Optimal. Leaf size=100 \[ \frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}-\frac{d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b} \]
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Rubi [A] time = 0.0769487, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2565, 321, 329, 298, 203, 206} \[ \frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}-\frac{d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 321
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^{9/2}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac{d \operatorname{Subst}\left (\int \frac{x^{5/2}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b}\\ &=\frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac{d^3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b}\\ &=\frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{d^2}} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b}\\ &=\frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac{d^5 \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b}+\frac{d^5 \operatorname{Subst}\left (\int \frac{1}{d+x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b}\\ &=\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}-\frac{d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b}+\frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}\\ \end{align*}
Mathematica [A] time = 0.19088, size = 83, normalized size = 0.83 \[ \frac{d^4 \sqrt{d \cos (a+b x)} \left (2 \left (3 \cos ^2(a+b x)+7\right ) \cos ^{\frac{3}{2}}(a+b x)+21 \tan ^{-1}\left (\sqrt{\cos (a+b x)}\right )-21 \tanh ^{-1}\left (\sqrt{\cos (a+b x)}\right )\right )}{21 b \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 318, normalized size = 3.2 \begin{align*} -{\frac{16\,{d}^{4}}{7\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{6}}-{\frac{1}{2\,b}{d}^{{\frac{9}{2}}}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}-2\,d\cos \left ( 1/2\,bx+a/2 \right ) -d}{\cos \left ( 1/2\,bx+a/2 \right ) +1}} \right ) }-{\frac{1}{2\,b}{d}^{{\frac{9}{2}}}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}+2\,d\cos \left ( 1/2\,bx+a/2 \right ) -d}{\cos \left ( 1/2\,bx+a/2 \right ) -1}} \right ) }+{\frac{24\,{d}^{4}}{7\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}}-{\frac{64\,{d}^{4}}{21\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}-{\frac{{d}^{5}}{b}\ln \left ( 2\,{\frac{\sqrt{-d}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}-d}{\cos \left ( 1/2\,bx+a/2 \right ) }} \right ){\frac{1}{\sqrt{-d}}}}+{\frac{20\,{d}^{4}}{21\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.80212, size = 836, normalized size = 8.36 \begin{align*} \left [\frac{42 \, \sqrt{-d} d^{4} \arctan \left (\frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{-d}}{d \cos \left (b x + a\right ) + d}\right ) + 21 \, \sqrt{-d} d^{4} \log \left (-\frac{d \cos \left (b x + a\right )^{2} + 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{-d}{\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \,{\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt{d \cos \left (b x + a\right )}}{84 \, b}, -\frac{42 \, d^{\frac{9}{2}} \arctan \left (\frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{d}}{d \cos \left (b x + a\right ) - d}\right ) - 21 \, d^{\frac{9}{2}} \log \left (-\frac{d \cos \left (b x + a\right )^{2} - 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{d}{\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) - 8 \,{\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt{d \cos \left (b x + a\right )}}{84 \, b}\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 \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \csc \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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