Optimal. Leaf size=222 \[ -\frac{a^2 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
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Rubi [A] time = 0.312573, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3878, 3872, 2873, 2635, 2641, 2564, 321, 329, 212, 206, 203, 2566} \[ -\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3878
Rule 3872
Rule 2873
Rule 2635
Rule 2641
Rule 2564
Rule 321
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2566
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx &=\frac{\int (a+a \sec (c+d x))^2 \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{\int \left (a^2 \sin ^{\frac{3}{2}}(c+d x)+2 a^2 \sec (c+d x) \sin ^{\frac{3}{2}}(c+d x)+a^2 \sec ^2(c+d x) \sin ^{\frac{3}{2}}(c+d x)\right ) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{a^2 \int \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a^2 \int \sec ^2(c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2\right ) \int \sec (c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}-\frac{a^2 \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{x^{3/2}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}-\frac{a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.61748, size = 164, normalized size = 0.74 \[ \frac{2 a^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \tan (c+d x) \sqrt{e \csc (c+d x)} \sec ^4\left (\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (-6 \sqrt{\cos ^2(c+d x)} \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},\csc ^2(c+d x)\right )+3 \sqrt{-\cot ^2(c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},\csc ^2(c+d x)\right )+\sin ^2(c+d x) \sqrt{-\cot ^2(c+d x)} \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{3}{2},\frac{1}{4},\csc ^2(c+d x)\right )+3\right )}{3 d e^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.218, size = 763, normalized size = 3.4 \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{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt{e \csc \left (d x + c\right )}}{e^{2} \csc \left (d x + c\right )^{2}}, x\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{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \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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