Integrand size = 45, antiderivative size = 217 \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt {a} (3 i A-2 B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}-\frac {(3 i A-2 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f} \]
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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} \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt {a} c^{5/2} (-2 B+3 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}-\frac {c^2 (-2 B+3 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {c (-2 B+3 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f} \]
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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+B x) (c-i c x)^{3/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}+\frac {(a (3 A+2 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 )}{3 f} \\ & = -\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}+\frac {\left (a (3 A+2 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 i A-2 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}+\frac {\left (a (3 A+2 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 i A-2 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}-\frac {\left ((3 i A-2 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 )}{f} \\ & = -\frac {(3 i A-2 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}-\frac {\left ((3 i A-2 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 )}{f} \\ & = -\frac {\sqrt {a} (3 i A-2 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 )}{f}-\frac {(3 i A-2 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(3 i A-2 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f} \\ \end{align*}
Time = 5.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.69 \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {6 \sqrt {a} (-3 i A+2 B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )-c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} \left (12 i A-10 B+3 (A+2 i B) \tan (e+f x)+2 B \tan ^2(e+f x)\right )}{6 f} \]
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Time = 0.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +3 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\) | \(285\) |
default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +3 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\) | \(285\) |
parts | \(-\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} \left (4 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+\tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-3 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )\right )}{2 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}+\frac {B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} \left (3 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -3 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-\tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+5 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{3 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}\) | \(340\) |
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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. \(2 (164) = 328\).
Time = 0.27 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.52 \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {3 \, \sqrt {\frac {{\left (9 \, A^{2} + 12 i \, A B - 4 \, B^{2}\right )} a c^{5}}{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 )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (3 i \, A - 2 \, 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 (9 \, A^{2} + 12 i \, A B - 4 \, B^{2}\right )} a c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-3 i \, A + 2 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A + 2 \, B\right )} c^{2}}\right ) - 3 \, \sqrt {\frac {{\left (9 \, A^{2} + 12 i \, A B - 4 \, B^{2}\right )} a c^{5}}{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 )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (3 i \, A - 2 \, 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 (9 \, A^{2} + 12 i \, A B - 4 \, B^{2}\right )} a c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-3 i \, A + 2 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A + 2 \, B\right )} c^{2}}\right ) - 4 \, {\left (3 \, {\left (3 i \, A - 2 \, B\right )} c^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 8 \, {\left (3 i \, A - 2 \, B\right )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (5 i \, A - 6 \, 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}}}{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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\[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int \sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (164) = 328\).
Time = 0.85 (sec) , antiderivative size = 1085, normalized size of antiderivative = 5.00 \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]
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\[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} \sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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