Integrand size = 43, antiderivative size = 140 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {8 a^3 (i A+B)}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {8 a^3 (i A+2 B)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^3 (i A+5 B)}{c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f} \]
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Time = 0.24 (sec) , antiderivative size = 140, 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} \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 a^3 (5 B+i A)}{c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f} \]
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Rule 78
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^2 (A+B x)}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (\frac {4 a^2 (A-i B)}{(c-i c x)^{7/2}}-\frac {4 a^2 (A-2 i B)}{c (c-i c x)^{5/2}}+\frac {a^2 (A-5 i B)}{c^2 (c-i c x)^{3/2}}+\frac {i a^2 B}{c^3 \sqrt {c-i c x}}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {8 a^3 (i A+B)}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {8 a^3 (i A+2 B)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^3 (i A+5 B)}{c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f} \\ \end{align*}
Time = 6.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {2 a^3 \left (7 i A+62 B+5 (2 A-31 i B) \tan (e+f x)-15 i (A-8 i B) \tan ^2(e+f x)+15 i B \tan ^3(e+f x)\right )}{15 c^2 f (i+\tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (i \sqrt {c -i c \tan \left (f x +e \right )}\, B +\frac {4 c^{2} \left (-2 i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {c \left (-5 i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {4 c^{3} \left (-i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f \,c^{3}}\) | \(105\) |
default | \(\frac {2 i a^{3} \left (i \sqrt {c -i c \tan \left (f x +e \right )}\, B +\frac {4 c^{2} \left (-2 i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {c \left (-5 i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {4 c^{3} \left (-i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f \,c^{3}}\) | \(105\) |
parts | \(\frac {2 i a^{3} A c \left (-\frac {1}{8 c^{3} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {1}{12 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {1}{10 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}\right )}{f}+\frac {a^{3} \left (3 i A +B \right ) \left (-\frac {1}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {1}{4 c^{2} \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {1}{6 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}\right )}{f}+\frac {2 B \,a^{3} \left (-\sqrt {c -i c \tan \left (f x +e \right )}-\frac {17 c}{8 \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {7 c^{2}}{12 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {c^{3}}{10 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16}\right )}{f \,c^{3}}-\frac {6 i a^{3} \left (-i B +A \right ) \left (-\frac {1}{4 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {1}{8 c \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {c}{10 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {3}{2}}}\right )}{f c}-\frac {2 a^{3} \left (i A +3 B \right ) \left (\frac {7}{8 \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {5 c}{12 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {c^{2}}{10 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}\right )}{f \,c^{2}}\) | \(521\) |
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Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {\sqrt {2} {\left (3 \, {\left (i \, A + B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - {\left (i \, A + 11 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, {\left (i \, A + 11 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (i \, A + 11 \, B\right )} a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, c^{3} f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=- i a^{3} \left (\int \frac {i A}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 A \tan {\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan ^{3}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 B \tan ^{2}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {B \tan ^{4}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i A \tan ^{2}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {i B \tan {\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i B \tan ^{3}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 i \, {\left (-\frac {15 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} B a^{3}}{c^{2}} + \frac {15 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} {\left (A - 5 i \, B\right )} a^{3} - 20 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (A - 2 i \, B\right )} a^{3} c + 12 \, {\left (A - i \, B\right )} a^{3} c^{2}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c}\right )}}{15 \, c f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 9.89 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.49 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {a^3\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,8{}\mathrm {i}+88\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,4{}\mathrm {i}-A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}+44\,B\,\cos \left (2\,e+2\,f\,x\right )-11\,B\,\cos \left (4\,e+4\,f\,x\right )+3\,B\,\cos \left (6\,e+6\,f\,x\right )-4\,A\,\sin \left (2\,e+2\,f\,x\right )+A\,\sin \left (4\,e+4\,f\,x\right )-3\,A\,\sin \left (6\,e+6\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,44{}\mathrm {i}-B\,\sin \left (4\,e+4\,f\,x\right )\,11{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}\right )}{15\,c^3\,f} \]
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