Optimal. Leaf size=176 \[ \frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.370926, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3558, 3595, 3592, 3526, 3480, 206} \[ \frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3558
Rule 3595
Rule 3592
Rule 3526
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac{\int \frac{\tan ^2(c+d x) \left (-3 a+\frac{11}{2} i a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\int \frac{\tan (c+d x) \left (-17 i a^2-\frac{83}{4} a^2 \tan (c+d x)\right )}{\sqrt{a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{\int \frac{\frac{83 a^2}{4}-17 i a^2 \tan (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{\int \sqrt{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}-\frac{i \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.51176, size = 149, normalized size = 0.85 \[ \frac{e^{-6 i (c+d x)} \left (i \left (1+e^{2 i (c+d x)}\right ) \left (-26 e^{2 i (c+d x)}+194 e^{4 i (c+d x)}+463 e^{6 i (c+d x)}+3\right ) \sec ^2(c+d x)-\frac{60 i e^{7 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}\right )}{240 a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 111, normalized size = 0.6 \begin{align*}{\frac{2\,i}{d{a}^{3}} \left ( \sqrt{a+ia\tan \left ( dx+c \right ) }-{\frac{\sqrt{2}}{16}\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ) }-{\frac{7\,{a}^{2}}{12} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{17\,a}{8}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{{a}^{3}}{10} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23203, size = 865, normalized size = 4.91 \begin{align*} \frac{{\left (-15 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 15 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (463 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 194 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 26 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{120 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{4}{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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