Integrand size = 18, antiderivative size = 80 \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {12 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{25 b^2}+\frac {4 \sin (a+b x)}{25 b^2 \sec ^{\frac {3}{2}}(a+b x)} \]
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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, 2719} \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \sin (a+b x)}{25 b^2 \sec ^{\frac {3}{2}}(a+b x)}+\frac {12 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}-\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)} \]
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Rule 2719
Rule 3854
Rule 3856
Rule 4297
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sec ^{\frac {5}{2}}(a+b x)} \, dx}{5 b} \\ & = -\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {4 \sin (a+b x)}{25 b^2 \sec ^{\frac {3}{2}}(a+b x)}+\frac {6 \int \frac {1}{\sqrt {\sec (a+b x)}} \, dx}{25 b} \\ & = -\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {4 \sin (a+b x)}{25 b^2 \sec ^{\frac {3}{2}}(a+b x)}+\frac {\left (6 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{25 b} \\ & = -\frac {2 x}{5 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {12 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{25 b^2}+\frac {4 \sin (a+b x)}{25 b^2 \sec ^{\frac {3}{2}}(a+b x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.83 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.21 \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\frac {-5 b x-5 b x \cos (2 (a+b x))+\sec (a+b x) \sin (3 (a+b x))+13 \tan (a+b x)-6 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\tan ^2(a+b x)\right ) \sqrt [4]{\sec ^2(a+b x)} \tan (a+b x)}{25 b^2 \sqrt {\sec (a+b x)}} \]
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\[\int \frac {x \sin \left (x b +a \right )}{\sec \left (x b +a \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \sin {\left (a + b x \right )}}{\sec ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sec \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sec \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\sin \left (a+b\,x\right )}{{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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