Optimal. Leaf size=74 \[ \frac {d x \sqrt {c^2 x^2-1}}{2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {\sqrt {c^2 x^2-1} \tanh ^{-1}(c x)}{2 c \sqrt {d-c^2 d x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {23, 199, 208} \begin {gather*} \frac {x \left (d-c^2 d x^2\right )^{3/2}}{2 d^2 \left (1-c^2 x^2\right ) \left (c^2 x^2-1\right )^{3/2}}+\frac {\left (d-c^2 d x^2\right )^{3/2} \tanh ^{-1}(c x)}{2 c d^2 \left (c^2 x^2-1\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 199
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (d-c^2 d x^2\right )^{3/2} \int \frac {1}{\left (d-c^2 d x^2\right )^2} \, dx}{\left (-1+c^2 x^2\right )^{3/2}}\\ &=\frac {x \left (d-c^2 d x^2\right )^{3/2}}{2 d^2 \left (1-c^2 x^2\right ) \left (-1+c^2 x^2\right )^{3/2}}+\frac {\left (d-c^2 d x^2\right )^{3/2} \int \frac {1}{d-c^2 d x^2} \, dx}{2 d \left (-1+c^2 x^2\right )^{3/2}}\\ &=\frac {x \left (d-c^2 d x^2\right )^{3/2}}{2 d^2 \left (1-c^2 x^2\right ) \left (-1+c^2 x^2\right )^{3/2}}+\frac {\left (d-c^2 d x^2\right )^{3/2} \tanh ^{-1}(c x)}{2 c d^2 \left (-1+c^2 x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 0.73 \begin {gather*} \frac {\left (c^2 x^2-1\right ) \tanh ^{-1}(c x)-c x}{2 c \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 0.08, size = 79, normalized size = 1.07 \begin {gather*} \frac {i \sqrt {d} \sqrt {c^2 x^2-1} \left (\frac {i x}{2 \sqrt {d} \left (c^2 x^2-1\right )}-\frac {i \tanh ^{-1}(c x)}{2 c \sqrt {d}}\right )}{\sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.61, size = 303, normalized size = 4.09 \begin {gather*} \left [\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x - {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right )}{8 \, {\left (c^{5} d x^{4} - 2 \, c^{3} d x^{2} + c d\right )}}, \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x - {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right )}{4 \, {\left (c^{5} d x^{4} - 2 \, c^{3} d x^{2} + c d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-c^{2} d x^{2} + d} {\left (c^{2} x^{2} - 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 94, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {-\left (c^{2} x^{2}-1\right ) d}\, \left (-c^{2} x^{2} \ln \left (c x -1\right )+c^{2} x^{2} \ln \left (c x +1\right )-2 c x +\ln \left (c x -1\right )-\ln \left (c x +1\right )\right )}{4 \sqrt {c^{2} x^{2}-1}\, \left (c x +1\right ) \left (c x -1\right ) c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-c^{2} d x^{2} + d} {\left (c^{2} x^{2} - 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {d-c^2\,d\,x^2}\,{\left (c^2\,x^2-1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________