Optimal. Leaf size=165 \[ -\frac{32 i \sqrt{a+i a \tan (c+d x)}}{35 a d e^2 \sqrt{e \sec (c+d x)}}+\frac{16 i}{35 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}+\frac{2 i}{7 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.290697, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3502, 3497, 3488} \[ -\frac{32 i \sqrt{a+i a \tan (c+d x)}}{35 a d e^2 \sqrt{e \sec (c+d x)}}+\frac{16 i}{35 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}+\frac{2 i}{7 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{2 i}{7 d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{6 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx}{7 a}\\ &=\frac{2 i}{7 d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}+\frac{24 \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx}{35 e^2}\\ &=\frac{2 i}{7 d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{16 i}{35 d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}+\frac{16 \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{35 a e^2}\\ &=\frac{2 i}{7 d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{16 i}{35 d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}-\frac{32 i \sqrt{a+i a \tan (c+d x)}}{35 a d e^2 \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.339108, size = 79, normalized size = 0.48 \[ -\frac{i (\cos (2 (c+d x))+35 i \tan (c+d x)+3 i \sin (3 (c+d x)) \sec (c+d x)+17)}{35 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.332, size = 115, normalized size = 0.7 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( 5\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -16\,i \right ) }{35\,ad{e}^{5}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9702, size = 240, normalized size = 1.45 \begin{align*} \frac{5 i \, \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) - 7 i \, \cos \left (\frac{5}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) + 35 i \, \cos \left (\frac{3}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) - 105 i \, \cos \left (\frac{1}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 \, \sin \left (\frac{5}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac{3}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) + 105 \, \sin \left (\frac{1}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right )}{140 \, \sqrt{a} d e^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08237, size = 304, normalized size = 1.84 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-7 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 112 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 40 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac{7}{2} i \, d x - \frac{7}{2} i \, c\right )}}{140 \, a d e^{3}} \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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{5}{2}} \sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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