Optimal. Leaf size=34 \[ \frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]
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Rubi [A] time = 0.062903, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3088, 848, 37} \[ \frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 848
Rule 37
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^5 (i a+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{i}{a}+\frac{x}{a}\right )^3}{x^5} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{i (i-\cot (c+d x))^4 \tan ^4(c+d x)}{4 a^3 d}\\ \end{align*}
Mathematica [B] time = 0.465381, size = 90, normalized size = 2.65 \[ -\frac{i \sec (c) \sec ^4(c+d x) (2 i \sin (c+2 d x)-2 i \sin (3 c+2 d x)+i \sin (3 c+4 d x)+2 \cos (c+2 d x)+2 \cos (3 c+2 d x)-3 i \sin (c)+3 \cos (c))}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.197, size = 47, normalized size = 1.4 \begin{align*}{\frac{\tan \left ( dx+c \right ) +{\frac{i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}- \left ( \tan \left ( dx+c \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08891, size = 324, normalized size = 9.53 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{8 i \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{7 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{3 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{\sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{{\left (a^{3} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.448625, size = 177, normalized size = 5.21 \begin{align*} \frac{4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21989, size = 63, normalized size = 1.85 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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