Optimal. Leaf size=37 \[ -\frac {\cos (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{f (a-b)} \]
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Rubi [A] time = 0.05, antiderivative size = 37, 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 = {3664, 264} \[ -\frac {\cos (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{f (a-b)} \]
Antiderivative was successfully verified.
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Rule 264
Rule 3664
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{(a-b) f}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 52, normalized size = 1.41 \[ \frac {\cos (e+f x) \sqrt {\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{\sqrt {2} f (b-a)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 45, normalized size = 1.22 \[ -\frac {\sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a - b\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 78, normalized size = 2.11 \[ -\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{f \sqrt {\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right ) \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 35, normalized size = 0.95 \[ -\frac {\sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a - b\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sin \left (e+f\,x\right )}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\sin {\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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