Integrand size = 26, antiderivative size = 292 \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {429 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{1024 \sqrt {2} \sqrt {a} d}+\frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i}{1024 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53, 65, 212} \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}-\frac {429 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{1024 \sqrt {2} \sqrt {a} d}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i}{1024 d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 44
Rule 53
Rule 65
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {\left (13 i a^6\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{12 d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {\left (143 i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{96 d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {\left (429 i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{128 d} \\ & = \frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {\left (429 i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{7/2}} \, dx,x,i a \tan (c+d x)\right )}{256 d} \\ & = \frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}-\frac {\left (429 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{512 d} \\ & = \frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}-\frac {(429 i a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{1024 d} \\ & = \frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i}{1024 d \sqrt {a+i a \tan (c+d x)}}-\frac {(429 i) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{2048 d} \\ & = \frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i}{1024 d \sqrt {a+i a \tan (c+d x)}}-\frac {(429 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{1024 d} \\ & = -\frac {429 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{1024 \sqrt {2} \sqrt {a} d}+\frac {429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac {13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac {143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i}{1024 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 = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.18 \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i a^3 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},4,-\frac {5}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{56 d (a+i a \tan (c+d x))^{7/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (234 ) = 468\).
Time = 9.44 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.31
method | result | size |
default | \(-\frac {-90090 i \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-45045 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 )-33280 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+12012 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}}-33280 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-90090 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-41184 \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}}+4576 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}}-41184 \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}}+4576 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}}-60060 \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}}+2560 i \left (\cos ^{7}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+2560 i \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-60060 \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 )-45045 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 )+12012 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}}+45045 \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 )}{215040 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 )}}\) | \(674\) |
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Time = 0.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {{\left (-45045 i \, \sqrt {\frac {1}{2}} a d \sqrt {\frac {1}{a d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 45045 i \, \sqrt {\frac {1}{2}} a d \sqrt {\frac {1}{a d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-280 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 2870 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 16345 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 27029 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 49792 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 11072 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2304 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 240 i\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{215040 \, a d} \]
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\[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\cos ^{6}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i \, {\left (45045 \, \sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (45045 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6} a - 240240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} a^{2} + 396396 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a^{3} - 164736 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{4} - 36608 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5} - 19968 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{6} - 15360 \, a^{7}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3}}\right )}}{430080 \, a d} \]
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\[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{6}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^6}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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