Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d} \]
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Rubi [A] time = 0.0943114, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3092, 3090, 3770, 2606, 8} \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3092
Rule 3090
Rule 3770
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac{i \int \sec ^2(c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac{i \int (i a \sec (c+d x)+a \sec (c+d x) \tan (c+d x)) \, dx}{a^2}\\ &=-\frac{i \int \sec (c+d x) \tan (c+d x) \, dx}{a}+\frac{\int \sec (c+d x) \, dx}{a}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{a d}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.193977, size = 35, normalized size = 1.13 \[ -\frac{i \left (\sec (c+d x)+2 i \tanh ^{-1}\left (\cos (c) \tan \left (\frac{d x}{2}\right )+\sin (c)\right )\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.145, size = 85, normalized size = 2.7 \begin{align*}{\frac{-i}{ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{i}{ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17606, size = 112, normalized size = 3.61 \begin{align*} \frac{\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac{2}{-i \, a + \frac{i \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.474333, size = 217, normalized size = 7. \begin{align*} \frac{{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) -{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 2 i \, e^{\left (i \, d x + i \, c\right )}}{a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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 [B] time = 1.15208, size = 81, normalized size = 2.61 \begin{align*} \frac{\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac{2 i}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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