Optimal. Leaf size=375 \[ -\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) \text{EllipticF}\left (c+d x-\frac{\pi }{4},2\right )}{3 d (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.328424, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {3900, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 30} \[ \frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac{\pi }{4}\right |2\right )}{3 d (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3900
Rule 3886
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 2611
Rule 2614
Rule 2573
Rule 2641
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx &=\frac{\int (a+a \sec (c+d x))^2 \tan ^{\frac{3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{\int \left (a^2 \tan ^{\frac{3}{2}}(c+d x)+2 a^2 \sec (c+d x) \tan ^{\frac{3}{2}}(c+d x)+a^2 \sec ^2(c+d x) \tan ^{\frac{3}{2}}(c+d x)\right ) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{a^2 \int \tan ^{\frac{3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \int \sec ^2(c+d x) \tan ^{\frac{3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left (2 a^2\right ) \int \sec (c+d x) \tan ^{\frac{3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac{\left (2 a^2\right ) \int \frac{\sec (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{3 (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \int \frac{1}{\sqrt{\tan (c+d x)}} \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \operatorname{Subst}\left (\int x^{3/2} \, dx,x,\tan (c+d x)\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac{\left (2 a^2 \cos ^{\frac{3}{2}}(c+d x)\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{\sin (c+d x)}} \, dx}{3 (e \cot (c+d x))^{3/2} \sin ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac{\left (2 a^2 \cot (c+d x) \csc (c+d x) \sqrt{\sin (2 c+2 d x)}\right ) \int \frac{1}{\sqrt{\sin (2 c+2 d x)}} \, dx}{3 (e \cot (c+d x))^{3/2}}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \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} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \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} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d (e \cot (c+d x))^{3/2} \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 5.92444, size = 127, normalized size = 0.34 \[ \frac{a^2 \sin ^2(c+d x) (\sec (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right ) \left (\text{Hypergeometric2F1}\left (-\frac{5}{4},1,-\frac{1}{4},-\cot ^2(c+d x)\right )+2 \left (\text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},-\tan ^2(c+d x)\right )+5 \cot ^2(c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )\right )\right )}{10 d e \sqrt{e \cot (c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.309, size = 721, normalized size = 1.9 \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(-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 [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{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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