Optimal. Leaf size=162 \[ -\frac{32 i \sqrt{a+i a \tan (c+d x)}}{77 a^3 d \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a^2 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{12 i}{77 a d (a+i a \tan (c+d x))^{3/2} \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{5/2} \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.301009, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3502, 3488} \[ -\frac{32 i \sqrt{a+i a \tan (c+d x)}}{77 a^3 d \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a^2 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{12 i}{77 a d (a+i a \tan (c+d x))^{3/2} \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{5/2} \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i}{11 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{6 \int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx}{11 a}\\ &=\frac{2 i}{11 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{12 i}{77 a d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{24 \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx}{77 a^2}\\ &=\frac{2 i}{11 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{12 i}{77 a d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a^2 d \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{16 \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{77 a^3}\\ &=\frac{2 i}{11 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{12 i}{77 a d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{16 i}{77 a^2 d \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{32 i \sqrt{a+i a \tan (c+d x)}}{77 a^3 d \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.400181, size = 102, normalized size = 0.63 \[ \frac{i \sec ^3(c+d x) (-22 i \sin (c+d x)+42 i \sin (3 (c+d x))-55 \cos (c+d x)+35 \cos (3 (c+d x)))}{154 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.32, size = 140, normalized size = 0.9 \begin{align*}{\frac{2\,\cos \left ( dx+c \right ) \left ( 28\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+28\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -9\,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 ) }{77\,d{a}^{3}e}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}}\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 = 1.9415, size = 240, normalized size = 1.48 \begin{align*} \frac{7 i \, \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 33 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{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 ) - 77 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 ) + 7 \, \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 33 \, \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{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 ) + 77 \, \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 )}{308 \, a^{\frac{5}{2}} d \sqrt{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12603, size = 271, normalized size = 1.67 \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 (8 i \, d x + 8 i \, c\right )} + 110 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 40 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-\frac{11}{2} i \, d x - \frac{11}{2} i \, c\right )}}{308 \, a^{3} d e} \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}{\sqrt{e \sec \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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