\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx\) [261]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 30, 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 = {} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} \left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 70.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{3}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

[In]

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

[Out]

Integral((f*x)**m*(a + b*acosh(c*x))**3/sqrt(-(c*x - 1)*(c*x + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} \left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^3*(f*x)^m/sqrt(-c^2*x^2 + 1), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3} \left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^3*(f*x)^m/sqrt(-c^2*x^2 + 1), x)

Mupad [N/A]

Not integrable

Time = 3.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^3}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3\,{\left (f\,x\right )}^m}{\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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