Optimal. Leaf size=70 \[ -\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {i \tan ^4(c+d x)}{2 a^2 d}-\frac {i \tan ^2(c+d x)}{a^2 d}+\frac {\tan (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3088, 848, 75} \[ -\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {i \tan ^4(c+d x)}{2 a^2 d}-\frac {i \tan ^2(c+d x)}{a^2 d}+\frac {\tan (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 75
Rule 848
Rule 3088
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6 (i a+a x)^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {i}{a}+\frac {x}{a}\right )^3 (i a+a x)}{x^6} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{a^2 x^6}-\frac {2 i}{a^2 x^5}-\frac {2 i}{a^2 x^3}+\frac {1}{a^2 x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {\tan (c+d x)}{a^2 d}-\frac {i \tan ^2(c+d x)}{a^2 d}-\frac {i \tan ^4(c+d x)}{2 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 77, normalized size = 1.10 \[ \frac {\sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)-5 i \cos (2 c+d x)+5 \sin (d x)-5 i \cos (d x))}{20 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 97, normalized size = 1.39 \[ \frac {40 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i}{5 \, {\left (a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 47, normalized size = 0.67 \[ -\frac {2 \, \tan \left (d x + c\right )^{5} + 5 i \, \tan \left (d x + c\right )^{4} + 10 i \, \tan \left (d x + c\right )^{2} - 10 \, \tan \left (d x + c\right )}{10 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 47, normalized size = 0.67 \[ \frac {\tan \left (d x +c \right )-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{2}-i \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 47, normalized size = 0.67 \[ -\frac {6 \, \tan \left (d x + c\right )^{5} + 15 i \, \tan \left (d x + c\right )^{4} + 30 i \, \tan \left (d x + c\right )^{2} - 30 \, \tan \left (d x + c\right )}{30 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 76, normalized size = 1.09 \[ -\frac {\sin \left (c+d\,x\right )\,\left (-4\,{\cos \left (c+d\,x\right )}^4+\frac {5{}\mathrm {i}\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3}{2}-2\,{\cos \left (c+d\,x\right )}^2+\frac {5{}\mathrm {i}\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )}{2}+1\right )}{5\,a^2\,d\,{\cos \left (c+d\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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