Integrand size = 45, antiderivative size = 228 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {3 (2 i A-3 B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a} f}+\frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}} \]
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Time = 0.35 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 79, 52, 65, 223, 209} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {3 c^{5/2} (-3 B+2 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a} f}+\frac {3 c^2 (-3 B+2 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {c (-3 B+2 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}} \]
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Rule 52
Rule 65
Rule 79
Rule 209
Rule 223
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{3/2}}{(a+i a x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {((2 A+3 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {\left (3 (2 A+3 i B) c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {\left (3 (2 A+3 i B) c^3\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 f} \\ & = \frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}+\frac {\left (3 (2 i A-3 B) c^3\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 )}{a f} \\ & = \frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}+\frac {\left (3 (2 i A-3 B) c^3\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 )}{a f} \\ & = \frac {3 (2 i A-3 B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a} f}+\frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}} \\ \end{align*}
Time = 7.74 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {c^2 \sqrt {c-i c \tan (e+f x)} \left (-((2 A+3 i B) (i+\tan (e+f x)))-B (i+\tan (e+f x))^2+6 (2 i A-3 B) \left (1+\frac {\arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {1-i \tan (e+f x)}}\right )\right )}{2 f \sqrt {a+i a \tan (e+f x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (188 ) = 376\).
Time = 0.39 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.48
method | result | size |
derivativedivides | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (6 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+18 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+4 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-9 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-6 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+12 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+2 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-14 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +19 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(566\) |
default | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (6 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+18 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+4 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-9 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-6 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+12 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+2 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-14 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +19 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(566\) |
parts | \(\frac {i A \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (3 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c -3 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +6 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c -6 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-5 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}-\frac {B \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (9 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c -i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-9 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +18 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c -19 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+4 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-14 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(633\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (176) = 352\).
Time = 0.28 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.40 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=-\frac {3 \, \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2} 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 {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )}\right )}}{{\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2}}\right ) - 3 \, \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2} 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 {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )}\right )}}{{\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2}}\right ) + 4 \, {\left (3 \, {\left (-2 i \, A + 3 \, B\right )} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, {\left (-2 i \, A + 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, {\left (-i \, A + B\right )} c^{2}\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}}}{4 \, {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )}} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1319 vs. \(2 (176) = 352\).
Time = 0.66 (sec) , antiderivative size = 1319, normalized size of antiderivative = 5.79 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {i \, a \tan \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]
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