Optimal. Leaf size=47 \[ \frac{\sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}} \]
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Rubi [A] time = 0.0572691, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2601, 2572, 2639} \[ \frac{\sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2601
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx &=\frac{\sqrt{\sin (a+b x)} \int \sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)} \, dx}{\sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}}\\ &=\frac{\sin (a+b x) \int \sqrt{\sin (2 a+2 b x)} \, dx}{\sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ &=\frac{E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.125529, size = 60, normalized size = 1.28 \[ \frac{2 \sin (a+b x) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right )}{3 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.147, size = 531, normalized size = 11.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )}{\sqrt{d \tan \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (b x + a\right )} \sin \left (b x + a\right )}{d \tan \left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b x \right )}}{\sqrt{d \tan{\left (a + b x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )}{\sqrt{d \tan \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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