Integrand size = 18, antiderivative size = 53 \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\frac {2 x \sqrt {\sec (a+b x)}}{b}-\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4297, 3856, 2720} \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\frac {2 x \sqrt {\sec (a+b x)}}{b}-\frac {4 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b^2} \]
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Rule 2720
Rule 3856
Rule 4297
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sqrt {\sec (a+b x)}}{b}-\frac {2 \int \sqrt {\sec (a+b x)} \, dx}{b} \\ & = \frac {2 x \sqrt {\sec (a+b x)}}{b}-\frac {\left (2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{b} \\ & = \frac {2 x \sqrt {\sec (a+b x)}}{b}-\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{b^2} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\frac {2 \left (b x-2 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )\right ) \sqrt {\sec (a+b x)}}{b^2} \]
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\[\int x \sec \left (x b +a \right )^{\frac {3}{2}} \sin \left (x b +a \right )d x\]
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Exception generated. \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \sec \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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\[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \sec \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int x\,\sin \left (a+b\,x\right )\,{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{3/2} \,d x \]
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