Optimal. Leaf size=240 \[ -\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(-25 B+39 i A) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) (B+i A) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.769485, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {3596, 3598, 12, 3544, 205} \[ -\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(-25 B+39 i A) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) (B+i A) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3596
Rule 3598
Rule 12
Rule 3544
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{\int \frac{\frac{3}{2} a (3 A+i B)-3 a (i A-B) \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{\int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{3}{4} a^2 (21 A+11 i B)-3 a^2 (5 i A-3 B) \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{3 a^4}\\ &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{3}{8} a^3 (39 i A-25 B)-\frac{3}{4} a^3 (21 A+11 i B) \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{9 a^5}\\ &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(39 i A-25 B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{4 \int -\frac{9 a^4 (A-i B) \sqrt{a+i a \tan (c+d x)}}{16 \sqrt{\tan (c+d x)}} \, dx}{9 a^6}\\ &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(39 i A-25 B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}-\frac{(A-i B) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{4 a^2}\\ &=\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(39 i A-25 B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{(i A+B) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{2 d}\\ &=\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(39 i A-25 B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.64076, size = 244, normalized size = 1.02 \[ \frac{i e^{-2 i (c+d x)} \sec ^2(c+d x) \left (-3 i (A-i B) e^{3 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right )^{3/2} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )-i A \left (18 e^{2 i (c+d x)}-87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}+1\right )+B \left (12 e^{2 i (c+d x)}-51 e^{4 i (c+d x)}+38 e^{6 i (c+d x)}+1\right )\right )}{12 a d \left (-1+e^{2 i (c+d x)}\right ) \sqrt{\tan (c+d x)} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 1012, normalized size = 4.2 \begin{align*} \text{result too large to display} \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 = 2.22245, size = 1656, normalized size = 6.9 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47054, size = 261, normalized size = 1.09 \begin{align*} \frac{\left (i + 1\right ) \, \sqrt{-2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} +{\left (-\left (2 i - 2\right ) \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{3} + \left (2 i - 2\right ) \, a^{4}\right )} \sqrt{-2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}} \sqrt{i \, a \tan \left (d x + c\right ) + a} B}{2 \,{\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6} a - 5 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} a^{2} + 9 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a^{3} - 7 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{4} + 2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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