Optimal. Leaf size=71 \[ \frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4133, 459, 321, 205} \[ -\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}+\frac {\cos ^3(e+f x)}{3 a f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 459
Rule 4133
Rubi steps
\begin {align*} \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (1-x^2\right )}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x)}{3 a f}-\frac {(a+b) \operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a f}\\ &=-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f}+\frac {(b (a+b)) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a^2 f}\\ &=\frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f}\\ \end {align*}
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Mathematica [C] time = 1.39, size = 376, normalized size = 5.30 \[ \frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (3 \left (a^2+8 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )+3 \left (a^2+8 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )-3 a^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )-3 a^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )+\sqrt {a}}{\sqrt {b}}\right )+4 \sqrt {a} \sqrt {b} \cos (e+f x) (a \cos (2 (e+f x))-5 a-6 b)\right )}{48 a^{5/2} \sqrt {b} f \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 154, normalized size = 2.17 \[ \left [\frac {2 \, a \cos \left (f x + e\right )^{3} + 3 \, {\left (a + b\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 6 \, {\left (a + b\right )} \cos \left (f x + e\right )}{6 \, a^{2} f}, \frac {a \cos \left (f x + e\right )^{3} + 3 \, {\left (a + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 3 \, {\left (a + b\right )} \cos \left (f x + e\right )}{3 \, a^{2} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 89, normalized size = 1.25 \[ \frac {{\left (a b + b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} f} + \frac {a^{2} f^{5} \cos \left (f x + e\right )^{3} - 3 \, a^{2} f^{5} \cos \left (f x + e\right ) - 3 \, a b f^{5} \cos \left (f x + e\right )}{3 \, a^{3} f^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.93, size = 103, normalized size = 1.45 \[ \frac {\cos ^{3}\left (f x +e \right )}{3 a f}-\frac {\cos \left (f x +e \right )}{a f}-\frac {b \cos \left (f x +e \right )}{f \,a^{2}}+\frac {b \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{f a \sqrt {a b}}+\frac {b^{2} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{f \,a^{2} \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 63, normalized size = 0.89 \[ \frac {\frac {3 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (f x + e\right )}{a^{2}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 76, normalized size = 1.07 \[ \frac {{\cos \left (e+f\,x\right )}^3}{3\,a\,f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {b}{a^2}+\frac {1}{a}\right )}{f}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (a+b\right )}{b^2+a\,b}\right )\,\left (a+b\right )}{a^{5/2}\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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