Optimal. Leaf size=62 \[ -\frac {2 b \sec (e+f x)}{a^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4134, 271, 191} \[ -\frac {2 b \sec (e+f x)}{a^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 4134
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{a f}\\ &=-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {2 b \sec (e+f x)}{a^2 f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 64, normalized size = 1.03 \[ -\frac {\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) (a \cos (2 (e+f x))+a+4 b)}{4 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 67, normalized size = 1.08 \[ -\frac {{\left (a \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f} \]
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 [A] time = 0.29, size = 59, normalized size = 0.95 \[ \frac {-\frac {1}{a \sec \left (f x +e \right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {2 b \sec \left (f x +e \right )}{a^{2} \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 57, normalized size = 0.92 \[ -\frac {\frac {\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2}} + \frac {b}{\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} a^{2} \cos \left (f x + e\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.87, size = 155, normalized size = 2.50 \[ -\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {a+\frac {b}{{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )}^2}}\,\left (a+2\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}+8\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )}{2\,a^2\,f\,\left (a+2\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}+4\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )} \]
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 \[ \int \frac {\sin {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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