Optimal. Leaf size=33 \[ \text {Int}\left (\frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.56, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^m}{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.28, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} \left (f x\right )^{m}}{a^{2} c^{4} x^{4} - 2 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x\right )^{m}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c f^{m} x x^{m} + \sqrt {c x + 1} \sqrt {c x - 1} f^{m} x^{m}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} - \int \frac {{\left (c^{3} f^{m} {\left (m - 2\right )} x^{3} - c f^{m} {\left (m - 1\right )} x\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} x^{m} + {\left (2 \, c^{4} f^{m} {\left (m - 2\right )} x^{4} - c^{2} f^{m} {\left (3 \, m - 2\right )} x^{2} + f^{m} m\right )} \sqrt {c x + 1} \sqrt {c x - 1} x^{m} + {\left (c^{5} f^{m} {\left (m - 2\right )} x^{5} - c^{3} f^{m} {\left (2 \, m - 1\right )} x^{3} + c f^{m} {\left (m + 1\right )} x\right )} x^{m}}{{\left ({\left (b^{2} c^{5} x^{5} - b^{2} c^{3} x^{3}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{6} x^{6} - 2 \, b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{7} x^{7} - 3 \, b^{2} c^{5} x^{5} + 3 \, b^{2} c^{3} x^{3} - b^{2} c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (a b c^{5} x^{5} - a b c^{3} x^{3}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + 2 \, {\left (a b c^{6} x^{6} - 2 \, a b c^{4} x^{4} + a b c^{2} x^{2}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{7} x^{7} - 3 \, a b c^{5} x^{5} + 3 \, a b c^{3} x^{3} - a b c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (f\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________