Optimal. Leaf size=257 \[ \frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{5 (-1)^{3/4} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.862019, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3558, 3595, 3597, 3601, 3544, 205, 3599, 63, 217, 203} \[ \frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{5 (-1)^{3/4} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3558
Rule 3595
Rule 3597
Rule 3601
Rule 3544
Rule 205
Rule 3599
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{9}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac{\int \frac{\tan ^{\frac{5}{2}}(c+d x) \left (-\frac{7 a}{2}+6 i a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\int \frac{\tan ^{\frac{3}{2}}(c+d x) \left (-\frac{95 i a^2}{4}-\frac{55}{2} a^2 \tan (c+d x)\right )}{\sqrt{a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\int \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \left (\frac{615 a^3}{8}-\frac{315}{4} i a^3 \tan (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{\int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{315 i a^4}{8}+\frac{75}{2} a^4 \tan (c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{15 a^7}\\ &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{(5 i) \int \frac{(a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{2 a^4}-\frac{i \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{8 a^3}\\ &=-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}-\frac{\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 )}{4 a d}\\ &=-\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a^2 d}\\ &=-\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{1-i a x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^2 d}\\ &=\frac{5 (-1)^{3/4} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}\\ \end{align*}
Mathematica [A] time = 3.77506, size = 270, normalized size = 1.05 \[ -\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\tan (c+d x)} \left (-\sqrt{-1+e^{2 i (c+d x)}} \left (-28 e^{2 i (c+d x)}+252 e^{4 i (c+d x)}+403 e^{6 i (c+d x)}+3\right )+15 e^{5 i (c+d x)} \left (1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )+300 \sqrt{2} e^{5 i (c+d x)} \left (1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )\right )}{60 \sqrt{2} a^3 d \sqrt{-1+e^{2 i (c+d x)}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 1002, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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