Integrand size = 26, antiderivative size = 193 \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {35 i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a}{64 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.15 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53, 65, 212} \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {35 i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {35 i a}{64 d \sqrt {a+i a \tan (c+d x)}} \]
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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)^{5/2}} \, 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))^{3/2}}-\frac {\left (7 i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{5/2}} \, 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))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}-\frac {\left (35 i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{32 d} \\ & = \frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}-\frac {\left (35 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{64 d} \\ & = \frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a}{64 d \sqrt {a+i a \tan (c+d x)}}-\frac {(35 i a) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{128 d} \\ & = \frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a}{64 d \sqrt {a+i a \tan (c+d x)}}-\frac {(35 i a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{64 d} \\ & = -\frac {35 i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a}{64 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 = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.27 \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {i a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},3,-\frac {1}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{12 d (a+i a \tan (c+d x))^{3/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (154 ) = 308\).
Time = 109.91 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.13
method | result | size |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (112 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+105 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+105 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sin \left (d x +c \right )-16 \left (\cos ^{4}\left (d x +c \right )\right )+105 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+210 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+105 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )-105 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right )-105 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-70 \left (\cos ^{2}\left (d x +c \right )\right )\right )}{384 d}\) | \(411\) |
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Time = 0.25 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.42 \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {{\left (105 \, \sqrt {\frac {1}{2}} d \sqrt {-\frac {a}{d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {a}{d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 105 \, \sqrt {\frac {1}{2}} d \sqrt {-\frac {a}{d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {a}{d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-6 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 45 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 41 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 88 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{384 \, d} \]
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\[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \cos ^{4}{\left (c + d x \right )}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.91 \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {i \, {\left (105 \, \sqrt {2} a^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} - 350 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} + 224 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + 64 \, a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2}}\right )}}{768 \, a d} \]
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\[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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