Integrand size = 26, antiderivative size = 248 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {99 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53, 65, 212} \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {99 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 44
Rule 53
Rule 65
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {\left (11 i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{8 d} \\ & = -\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {\left (99 i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{32 d} \\ & = \frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {\left (99 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{7/2}} \, dx,x,i a \tan (c+d x)\right )}{64 d} \\ & = \frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}-\frac {(99 i a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{128 d} \\ & = \frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {(99 i) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{256 d} \\ & = \frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(99 i) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{512 a d} \\ & = \frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(99 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{256 a d} \\ & = -\frac {99 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {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 = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.21 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {i a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},3,-\frac {5}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{28 d (a+i a \tan (c+d x))^{7/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (197 ) = 394\).
Time = 8.62 (sec) , antiderivative size = 729, normalized size of antiderivative = 2.94
method | result | size |
default | \(-\frac {-11550 i \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-3465 i \arctan \left (\frac {\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-3520 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+3465 i \sec \left (d x +c \right ) \arctan \left (\frac {\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-3520 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+2376 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-5544 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-11550 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+960 i \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-5544 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+960 i \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-6930 i \cos \left (d x +c \right ) \arctan \left (\frac {\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+6930 \sin \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+6930 \arctan \left (\frac {\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )+2376 i \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+6930 \tan \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+3465 \tan \left (d x +c \right ) \arctan \left (\frac {\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )}{17920 d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1+i \tan \left (d x +c \right )\right ) a}\) | \(729\) |
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Time = 0.25 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {{\left (-3465 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3465 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-70 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 805 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 2833 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 4584 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1304 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 328 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 40 i\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{17920 \, a^{2} d} \]
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\[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {i \, {\left (\frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3465 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} - 11550 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a + 7392 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 2112 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} + 1408 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + 1280 \, a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}}\right )}}{35840 \, a d} \]
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\[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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