Integrand size = 43, antiderivative size = 105 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=\frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{7/2}}{7 c^2 f} \]
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Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3669, 78} \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=-\frac {2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {4 a^2 (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{7/2}}{7 c^2 f} \]
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Rule 78
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int (a+i a x) (A+B x) \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (2 a (A-i B) \sqrt {c-i c x}-\frac {a (A-3 i B) (c-i c x)^{3/2}}{c}-\frac {i a B (c-i c x)^{5/2}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{7/2}}{7 c^2 f} \\ \end{align*}
Time = 4.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \sec ^4(e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x))) (7 (7 i A+B)+(49 i A+37 B) \cos (2 (e+f x))+(-21 A+33 i B) \sin (2 (e+f x)))}{105 f \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {2 i a^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (3 i B c -c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (-i B c +c A \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f \,c^{2}}\) | \(84\) |
default | \(\frac {2 i a^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (3 i B c -c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (-i B c +c A \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f \,c^{2}}\) | \(84\) |
parts | \(\frac {2 i A \,a^{2} c \left (-\sqrt {c -i c \tan \left (f x +e \right )}+\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {a^{2} \left (2 i A +B \right ) \left (\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 c \sqrt {c -i c \tan \left (f x +e \right )}-2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}-\frac {2 B \,a^{2} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{2}}{3}-\sqrt {c -i c \tan \left (f x +e \right )}\, c^{3}+c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f \,c^{2}}-\frac {2 i a^{2} \left (-2 i B +A \right ) \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\sqrt {c -i c \tan \left (f x +e \right )}\, c^{2}-c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f c}\) | \(336\) |
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.13 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=-\frac {8 \, \sqrt {2} {\left (35 \, {\left (-i \, A - B\right )} a^{2} c e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, {\left (-7 i \, A - B\right )} a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (-7 i \, A - B\right )} a^{2} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{105 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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\[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=- a^{2} \left (\int \left (- A c \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- A c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- i A c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- i A c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- i B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- i B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=-\frac {2 i \, {\left (15 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} B a^{2} + 21 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - 3 i \, B\right )} a^{2} c - 70 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (A - i \, B\right )} a^{2} c^{2}\right )}}{105 \, c^{2} f} \]
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\[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Time = 11.70 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx=\frac {8\,a^2\,c\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left (A\,14{}\mathrm {i}+2\,B+A\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,49{}\mathrm {i}+A\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,35{}\mathrm {i}+7\,B\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+35\,B\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}{105\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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