Optimal. Leaf size=106 \[ \frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}}{3 d}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d} \]
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Rubi [A] time = 0.0894341, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3498, 3486, 3771, 2641} \[ \frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}}{3 d}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3498
Rule 3486
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx &=\frac{2 i \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac{1}{3} (5 a) \int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx\\ &=\frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac{1}{3} \left (5 a^2\right ) \int \sqrt{e \sec (c+d x)} \, dx\\ &=\frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac{1}{3} \left (5 a^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}\\ \end{align*}
Mathematica [A] time = 0.614256, size = 67, normalized size = 0.63 \[ \frac{2 a^2 (e \sec (c+d x))^{3/2} \left (-\sin (c+d x)+6 i \cos (c+d x)+5 \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 d e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.237, size = 201, normalized size = 1.9 \begin{align*}{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{3\,d\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}} \left ( 5\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +5\,i\cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +6\,i\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) \right ) } \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 \sqrt{e \sec \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{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*} \frac{\sqrt{2}{\left (14 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i \, a^{2}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 3 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}{\rm integral}\left (-\frac{5 i \, \sqrt{2} a^{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \, d}, x\right )}{3 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \sqrt{e \sec{\left (c + d x \right )}}\, dx + \int - \sqrt{e \sec{\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\, dx + \int 2 i \sqrt{e \sec{\left (c + d x \right )}} \tan{\left (c + d x \right )}\, dx\right ) \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 \sqrt{e \sec \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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