Integrand size = 26, antiderivative size = 137 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))} \]
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Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 43, 44, 65, 212} \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))} \]
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Rule 43
Rule 44
Rule 65
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {\sqrt {a+x}}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{4 d} \\ & = -\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac {\left (i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{16 d} \\ & = -\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac {\left (i a^4\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{8 d} \\ & = \frac {i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {i a^5 \sqrt {a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{3/2}}{12 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (110 ) = 220\).
Time = 123.98 (sec) , antiderivative size = 869, normalized size of antiderivative = 6.34
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (102) = 204\).
Time = 0.24 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.91 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {\sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d \log \left (\frac {4 \, {\left (a^{4} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3}}\right ) - \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d \log \left (\frac {4 \, {\left (a^{4} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3}}\right ) + \sqrt {2} {\left (-2 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 3 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{16 \, d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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none
Time = 0.47 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i \, {\left (\sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} + 2 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{6}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 4 \, a^{2}}\right )}}{32 \, a d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
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