Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d} \]
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Rubi [A] time = 0.0690455, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 43} \[ -\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^6(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{5/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.407978, size = 77, normalized size = 0.88 \[ \frac{2 \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} (-91 i \sin (2 (c+d x))+107 \cos (2 (c+d x))+44) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{693 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.536, size = 114, normalized size = 1.3 \begin{align*} -{\frac{256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}-256\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+32\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-160\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +14\,i\cos \left ( dx+c \right ) -126\,\sin \left ( dx+c \right ) }{693\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}\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 = 0.999381, size = 78, normalized size = 0.89 \begin{align*} -\frac{2 i \,{\left (63 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 308 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a + 396 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{2}\right )}}{693 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31028, size = 392, normalized size = 4.45 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-512 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 2816 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 6336 i \, e^{\left (6 i \, d x + 6 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{693 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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 \sqrt{i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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