Optimal. Leaf size=371 \[ -\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right )}{5 a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}} \]
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Rubi [A] time = 0.367597, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3900, 3888, 3881, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639} \[ -\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right )}{5 a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 3900
Rule 3888
Rule 3881
Rule 3884
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 2613
Rule 2615
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx &=\frac{\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{(e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=\frac{\int (-a+a \sec (c+d x)) \tan ^{\frac{5}{2}}(c+d x) \, dx}{a^2 (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{2 \int \left (-\frac{5 a}{2}+\frac{3}{2} a \sec (c+d x)\right ) \sqrt{\tan (c+d x)} \, dx}{5 a^2 (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{3 \int \sec (c+d x) \sqrt{\tan (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\int \sqrt{\tan (c+d x)} \, dx}{a (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{6 \int \cos (c+d x) \sqrt{\tan (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{\left (6 \cos ^{\frac{9}{2}}(c+d x)\right ) \int \sqrt{\cos (c+d x)} \sqrt{\sin (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \sin ^{\frac{9}{2}}(c+d x)}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{\left (6 \cos (c+d x) \cot ^4(c+d x)\right ) \int \sqrt{\sin (2 c+2 d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \sqrt{\sin (2 c+2 d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt{\sin (2 c+2 d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt{\sin (2 c+2 d x)}}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ &=-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt{\sin (2 c+2 d x)}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 19.2914, size = 261, normalized size = 0.7 \[ \frac{\sin ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (\sqrt{\sec ^2(c+d x)}+1\right ) \sqrt{e \cot (c+d x)} \left (8 \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2(c+d x)\right )-8 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2(c+d x)\right )+3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+6 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-6 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-8\right )}{6 a d e^5} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.28, size = 1505, normalized size = 4.1 \begin{align*} \text{result too large to display} \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{1}{\left (e \cot \left (d x + c\right )\right )^{\frac{9}{2}}{\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{1}{\left (e \cot \left (d x + c\right )\right )^{\frac{9}{2}}{\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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