Optimal. Leaf size=79 \[ \frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {\sqrt {\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b d \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2602, 2569, 2573, 2641} \[ \frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {\sqrt {\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b d \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2569
Rule 2573
Rule 2602
Rule 2641
Rubi steps
\begin {align*} \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {\sqrt {\sin (a+b x)} \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sqrt {\sin (a+b x)}} \, dx}{d \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=\frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{2 d \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=\frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {\left (\sec (a+b x) \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{2 d \sqrt {d \tan (a+b x)}}\\ &=\frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b d \sqrt {d \tan (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.77, size = 126, normalized size = 1.59 \[ \frac {\cos (2 (a+b x)) \tan ^{\frac {3}{2}}(a+b x) \sec (a+b x) \left (-\sqrt {\tan (a+b x)} \sqrt {\sec ^2(a+b x)}+\sqrt [4]{-1} \sec ^2(a+b x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right )\right )}{b \left (\tan ^2(a+b x)-1\right ) \sqrt {\sec ^2(a+b x)} (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right )}{d^{2} \tan \left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 196, normalized size = 2.48 \[ -\frac {\left (\cos \left (b x +a \right )+1\right )^{2} \left (-1+\cos \left (b x +a \right )\right ) \left (\sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+\cos \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{2 b \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sin \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sin \left (a+b\,x\right )}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sin {\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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