Integrand size = 26, antiderivative size = 201 \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {188 \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d} \]
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Time = 0.59 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3639, 3678, 3673, 3608, 3561, 212} \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {188 \sqrt {a+i a \tan (c+d x)}}{35 a d} \]
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Rule 212
Rule 3561
Rule 3608
Rule 3639
Rule 3673
Rule 3678
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\int \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-4 a+\frac {9}{2} i a \tan (c+d x)\right ) \, dx}{a^2} \\ & = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}-\frac {2 \int \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {27 i a^2}{2}-\frac {47}{4} a^2 \tan (c+d x)\right ) \, dx}{7 a^3} \\ & = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}-\frac {4 \int \tan (c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {47 a^3}{2}-\frac {223}{8} i a^3 \tan (c+d x)\right ) \, dx}{35 a^4} \\ & = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d}-\frac {4 \int \sqrt {a+i a \tan (c+d x)} \left (\frac {223 i a^3}{8}+\frac {47}{2} a^3 \tan (c+d x)\right ) \, dx}{35 a^4} \\ & = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {188 \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d}-\frac {i \int \sqrt {a+i a \tan (c+d x)} \, dx}{2 a} \\ & = -\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {188 \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d}-\frac {\text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {\tan ^4(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {188 \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {47 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {9 i \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 a d}+\frac {223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {-341-118 i \tan (c+d x)-82 \tan ^2(c+d x)+6 i \tan ^3(c+d x)+30 \tan ^4(c+d x)}{105 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.78 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {6 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {8 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-\frac {a^{4}}{\sqrt {a +i a \tan \left (d x +c \right )}}-\frac {a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2}}{a^{4} d}\) | \(131\) |
default | \(\frac {\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {6 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {8 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-\frac {a^{4}}{\sqrt {a +i a \tan \left (d x +c \right )}}-\frac {a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2}}{a^{4} d}\) | \(131\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (160) = 320\).
Time = 0.25 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.94 \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {105 \, \sqrt {2} {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} + 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {1}{a d^{2}}} \log \left (4 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 105 \, \sqrt {2} {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} + 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {1}{a d^{2}}} \log \left (-4 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (353 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 1708 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2030 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1260 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105\right )}}{420 \, {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} + 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )}} \]
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\[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {105 \, \sqrt {2} a^{\frac {11}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + 120 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 504 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3} + 1120 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4} - 1680 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{5} - \frac {420 \, a^{6}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}}{420 \, a^{6} d} \]
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\[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{5}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Time = 0.48 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72 \[ \int \frac {\tan ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {1}{d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}-\frac {4\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{a\,d}+\frac {8\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a^2\,d}-\frac {6\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,a^3\,d}+\frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{7\,a^4\,d}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,1{}\mathrm {i}}{2\,\sqrt {a}\,d} \]
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