\(\int \frac {x^3}{(1-c^2 x^2)^{3/2} (a+b \text {arccosh}(c x))^2} \, dx\) [350]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2},x\right ) \]

[Out]

Unintegrable(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Int[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Defer[Int][x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 24.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {x^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]

[In]

int(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{3}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)*x^3/(a^2*c^4*x^4 - 2*a^2*c^2*x^2 + (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arccosh(c*x
)^2 + a^2 + 2*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arccosh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 64.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{3}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(x**3/(-c**2*x**2+1)**(3/2)/(a+b*acosh(c*x))**2,x)

[Out]

Integral(x**3/((-(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 531, normalized size of antiderivative = 18.96 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{3}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

(c*x^4 + sqrt(c*x + 1)*sqrt(c*x - 1)*x^3)/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*x^2 - b^2*c)*sqrt(c*x
+ 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x + (a*b*c^3*x
^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) - integrate((c^5*x^7 - 5*c^3*x^5 + 4*c*x^3 + (c^3*x^5 - 2*c*x^3)*(c
*x + 1)*(c*x - 1) + (2*c^4*x^6 - 7*c^2*x^4 + 3*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((b^2*c^5*x^4 - b^2*c^3*x^2)
*(c*x + 1)^(3/2)*(c*x - 1) + 2*(b^2*c^6*x^5 - 2*b^2*c^4*x^3 + b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^7*x^
6 - 3*b^2*c^5*x^4 + 3*b^2*c^3*x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)
) + ((a*b*c^5*x^4 - a*b*c^3*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^6*x^5 - 2*a*b*c^4*x^3 + a*b*c^2*x)*(c*x
+ 1)*sqrt(c*x - 1) + (a*b*c^7*x^6 - 3*a*b*c^5*x^4 + 3*a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^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 = 3.46 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^3}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

[In]

int(x^3/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(3/2)),x)

[Out]

int(x^3/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(3/2)), x)