Optimal. Leaf size=120 \[ \frac{2 i}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x))^2 \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.158404, antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3515, 3500, 3769, 3771, 2641} \[ \frac{4 i \cos ^2(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \cos (c+d x)}}+\frac{2 \sin (c+d x) \cos (c+d x)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3500
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cos (c+d x)} (a+i a \tan (c+d x))^2} \, dx &=\frac{\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx}{\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{4 i \cos ^2(c+d x)}{7 d \sqrt{e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (3 e^2\right ) \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 i \cos ^2(c+d x)}{7 d \sqrt{e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\int \sqrt{e \sec (c+d x)} \, dx}{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 i \cos ^2(c+d x)}{7 d \sqrt{e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\sqrt{\cos (c+d x)} \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{7 a^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 i \cos ^2(c+d x)}{7 d \sqrt{e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.640359, size = 158, normalized size = 1.32 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )-i \cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sqrt{\cos (c+d x)} \left (4 i \sin ^3\left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )+2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (\sin \left (\frac{3}{2} (c+d x)\right )-i \cos \left (\frac{3}{2} (c+d x)\right )\right )\right )}{7 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\tan (c+d x)-i)^2 \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.564, size = 240, normalized size = 2. \begin{align*}{\frac{2}{7\,{a}^{2}d} \left ( 32\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}-32\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-64\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}+48\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +48\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}-28\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -16\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}-\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) +6\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +2\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \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: RuntimeError} \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{{\left (7 \, a^{2} d e e^{\left (3 i \, d x + 3 i \, c\right )}{\rm integral}\left (-\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{7 \,{\left (a^{2} d e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e\right )}}, x\right ) + \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e}{\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{7 \, a^{2} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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}{\sqrt{e \cos \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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