Optimal. Leaf size=174 \[ \frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.353426, antiderivative size = 174, 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, 3527, 3480, 206} \[ \frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3558
Rule 3595
Rule 3592
Rule 3527
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}-\frac{\int \frac{\tan ^2(c+d x) \left (-3 a+\frac{9}{2} i a \tan (c+d x)\right )}{\sqrt{a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\int \tan (c+d x) \sqrt{a+i a \tan (c+d x)} \left (-15 i a^2-\frac{63}{4} a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}+\frac{\int \sqrt{a+i a \tan (c+d x)} \left (\frac{63 a^2}{4}-15 i a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}+\frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}+\frac{\int \sqrt{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}+\frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 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 )}{2 a d}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}+\frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.18564, size = 165, normalized size = 0.95 \[ -\frac{i e^{-4 i (c+d x)} \sec ^2(c+d x) \left (\left (18 e^{2 i (c+d x)}+87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}-1\right ) \sqrt{1+e^{2 i (c+d x)}}+3 e^{3 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^2 \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{24 a d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 113, normalized size = 0.7 \begin{align*}{\frac{2\,i}{d{a}^{3}} \left ({\frac{1}{3} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-2\,a\sqrt{a+ia\tan \left ( dx+c \right ) }-{\frac{\sqrt{2}}{8}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ) }-{\frac{7\,{a}^{2}}{4}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{{a}^{3}}{6} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{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 = 2.24789, size = 986, normalized size = 5.67 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-52 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 87 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (i \, d x + i \, c\right )} + \sqrt{\frac{1}{2}}{\left (-3 i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt{\frac{1}{a^{3} d^{2}}} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} 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{\frac{1}{2}}{\left (3 i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt{\frac{1}{a^{3} d^{2}}} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} 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 )}{12 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \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{3}{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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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