Optimal. Leaf size=135 \[ -\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}+\frac{9 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}+\frac{9 d^{11/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{9/2}}{2 b} \]
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Rubi [A] time = 0.0894689, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2565, 288, 321, 329, 212, 206, 203} \[ -\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}+\frac{9 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}+\frac{9 d^{11/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{9/2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 288
Rule 321
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{11/2} \csc ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^{11/2}}{\left (1-\frac{x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}+\frac{(9 d) \operatorname{Subst}\left (\int \frac{x^{7/2}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b}\\ &=-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}+\frac{\left (9 d^3\right ) \operatorname{Subst}\left (\int \frac{x^{3/2}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b}\\ &=-\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}+\frac{\left (9 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b}\\ &=-\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}+\frac{\left (9 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^4}{d^2}} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{2 b}\\ &=-\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}+\frac{\left (9 d^6\right ) \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{4 b}+\frac{\left (9 d^6\right ) \operatorname{Subst}\left (\int \frac{1}{d+x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{4 b}\\ &=\frac{9 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}+\frac{9 d^{11/2} \tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{4 b}-\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d (d \cos (a+b x))^{9/2} \csc ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 2.07647, size = 137, normalized size = 1.01 \[ \frac{d (d \cos (a+b x))^{9/2} \left (-\frac{21}{2} \left (8 \sqrt{\cos (a+b x)}+\log \left (1-\sqrt{\cos (a+b x)}\right )-\log \left (\sqrt{\cos (a+b x)}+1\right )\right )+45 \tan ^{-1}\left (\sqrt{\cos (a+b x)}\right )-2 \sqrt{\cos (a+b x)} \left (2 \cos (2 (a+b x))+5 \csc ^2(a+b x)\right )+24 \tanh ^{-1}\left (\sqrt{\cos (a+b x)}\right )\right )}{20 b \cos ^{\frac{9}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.221, size = 433, normalized size = 3.2 \begin{align*} -{\frac{8\,{d}^{5}}{5\,b} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d-d}}+{\frac{8\,{d}^{5}}{5\,b} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d-d}}+{\frac{8\,{d}^{5}}{5\,b}\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d-d}}-6\,{\frac{{d}^{5}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}{b}}+{\frac{9}{8\,b}{d}^{{\frac{11}{2}}}\ln \left ({ \left ( 4\,d\cos \left ( 1/2\,bx+a/2 \right ) +2\,\sqrt{d}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}-2\,d \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -1 \right ) ^{-1}} \right ) }+{\frac{{d}^{5}}{16\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -1 \right ) ^{-1}}+{\frac{9}{8\,b}{d}^{{\frac{11}{2}}}\ln \left ({ \left ( -4\,d\cos \left ( 1/2\,bx+a/2 \right ) +2\,\sqrt{d}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d}-2\,d \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) ^{-1}} \right ) }-{\frac{9\,{d}^{6}}{4\,b}\ln \left ({ \left ( -2\,d+2\,\sqrt{-d}\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d-d} \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}} \right ){\frac{1}{\sqrt{-d}}}}-{\frac{{d}^{5}}{16\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{d}^{5}}{8\,b}\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d-d} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-2}} \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 = 3.56949, size = 1061, normalized size = 7.86 \begin{align*} \left [-\frac{90 \,{\left (d^{5} \cos \left (b x + a\right )^{2} - d^{5}\right )} \sqrt{-d} \arctan \left (\frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{-d}}{d \cos \left (b x + a\right ) + d}\right ) - 45 \,{\left (d^{5} \cos \left (b x + a\right )^{2} - d^{5}\right )} \sqrt{-d} \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 (4 \, d^{5} \cos \left (b x + a\right )^{4} + 36 \, d^{5} \cos \left (b x + a\right )^{2} - 45 \, d^{5}\right )} \sqrt{d \cos \left (b x + a\right )}}{80 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}}, -\frac{90 \,{\left (d^{5} \cos \left (b x + a\right )^{2} - d^{5}\right )} \sqrt{d} \arctan \left (\frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{d}}{d \cos \left (b x + a\right ) - d}\right ) - 45 \,{\left (d^{5} \cos \left (b x + a\right )^{2} - d^{5}\right )} \sqrt{d} \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 (4 \, d^{5} \cos \left (b x + a\right )^{4} + 36 \, d^{5} \cos \left (b x + a\right )^{2} - 45 \, d^{5}\right )} \sqrt{d \cos \left (b x + a\right )}}{80 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}}\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{11}{2}} \csc \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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