Optimal. Leaf size=34 \[ \frac {\tan (c+d x)}{a d}-\frac {i \sec ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.11, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3092, 3090, 3767, 8, 2606, 30} \[ \frac {\tan (c+d x)}{a d}-\frac {i \sec ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 3090
Rule 3092
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac {i \int \sec ^3(c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac {i \int \left (i a \sec ^2(c+d x)+a \sec ^2(c+d x) \tan (c+d x)\right ) \, dx}{a^2}\\ &=-\frac {i \int \sec ^2(c+d x) \tan (c+d x) \, dx}{a}+\frac {\int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac {i \operatorname {Subst}(\int x \, dx,x,\sec (c+d x))}{a d}-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=-\frac {i \sec ^2(c+d x)}{2 a d}+\frac {\tan (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 35, normalized size = 1.03 \[ -\frac {i \sec (c+d x) (\sec (c+d x)+2 i \sec (c) \sin (d x))}{2 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 33, normalized size = 0.97 \[ \frac {2 i}{a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 27, normalized size = 0.79 \[ -\frac {i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 26, normalized size = 0.76 \[ \frac {\tan \left (d x +c \right )-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 108, normalized size = 3.18 \[ \frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{{\left (a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 25, normalized size = 0.74 \[ -\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,d} \]
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 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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