Integrand size = 35, antiderivative size = 255 \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {35 i a^{9/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f} \]
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Time = 0.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3604, 49, 52, 65, 223, 209} \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {35 i a^{9/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rule 49
Rule 52
Rule 65
Rule 209
Rule 223
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{7/2}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {\left (7 a^2\right ) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {\left (35 a^3\right ) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 c f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f}+\frac {\left (35 a^4\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 c f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f}+\frac {\left (35 a^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 c f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f}-\frac {\left (35 i a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{c f} \\ & = -\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f}-\frac {\left (35 i a^4\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{c f} \\ & = -\frac {35 i a^{9/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {35 i a^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c^2 f}+\frac {35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 9.65 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.05 \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {i a \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {1}{2} (1+i \tan (e+f x))\right ) \sqrt {1-i \tan (e+f x)} (a+i a \tan (e+f x))^{7/2}}{7 \sqrt {2} c f \sqrt {c-i c \tan (e+f x)}}+\frac {2 a \left (\frac {i a \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {a+i a \tan (e+f x)}{2 a}\right ) (a+i a \tan (e+f x))^{5/2} \sqrt {\frac {c-i c \tan (e+f x)}{c}}}{5 \sqrt {2} c \sqrt {c-i c \tan (e+f x)}}+2 a \left (\frac {2 i a^2 (-i+\tan (e+f x)) \sqrt {i a (-i+\tan (e+f x))}}{3 c (i+\tan (e+f x)) \sqrt {-i c (i+\tan (e+f x))}}+\frac {2 i \sqrt {2} a^3 \sqrt {\frac {c-i c \tan (e+f x)}{c}} \left (-1+\frac {a+i a \tan (e+f x)}{2 a}\right ) \left (\frac {\arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a} \sqrt {1-\frac {a+i a \tan (e+f x)}{2 a}}}+\frac {a+i a \tan (e+f x)}{2 a \left (-1+\frac {a+i a \tan (e+f x)}{2 a}\right )}\right )}{c \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} \sqrt {1-\frac {a+i a \tan (e+f x)}{2 a}}}\right )\right )}{f} \]
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Time = 0.81 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{4} \left (315 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+27 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{3}\left (f x +e \right )\right )+105 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{4}\left (f x +e \right )\right )-105 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -393 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-315 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-259 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+164 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(409\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{4} \left (315 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+27 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{3}\left (f x +e \right )\right )+105 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{4}\left (f x +e \right )\right )-105 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -393 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-315 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-259 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+164 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(409\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (189) = 378\).
Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.69 \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {105 \, \sqrt {\frac {a^{9}}{c^{3} f^{2}}} {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (a^{4} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{4} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {a^{9}}{c^{3} f^{2}}} {\left (i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c^{2} f\right )}\right )}}{a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{4}}\right ) - 105 \, \sqrt {\frac {a^{9}}{c^{3} f^{2}}} {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (a^{4} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{4} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {a^{9}}{c^{3} f^{2}}} {\left (-i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{2} f\right )}\right )}}{a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{4}}\right ) - 4 \, {\left (8 i \, a^{4} e^{\left (7 i \, f x + 7 i \, e\right )} - 56 i \, a^{4} e^{\left (5 i \, f x + 5 i \, e\right )} - 175 i \, a^{4} e^{\left (3 i \, f x + 3 i \, e\right )} - 105 i \, a^{4} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f\right )}} \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (189) = 378\).
Time = 0.44 (sec) , antiderivative size = 915, normalized size of antiderivative = 3.59 \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {9}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{9/2}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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