Optimal. Leaf size=191 \[ \frac{4 i e^2}{33 d \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 a^4 d}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.191723, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3502, 3500, 3769, 3771, 2641} \[ \frac{4 i e^2}{33 d \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 a^4 d}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^4} \, dx &=\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{7 \int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx}{15 a}\\ &=\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{7 \int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx}{33 a^2}\\ &=\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{e^2 \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{11 a^4}\\ &=\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \sqrt{e \sec (c+d x)} \, dx}{33 a^4}\\ &=\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{33 a^4}\\ &=\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 a^4 d}+\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.486795, size = 137, normalized size = 0.72 \[ \frac{\sec ^4(c+d x) \sqrt{e \sec (c+d x)} \left (i (54 i \sin (2 (c+d x))+37 i \sin (4 (c+d x))+112 \cos (2 (c+d x))+48 \cos (4 (c+d x))+64)+40 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (4 (c+d x))+i \sin (4 (c+d x)))\right )}{660 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.356, size = 252, normalized size = 1.3 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{165\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}} \left ( 88\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}+88\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}-60\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-16\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \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 ) +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}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +5\,\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(-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 (1320 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )}{\rm integral}\left (-\frac{i \, \sqrt{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 )}}{33 \, a^{4} d}, x\right ) + \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (85 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 166 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 128 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 58 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{1320 \, a^{4} d} \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{\sqrt{e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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