Optimal. Leaf size=57 \[ \frac{4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i}{a^3 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.0736954, antiderivative size = 57, 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} \[ \frac{4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i}{a^3 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (\frac{2 a}{(a+x)^{5/2}}-\frac{1}{(a+x)^{3/2}}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac{4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i}{a^3 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.205322, size = 80, normalized size = 1.4 \[ \frac{2 (1+3 i \tan (c+d x)) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{3 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.271, size = 88, normalized size = 1.5 \begin{align*}{\frac{2\,\cos \left ( dx+c \right ) \left ( 4\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -5\,i\cos \left ( dx+c \right ) -3\,\sin \left ( dx+c \right ) \right ) }{3\,{a}^{4}d}\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 = 0.977343, size = 43, normalized size = 0.75 \begin{align*} -\frac{2 i \,{\left (3 i \, a \tan \left (d x + c\right ) + a\right )}}{3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04964, size = 177, normalized size = 3.11 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-2 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{4} d} \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{\sec \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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