Optimal. Leaf size=169 \[ \frac{32 i a^3 \sqrt{e \sec (c+d x)}}{77 d e^6 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{77 d e^4 (e \sec (c+d x))^{3/2}}-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}} \]
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Rubi [A] time = 0.311941, antiderivative size = 169, 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 = {3497, 3488} \[ \frac{32 i a^3 \sqrt{e \sec (c+d x)}}{77 d e^6 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{77 d e^4 (e \sec (c+d x))^{3/2}}-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{11/2}} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}}+\frac{(6 a) \int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx}{11 e^2}\\ &=-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (24 a^2\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{77 e^4}\\ &=-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{77 d e^4 (e \sec (c+d x))^{3/2}}-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (16 a^3\right ) \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{77 e^6}\\ &=\frac{32 i a^3 \sqrt{e \sec (c+d x)}}{77 d e^6 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{77 d e^4 (e \sec (c+d x))^{3/2}}-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.645988, size = 121, normalized size = 0.72 \[ \frac{a^2 \sqrt{a+i a \tan (c+d x)} (-22 \sin (c+d x)+42 \sin (3 (c+d x))-55 i \cos (c+d x)+35 i \cos (3 (c+d x))) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{154 d e^5 (\cos (d x)+i \sin (d x))^2 \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.477, size = 132, normalized size = 0.8 \begin{align*} -{\frac{2\,{a}^{2} \left ( 14\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}-14\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -8\,i\cos \left ( dx+c \right ) -16\,\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{77\,d{e}^{11}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.91745, size = 167, normalized size = 0.99 \begin{align*} \frac{{\left (-7 i \, a^{2} \cos \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) - 33 i \, a^{2} \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) - 77 i \, a^{2} \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 77 i \, a^{2} \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7 \, a^{2} \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 33 \, a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 77 \, a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 77 \, a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{308 \, d e^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34185, size = 288, normalized size = 1.7 \begin{align*} \frac{{\left (-7 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 40 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 110 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 77 i \, a^{2}\right )} \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}} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{308 \, d e^{6}} \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{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\left (e \sec \left (d x + c\right )\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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