Optimal. Leaf size=209 \[ -\frac{256 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt{e \sec (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac{128 i}{385 a d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.392568, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3502, 3497, 3488} \[ -\frac{256 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt{e \sec (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac{128 i}{385 a d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}} \]
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} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac{8 \int \frac{1}{(e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}} \, dx}{11 a}\\ &=\frac{2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{48 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx}{77 a^2}\\ &=\frac{2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac{192 \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx}{385 a e^2}\\ &=\frac{2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{128 i}{385 a d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac{128 \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{385 a^2 e^2}\\ &=\frac{2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}+\frac{128 i}{385 a d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}-\frac{256 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.471438, size = 100, normalized size = 0.48 \[ -\frac{(e \sec (c+d x))^{3/2} (880 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+660 \cos (2 (c+d x))+21 \cos (4 (c+d x))-385)}{1540 a d e^4 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.312, size = 142, normalized size = 0.7 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( 70\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+70\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +5\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+40\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+64\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -128\,i \right ) }{385\,{a}^{2}d{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 = 2.03733, size = 305, normalized size = 1.46 \begin{align*} \frac{35 i \, \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 220 i \, \cos \left (\frac{7}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) - 77 i \, \cos \left (\frac{5}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) + 770 i \, \cos \left (\frac{3}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) - 1540 i \, \cos \left (\frac{1}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 220 \, \sin \left (\frac{7}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) + 77 \, \sin \left (\frac{5}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) + 770 \, \sin \left (\frac{3}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right ) + 1540 \, \sin \left (\frac{1}{11} \, \arctan \left (\sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ), \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right )\right )\right )}{3080 \, a^{\frac{3}{2}} d e^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05023, size = 358, normalized size = 1.71 \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 (-77 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 1617 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 770 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 990 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 255 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i\right )} e^{\left (-\frac{11}{2} i \, d x - \frac{11}{2} i \, c\right )}}{3080 \, a^{2} 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}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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