Optimal. Leaf size=62 \[ \frac {4 \sin (c+d x)}{a^3 d}+\frac {4 i \cos (c+d x)}{a^3 d}+\frac {i \sec (c+d x)}{a^3 d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.16, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3092, 3090, 2637, 2638, 2592, 321, 206, 2590, 14} \[ \frac {4 \sin (c+d x)}{a^3 d}+\frac {4 i \cos (c+d x)}{a^3 d}+\frac {i \sec (c+d x)}{a^3 d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 206
Rule 321
Rule 2590
Rule 2592
Rule 2637
Rule 2638
Rule 3090
Rule 3092
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=\frac {i \int \sec ^2(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {i \int \left (-i a^3 \cos (c+d x)-3 a^3 \sin (c+d x)+3 i a^3 \sin (c+d x) \tan (c+d x)+a^3 \sin (c+d x) \tan ^2(c+d x)\right ) \, dx}{a^6}\\ &=\frac {i \int \sin (c+d x) \tan ^2(c+d x) \, dx}{a^3}-\frac {(3 i) \int \sin (c+d x) \, dx}{a^3}+\frac {\int \cos (c+d x) \, dx}{a^3}-\frac {3 \int \sin (c+d x) \tan (c+d x) \, dx}{a^3}\\ &=\frac {3 i \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x)}{a^3 d}-\frac {i \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=\frac {3 i \cos (c+d x)}{a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {i \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=-\frac {3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac {4 i \cos (c+d x)}{a^3 d}+\frac {i \sec (c+d x)}{a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 109, normalized size = 1.76 \[ -\frac {i \sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 \left ((\tan (c+d x)-5 i) (\cos (2 c-d x)+i \sin (2 c-d x))+6 (\cos (3 c)+i \sin (3 c)) \tanh ^{-1}\left (\cos (c) \tan \left (\frac {d x}{2}\right )+\sin (c)\right )\right )}{a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 112, normalized size = 1.81 \[ -\frac {3 \, {\left (e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3 \, {\left (e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i}{a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 110, normalized size = 1.77 \[ -\frac {\frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} - \frac {2 \, {\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )} a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 108, normalized size = 1.74 \[ -\frac {i}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3} d}+\frac {i}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3} d}+\frac {8}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 329, normalized size = 5.31 \[ \frac {{\left (6 \, \cos \left (3 \, d x + 3 \, c\right ) + 6 \, \cos \left (d x + c\right ) + 6 i \, \sin \left (3 \, d x + 3 \, c\right ) + 6 i \, \sin \left (d x + c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) + {\left (6 \, \cos \left (3 \, d x + 3 \, c\right ) + 6 \, \cos \left (d x + c\right ) + 6 i \, \sin \left (3 \, d x + 3 \, c\right ) + 6 i \, \sin \left (d x + c\right )\right )} \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) - {\left (-3 i \, \cos \left (3 \, d x + 3 \, c\right ) - 3 i \, \cos \left (d x + c\right ) + 3 \, \sin \left (3 \, d x + 3 \, c\right ) + 3 \, \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - {\left (3 i \, \cos \left (3 \, d x + 3 \, c\right ) + 3 i \, \cos \left (d x + c\right ) - 3 \, \sin \left (3 \, d x + 3 \, c\right ) - 3 \, \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 12 \, \cos \left (2 \, d x + 2 \, c\right ) + 12 i \, \sin \left (2 \, d x + 2 \, c\right ) + 8}{{\left (-2 i \, a^{3} \cos \left (3 \, d x + 3 \, c\right ) - 2 i \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} \sin \left (3 \, d x + 3 \, c\right ) + 2 \, a^{3} \sin \left (d x + c\right )\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 105, normalized size = 1.69 \[ -\frac {6\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,8{}\mathrm {i}}{a^3}-\frac {10{}\mathrm {i}}{a^3}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1{}\mathrm {i}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{- i \sin ^{3}{\left (c + d x \right )} - 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} + 3 i \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \cos ^{3}{\left (c + d x \right )}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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