Optimal. Leaf size=80 \[ \frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}}+\frac {4 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{9 b^2}-\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4212, 3769, 3771, 2641} \[ \frac {4 \sin (a+b x)}{9 b^2 \sqrt {\sec (a+b x)}}+\frac {4 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{9 b^2}-\frac {2 x}{3 b \sec ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4212
Rubi steps
\begin {align*} \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 {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)} F\left (\left .\frac {1}{2} (a+b x)\right |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*}
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Mathematica [A] time = 0.24, size = 63, normalized size = 0.79 \[ \frac {\sqrt {\sec (a+b x)} \left (2 \sin (2 (a+b x))-6 b x \cos ^2(a+b x)+4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{9 b^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\sqrt {\sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x +a \right )}{\sqrt {\sec \left (b x +a \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\sqrt {\sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sin \left (a+b\,x\right )}{\sqrt {\frac {1}{\cos \left (a+b\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin {\left (a + b x \right )}}{\sqrt {\sec {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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