Optimal. Leaf size=105 \[ \frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{16 \sin (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}-\frac{8 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.0799988, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4303, 4304, 4292} \[ \frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{16 \sin (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}-\frac{8 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4303
Rule 4304
Rule 4292
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx &=-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{6}{7} \int \frac{\sin (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{24}{35} \int \frac{\cos (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{16}{35} \int \frac{\sin (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{16 \sin (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.14338, size = 67, normalized size = 0.64 \[ \frac{\sqrt{\sin (2 (a+b x))} (-10 \cos (2 (a+b x))-4 \cos (4 (a+b x))+4 \cos (6 (a+b x))+5) \csc ^4(a+b x) \sec ^3(a+b x)}{560 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 183.381, size = 222, normalized size = 2.1 \begin{align*}{\frac{1}{2688\,b}\sqrt{-{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) \left ( 3\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}+40\,\sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1}\sqrt{-2\,\tan \left ( 1/2\,bx+a/2 \right ) +2}\sqrt{-\tan \left ( 1/2\,bx+a/2 \right ) }{\it EllipticF} \left ( \sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1},1/2\,\sqrt{2} \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-26\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}+26\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-3 \right ) \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3}{\frac{1}{\sqrt{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) }}}} \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{\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.534884, size = 320, normalized size = 3.05 \begin{align*} \frac{128 \, \cos \left (b x + a\right )^{7} - 256 \, \cos \left (b x + a\right )^{5} + 128 \, \cos \left (b x + a\right )^{3} + \sqrt{2}{\left (128 \, \cos \left (b x + a\right )^{6} - 224 \, \cos \left (b x + a\right )^{4} + 84 \, \cos \left (b x + a\right )^{2} + 7\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{560 \,{\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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