Integrand size = 26, antiderivative size = 204 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {9 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac {9 i}{32 a^2 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.17 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53, 65, 212} \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {9 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{32 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {\left (9 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{4 d} \\ & = \frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}-\frac {(9 i a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{7/2}} \, dx,x,i a \tan (c+d x)\right )}{8 d} \\ & = \frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}-\frac {(9 i) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{16 d} \\ & = \frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}-\frac {(9 i) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{32 a d} \\ & = \frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac {9 i}{32 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {(9 i) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{64 a^2 d} \\ & = \frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac {9 i}{32 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {(9 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{32 a^2 d} \\ & = -\frac {9 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac {9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac {9 i}{32 a^2 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 = 51, normalized size of antiderivative = 0.25 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {i a \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},2,-\frac {5}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{14 d (a+i a \tan (c+d x))^{7/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (160 ) = 320\).
Time = 10.47 (sec) , antiderivative size = 780, normalized size of antiderivative = 3.82
method | result | size |
default | \(\frac {2184 i \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+720 \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}}+630 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 )+720 \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 )-400 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}}+2184 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-1260 \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 )-1680 \sin \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-630 i \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-945 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 )-630 \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 )-1680 \tan \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-630 i \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+1260 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 )+315 \tan \left (d x +c \right ) \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 )-400 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}}-315 i \left (\sec ^{2}\left (d x +c \right )\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 )}{2240 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 )^{2} a^{2}}\) | \(780\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (147) = 294\).
Time = 0.25 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {{\left (-315 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 315 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-35 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 353 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 544 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 214 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 68 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{2240 \, a^{3} d} \]
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\[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.44 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {i \, {\left (\frac {4 \, {\left (315 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} - 420 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a - 168 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{2} - 144 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{3} - 160 \, a^{4}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}} + \frac {315 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}}\right )}}{4480 \, a d} \]
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\[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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