Integrand size = 45, antiderivative size = 217 \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {a^{5/2} (3 i A+2 B) \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f} \]
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Time = 0.33 (sec) , antiderivative size = 217, 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, 81, 52, 65, 223, 209} \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {a^{5/2} \sqrt {c} (2 B+3 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a^2 (2 B+3 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (2 B+3 i A) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f} \]
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Rule 52
Rule 65
Rule 81
Rule 209
Rule 223
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{3/2} (A+B x)}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f}+\frac {(a (3 A-2 i B) c) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 f} \\ & = \frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f}+\frac {\left (a^2 (3 A-2 i B) c\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f}+\frac {\left (a^3 (3 A-2 i B) c\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 {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f}-\frac {\left (a^2 (3 i A+2 B) c\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 )}{f} \\ & = \frac {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f}-\frac {\left (a^2 (3 i A+2 B) c\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 )}{f} \\ & = -\frac {a^{5/2} (3 i A+2 B) \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f} \\ \end{align*}
Time = 5.74 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.91 \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {a^{5/2} c (i+\tan (e+f x)) \left (6 (3 A-2 i B) \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}+\sqrt {a} \sqrt {1-i \tan (e+f x)} (-i+\tan (e+f x)) \left (-12 i A-10 B+3 (A-2 i B) \tan (e+f x)+2 B \tan ^2(e+f x)\right )\right )}{6 f \sqrt {1-i \tan (e+f x)} \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.34 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -3 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+10 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\) | \(285\) |
default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -3 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+10 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\) | \(285\) |
parts | \(\frac {A \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (4 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-\tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+3 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )\right )}{2 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}-\frac {B \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (3 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -3 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-5 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{3 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}\) | \(340\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (165) = 330\).
Time = 0.28 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.52 \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {3 \, \sqrt {\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{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 (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (3 i \, A + 2 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (3 i \, A + 2 \, B\right )} a^{2}}\right ) - 3 \, \sqrt {\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{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 (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (3 i \, A + 2 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (3 i \, A + 2 \, B\right )} a^{2}}\right ) + 4 \, {\left (3 \, {\left (-5 i \, A - 6 \, B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 8 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{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}}}{12 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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\[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (165) = 330\).
Time = 0.58 (sec) , antiderivative size = 1087, normalized size of antiderivative = 5.01 \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\text {Too large to display} \]
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\[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
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