 1)*b^2*c*sqrt(sqrt(c*x + 1) + sqrt(
-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + (c*x + 1)*sqrt(-c*x + 1)*b^2*c - (-c*x + 1)^(3/2)*b^2*c)*lo
g(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))) - integrate(-2
*(2*(c*x + 1)*sqrt(-c*x + 1)*(sqrt(c*x + 1) + sqrt(-c*x + 1))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) + ((c*x + 1)^2
+ 2*(c*x + 1)*(c*x - 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)))/(2*(a*b*c
^2*x^2 - a*b)*(c*x + 1)^2*sqrt(-c*x + 1) - 4*(a*b*c^2*x^2 - a*b)*(c*x + 1)*(-c*x + 1)^(3/2) + 2*(a*b*c^2*x^2 -
a*b)*(-c*x + 1)^(5/2) + ((b^2*c^2*x^2 - b^2)*(-c*x + 1)^(3/2)*log(c*x + 1) - 2*(a*b*c^2*x^2 - a*b)*(-c*x + 1)
^(3/2))*(sqrt(c*x + 1) + sqrt(-c*x + 1))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) + 2*(2*(a*b*c^2*x^2 - a*b)*(c*x + 1)
*(c*x - 1) + 2*(a*b*c^2*x^2 - a*b)*(c*x - 1)^2 - ((b^2*c^2*x^2 - b^2)*(c*x + 1)*(c*x - 1) + (b^2*c^2*x^2 - b^2
)*(c*x - 1)^2)*log(c*x + 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) - ((b^
2*c^2*x^2 - b^2)*(c*x + 1)^2*sqrt(-c*x + 1) - 2*(b^2*c^2*x^2 - b^2)*(c*x + 1)*(-c*x + 1)^(3/2) + (b^2*c^2*x^2
- b^2)*(-c*x + 1)^(5/2))*log(c*x + 1) - 2*((b^2*c^2*x^2 - b^2)*(-c*x + 1)^(3/2)*(sqrt(c*x + 1) + sqrt(-c*x + 1
))*(sqrt(c*x + 1) - sqrt(-c*x + 1)) - (b^2*c^2*x^2 - b^2)*(c*x + 1)^2*sqrt(-c*x + 1) + 2*(b^2*c^2*x^2 - b^2)*(
c*x + 1)*(-c*x + 1)^(3/2) - (b^2*c^2*x^2 - b^2)*(-c*x + 1)^(5/2) - 2*((b^2*c^2*x^2 - b^2)*(c*x + 1)*(c*x - 1)
+ (b^2*c^2*x^2 - b^2)*(c*x - 1)^2)*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)))
*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))), x)
Giac [F(-2)]
Exception generated. \[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate(1/(-c^2*x^2+1)/(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Mupad [N/A]
Not integrable
Time = 3.60 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98
\[
\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=-\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x
\]
[In]
int(-1/((a + b*acosh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2*(c^2*x^2 - 1)),x)
[Out]
-int(1/((a + b*acosh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2*(c^2*x^2 - 1)), x)