Optimal. Leaf size=59 \[ \frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d} \]
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Rubi [A] time = 0.0622915, antiderivative size = 59, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d}+\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.267442, size = 58, normalized size = 0.98 \[ \frac{2 \sqrt{a+i a \tan (c+d x)} \left (8 (\tan (c+d x)-i)+(5 \tan (c+d x)-i) \sec ^2(c+d x)\right )}{35 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.34, size = 87, normalized size = 1.5 \begin{align*} -{\frac{16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +2\,i\cos \left ( dx+c \right ) -10\,\sin \left ( dx+c \right ) }{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\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 = 0.947808, size = 54, normalized size = 0.92 \begin{align*} \frac{2 i \,{\left (5 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 14 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a\right )}}{35 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.32816, size = 270, normalized size = 4.58 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-32 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 112 i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{35 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \sec ^{4}{\left (c + d x \right )}\, dx \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 )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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