Optimal. Leaf size=79 \[ \frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b d \sqrt {d \tan (a+b x)}} \]
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Rubi [A]
time = 0.07, 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 = {2682, 2649,
2653, 2720} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2649
Rule 2653
Rule 2682
Rule 2720
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] Result contains complex when optimal does not.
time = 10.82, size = 126, normalized size = 1.59 \begin {gather*} \frac {\cos (2 (a+b x)) \sec (a+b x) \left (\sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right ) \sec ^2(a+b x)-\sqrt {\sec ^2(a+b x)} \sqrt {\tan (a+b x)}\right ) \tan ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sec ^2(a+b x)} (d \tan (a+b x))^{3/2} \left (-1+\tan ^2(a+b x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs.
\(2(96)=192\).
time = 0.32, size = 197, normalized size = 2.49
method | result | size |
default | \(\frac {\left (\cos \left (b x +a \right )+1\right )^{2} \left (-1+\cos \left (b x +a \right )\right ) \left (-\sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-\cos \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{2 b \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\) | \(197\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sin {\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sin \left (a+b\,x\right )}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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