Optimal. Leaf size=206 \[ \frac{256 i \sqrt{e \sec (c+d x)}}{315 d e^4 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac{32 i}{105 d e^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac{2 i}{9 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.387039, antiderivative size = 206, 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} \[ \frac{256 i \sqrt{e \sec (c+d x)}}{315 d e^4 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac{32 i}{105 d e^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac{2 i}{9 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{7/2}} \]
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))^{7/2} \sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{8 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx}{9 a}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac{16 \int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx}{21 e^2}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac{64 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{105 a e^2}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac{128 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{315 e^4}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{256 i \sqrt{e \sec (c+d x)}}{315 d e^4 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.367011, size = 87, normalized size = 0.42 \[ \frac{\sqrt{e \sec (c+d x)} (336 \sin (2 (c+d x))+40 \sin (4 (c+d x))-84 i \cos (2 (c+d x))-5 i \cos (4 (c+d x))+945 i)}{1260 d e^4 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.352, size = 132, normalized size = 0.6 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ( 35\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+35\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+48\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +64\,i\cos \left ( dx+c \right ) +128\,\sin \left ( dx+c \right ) \right ) }{315\,ad{e}^{7}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{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 = 2.02318, size = 305, normalized size = 1.48 \begin{align*} \frac{35 i \, \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) - 45 i \, \cos \left (\frac{7}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 252 i \, \cos \left (\frac{5}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) - 420 i \, \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 1890 i \, \cos \left (\frac{1}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 45 \, \sin \left (\frac{7}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 252 \, \sin \left (\frac{5}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 420 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 1890 \, \sin \left (\frac{1}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right )}{2520 \, \sqrt{a} d e^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19531, size = 354, normalized size = 1.72 \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 (-45 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 465 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 1470 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2142 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 287 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i\right )} e^{\left (-\frac{9}{2} i \, d x - \frac{9}{2} i \, c\right )}}{2520 \, a d e^{4}} \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{7}{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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