Integrand size = 18, antiderivative size = 80 \[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=-\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)}+\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{9 b^2}+\frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}} \]
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Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4297, 3854, 3856, 2720} \[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}}+\frac {4 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{9 b^2}-\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)} \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4297
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sec ^{\frac {3}{2}}(a+b x)} \, dx}{3 b} \\ & = -\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)}+\frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}}+\frac {2 \int \sqrt {\sec (a+b x)} \, dx}{9 b} \\ & = -\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)}+\frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}}+\frac {\left (2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{9 b} \\ & = -\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)}+\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{9 b^2}+\frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\frac {\sqrt {\sec (a+b x)} \left (-6 b x \cos ^2(a+b x)+4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+2 \sin (2 (a+b x))\right )}{9 b^2} \]
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\[\int \frac {x \sin \left (x b +a \right )}{\sqrt {\sec \left (x b +a \right )}}d x\]
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Exception generated. \[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\int \frac {x \sin {\left (a + b x \right )}}{\sqrt {\sec {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sqrt {\sec \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sqrt {\sec \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {x \sin (a+b x)}{\sqrt {\sec (a+b x)}} \, dx=\int \frac {x\,\sin \left (a+b\,x\right )}{\sqrt {\frac {1}{\cos \left (a+b\,x\right )}}} \,d x \]
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