Optimal. Leaf size=59 \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^3}{3 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
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Rubi [A] time = 0.10, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {5157, 5155} \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^3}{3 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
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Rule 5155
Rule 5157
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^2}{\sqrt {d-\frac {c^2 d x^2}{a}}} \, dx &=\frac {\sqrt {a-c^2 x^2} \int \frac {\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^2}{\sqrt {a-c^2 x^2}} \, dx}{\sqrt {d-\frac {c^2 d x^2}{a}}}\\ &=\frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^3}{3 c \sqrt {d-\frac {c^2 d x^2}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 1.00 \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^3}{3 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a \sqrt {-\frac {c^{2} d x^{2} - a d}{a}} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )^{2}}{c^{2} d x^{2} - a d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )^{2}}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 72, normalized size = 1.22 \[ -\frac {\sqrt {-\frac {d \left (c^{2} x^{2}-a \right )}{a}}\, \sqrt {-c^{2} x^{2}+a}\, \arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2}+a}}\right )^{3} a}{3 d \left (c^{2} x^{2}-a \right ) c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )^{2}}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {atan}\left (\frac {c\,x}{\sqrt {a-c^2\,x^2}}\right )}^2}{\sqrt {d-\frac {c^2\,d\,x^2}{a}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}^{2}{\left (\frac {c x}{\sqrt {a - c^{2} x^{2}}} \right )}}{\sqrt {- d \left (-1 + \frac {c^{2} x^{2}}{a}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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