Optimal. Leaf size=234 \[ -\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) (B+(2+i) A) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) (B+(2+i) A) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}-\frac{((3+i) A-(1+i) B) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{8 \sqrt{2} a d} \]
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Rubi [A] time = 0.2913, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) (B+(2+i) A) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) (B+(2+i) A) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}-\frac{((3+i) A-(1+i) B) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{8 \sqrt{2} a d} \]
Antiderivative was successfully verified.
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Rule 3596
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))} \, dx &=\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+\frac{\int \frac{\frac{1}{2} a (3 A-i B)-\frac{1}{2} a (i A-B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{2 a^2}\\ &=\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} a (3 A-i B)-\frac{1}{2} a (i A-B) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a^2 d}\\ &=\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+\frac{((3+i) A-(1+i) B) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{4 a d}+\frac{\left (\left (\frac{1}{4}-\frac{i}{4}\right ) ((2+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d}\\ &=\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}-\frac{((3+i) A-(1+i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{8 \sqrt{2} a d}-\frac{((3+i) A-(1+i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{8 \sqrt{2} a d}+\frac{\left (\left (\frac{1}{8}-\frac{i}{8}\right ) ((2+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d}+\frac{\left (\left (\frac{1}{8}-\frac{i}{8}\right ) ((2+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d}\\ &=-\frac{((3+i) A-(1+i) B) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{8 \sqrt{2} a d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+-\frac{\left (\left (\frac{1}{4}-\frac{i}{4}\right ) ((2+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}+\frac{\left (\left (\frac{1}{4}-\frac{i}{4}\right ) ((2+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}\\ &=-\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) ((2+i) A+B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) ((2+i) A+B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a d}-\frac{((3+i) A-(1+i) B) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{8 \sqrt{2} a d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.98748, size = 199, normalized size = 0.85 \[ \frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left (4 (A+i B) \sin (c+d x) (\cos (d x)-i \sin (d x))+(1+i) (-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left ((B+(2+i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(i B-(1+2 i) A) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 192, normalized size = 0.8 \begin{align*}{\frac{B}{2\,ad \left ( \tan \left ( dx+c \right ) -i \right ) }\sqrt{\tan \left ( dx+c \right ) }}-{\frac{{\frac{i}{2}}A}{ad \left ( \tan \left ( dx+c \right ) -i \right ) }\sqrt{\tan \left ( dx+c \right ) }}-{\frac{2\,iA}{ad \left ( \sqrt{2}-i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}-i\sqrt{2}}} \right ) }+{\frac{iA}{ad \left ( \sqrt{2}+i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}+i\sqrt{2}}} \right ) }+{\frac{B}{ad \left ( \sqrt{2}+i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}+i\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83499, size = 1513, normalized size = 6.47 \begin{align*} \text{result too large to display} \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 = 1.22894, size = 132, normalized size = 0.56 \begin{align*} -\frac{\left (i - 1\right ) \, \sqrt{2} A \arctan \left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\tan \left (d x + c\right )}\right )}{2 \, a d} - \frac{\left (i - 1\right ) \, \sqrt{2}{\left (i \, A + B\right )} \arctan \left (-\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\tan \left (d x + c\right )}\right )}{4 \, a d} - \frac{i \, A \sqrt{\tan \left (d x + c\right )} - B \sqrt{\tan \left (d x + c\right )}}{2 \, a d{\left (\tan \left (d x + c\right ) - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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