Integrand size = 26, antiderivative size = 172 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {28 i \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d} \]
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Time = 0.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3639, 3678, 3673, 3608, 3561, 212} \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}+\frac {28 i \sqrt {a+i a \tan (c+d x)}}{5 a d} \]
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Rule 212
Rule 3561
Rule 3608
Rule 3639
Rule 3673
Rule 3678
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\int \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-3 a+\frac {7}{2} i a \tan (c+d x)\right ) \, dx}{a^2} \\ & = -\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {2 \int \tan (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-7 i a^2-\frac {23}{4} a^2 \tan (c+d x)\right ) \, dx}{5 a^3} \\ & = -\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}-\frac {2 \int \sqrt {a+i a \tan (c+d x)} \left (\frac {23 a^2}{4}-7 i a^2 \tan (c+d x)\right ) \, dx}{5 a^3} \\ & = -\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {28 i \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}+\frac {\int \sqrt {a+i a \tan (c+d x)} \, dx}{2 a} \\ & = -\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {28 i \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}-\frac {i \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d} \\ & = -\frac {i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^3(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {28 i \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {7 i \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 a d}-\frac {23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {61 i-38 \tan (c+d x)+2 i \tan ^2(c+d x)+6 \tan ^3(c+d x)}{15 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.80 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}+\frac {a^{3}}{2 \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4}\right )}{d \,a^{3}}\) | \(113\) |
default | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}+\frac {a^{3}}{2 \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4}\right )}{d \,a^{3}}\) | \(113\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (131) = 262\).
Time = 0.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.99 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} {\left (i \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 i \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + i \, a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {1}{a d^{2}}} \log \left (4 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 15 \, \sqrt {2} {\left (-i \, a d e^{\left (5 i \, d x + 5 i \, c\right )} - 2 i \, a d e^{\left (3 i \, d x + 3 i \, c\right )} - i \, a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {1}{a d^{2}}} \log \left (-4 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-103 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 205 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 165 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i\right )}}{60 \, {\left (a d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )}} \]
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\[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\tan ^{4}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i \, {\left (15 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + 24 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 80 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 240 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4} + \frac {60 \, a^{5}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}}{60 \, a^{5} d} \]
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\[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{4}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Time = 4.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {1{}\mathrm {i}}{d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}+\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{a\,d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,4{}\mathrm {i}}{3\,a^2\,d}+\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,2{}\mathrm {i}}{5\,a^3\,d}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,1{}\mathrm {i}}{2\,\sqrt {-a}\,d} \]
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