Optimal. Leaf size=76 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} f \sqrt {a+b}}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f} \]
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Rubi [A] time = 0.09, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4147, 390, 208} \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} f \sqrt {a+b}}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f} \]
Antiderivative was successfully verified.
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Rule 208
Rule 390
Rule 4147
Rubi steps
\begin {align*} \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a-b}{a^2}-\frac {x^2}{a}+\frac {b^2}{a^2 \left (a+b-a x^2\right )}\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{a^2 f}\\ &=\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b} f}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 105, normalized size = 1.38 \[ \frac {a^{3/2} \sin (3 (e+f x))+\frac {6 b^2 \left (\log \left (\sqrt {a+b}+\sqrt {a} \sin (e+f x)\right )-\log \left (\sqrt {a+b}-\sqrt {a} \sin (e+f x)\right )\right )}{\sqrt {a+b}}+3 \sqrt {a} (3 a-4 b) \sin (e+f x)}{12 a^{5/2} f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 230, normalized size = 3.03 \[ \left [\frac {3 \, \sqrt {a^{2} + a b} b^{2} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{4} + a^{3} b\right )} f}, -\frac {3 \, \sqrt {-a^{2} - a b} b^{2} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) - {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{4} + a^{3} b\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 89, normalized size = 1.17 \[ -\frac {\frac {3 \, b^{2} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} a^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3} - 3 \, a^{2} \sin \left (f x + e\right ) + 3 \, a b \sin \left (f x + e\right )}{a^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.48, size = 70, normalized size = 0.92 \[ \frac {-\frac {\frac {a \left (\sin ^{3}\left (f x +e \right )\right )}{3}-a \sin \left (f x +e \right )+b \sin \left (f x +e \right )}{a^{2}}+\frac {b^{2} \arctanh \left (\frac {a \sin \left (f x +e \right )}{\sqrt {\left (a +b \right ) a}}\right )}{a^{2} \sqrt {\left (a +b \right ) a}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 88, normalized size = 1.16 \[ -\frac {\frac {3 \, b^{2} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} + \frac {2 \, {\left (a \sin \left (f x + e\right )^{3} - 3 \, {\left (a - b\right )} \sin \left (f x + e\right )\right )}}{a^{2}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 72, normalized size = 0.95 \[ \frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )}{a^{5/2}\,f\,\sqrt {a+b}}-\frac {{\sin \left (e+f\,x\right )}^3}{3\,a\,f}-\frac {\sin \left (e+f\,x\right )\,\left (\frac {a+b}{a^2}-\frac {2}{a}\right )}{f} \]
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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