Optimal. Leaf size=34 \[ -\frac {\cos (e+f x)}{f (a+b) \sqrt {a-b \cos ^2(e+f x)+b}} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3186, 191} \[ -\frac {\cos (e+f x)}{f (a+b) \sqrt {a-b \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x)}{(a+b) f \sqrt {a+b-b \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 41, normalized size = 1.21 \[ -\frac {\sqrt {2} \cos (e+f x)}{f (a+b) \sqrt {2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 57, normalized size = 1.68 \[ \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \cos \left (f x + e\right )}{{\left (a b + b^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} + 2 \, a b + b^{2}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 53, normalized size = 1.56 \[ \frac {\sqrt {-{\left (\cos \left (f x + e\right )^{2} - 1\right )} b + a} \cos \left (f x + e\right )}{{\left ({\left (\cos \left (f x + e\right )^{2} - 1\right )} b - a\right )} {\left (a + b\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 31, normalized size = 0.91 \[ -\frac {\cos \left (f x +e \right )}{\left (a +b \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 32, normalized size = 0.94 \[ -\frac {\cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.18, size = 119, normalized size = 3.50 \[ -\frac {\sqrt {2}\,\sqrt {2\,a+b-b\,\cos \left (2\,e+2\,f\,x\right )}\,\left (4\,a\,\cos \left (e+f\,x\right )+b\,\cos \left (e+f\,x\right )-b\,\cos \left (3\,e+3\,f\,x\right )\right )}{f\,\left (a+b\right )\,\left (8\,a\,b+8\,a^2+3\,b^2-4\,b^2\,\cos \left (2\,e+2\,f\,x\right )+b^2\,\cos \left (4\,e+4\,f\,x\right )-8\,a\,b\,\cos \left (2\,e+2\,f\,x\right )\right )} \]
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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