Integrand size = 18, antiderivative size = 80 \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=-\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{15 b^2}+\frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {4 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{15 b^2} \]
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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, 3853, 3856, 2720} \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=-\frac {4 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {4 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{15 b^2}+\frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rule 2720
Rule 3853
Rule 3856
Rule 4297
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sec ^{\frac {5}{2}}(a+b x) \, dx}{5 b} \\ & = \frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {4 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{15 b^2}-\frac {2 \int \sqrt {\sec (a+b x)} \, dx}{15 b} \\ & = \frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {4 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{15 b^2}-\frac {\left (2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{15 b} \\ & = -\frac {4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{15 b^2}+\frac {2 x \sec ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {4 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{15 b^2} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\frac {2 \sqrt {\sec (a+b x)} \left (-2 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+3 b x \sec ^2(a+b x)-2 \tan (a+b x)\right )}{15 b^2} \]
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\[\int x \sec \left (x b +a \right )^{\frac {7}{2}} \sin \left (x b +a \right )d x\]
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Exception generated. \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \sec \left (b x + a\right )^{\frac {7}{2}} \sin \left (b x + a\right ) \,d x } \]
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\[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \sec \left (b x + a\right )^{\frac {7}{2}} \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \sec ^{\frac {7}{2}}(a+b x) \sin (a+b x) \, dx=\int x\,\sin \left (a+b\,x\right )\,{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{7/2} \,d x \]
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