Integrand size = 23, antiderivative size = 62 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {2 b \sec (e+f x)}{a^2 f \sqrt {a+b \sec ^2(e+f x)}} \]
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Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4219, 277, 197} \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {2 b \sec (e+f x)}{a^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cos (e+f x)}{a f \sqrt {a+b \sec ^2(e+f x)}} \]
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Rule 197
Rule 277
Rule 4219
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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) \text {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*}
Time = 0.94 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) (a+4 b+a \cos (2 (e+f x))) \sec ^3(e+f x)}{4 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {1}{a \sec \left (f x +e \right ) \sqrt {a +b \sec \left (f x +e \right )^{2}}}-\frac {2 b \sec \left (f x +e \right )}{a^{2} \sqrt {a +b \sec \left (f x +e \right )^{2}}}}{f}\) | \(59\) |
default | \(\frac {-\frac {1}{a \sec \left (f x +e \right ) \sqrt {a +b \sec \left (f x +e \right )^{2}}}-\frac {2 b \sec \left (f x +e \right )}{a^{2} \sqrt {a +b \sec \left (f x +e \right )^{2}}}}{f}\) | \(59\) |
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\left (a \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f} \]
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\[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sin {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2}} + \frac {b}{\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} a^{2} \cos \left (f x + e\right )}}{f} \]
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Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2} + b} + \frac {b}{\sqrt {a \cos \left (f x + e\right )^{2} + b}}}{a^{2} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Time = 24.73 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.50 \[ \int \frac {\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {a+\frac {b}{{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )}^2}}\,\left (a+2\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}+8\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )}{2\,a^2\,f\,\left (a+2\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}+4\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )} \]
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