Integrand size = 45, antiderivative size = 102 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}} \]
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Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3669, 79, 37} \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {(-6 B+i A) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}} \]
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Rule 37
Rule 79
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)}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {(a (A+6 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}} \\ \end{align*}
Time = 6.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\frac {i a^3 \sec ^3(e+f x) (\cos (3 (e+f x))+i \sin (3 (e+f x))) (6 i A-B+(A+6 i B) \tan (e+f x))}{35 c^3 f (i+\tan (e+f x))^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (5 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+5 B \,{\mathrm e}^{6 i \left (f x +e \right )}+7 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-7 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{70 c^{3} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(106\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (i A \tan \left (f x +e \right )^{2}+5 i \tan \left (f x +e \right ) B -6 B \tan \left (f x +e \right )^{2}+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (i A \tan \left (f x +e \right )^{2}+5 i \tan \left (f x +e \right ) B -6 B \tan \left (f x +e \right )^{2}+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
parts | \(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (5 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}+6\right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-1+5 i \tan \left (f x +e \right )-6 \tan \left (f x +e \right )^{2}\right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(171\) |
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Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {{\left (5 \, {\left (i \, A + B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 2 \, {\left (6 i \, A - B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 7 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{70 \, c^{4} f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {7}{2}}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).
Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.63 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {70 \, {\left (5 \, {\left (A - i \, B\right )} a^{2} \cos \left (9 \, f x + 9 \, e\right ) + 2 \, {\left (6 \, A + i \, B\right )} a^{2} \cos \left (7 \, f x + 7 \, e\right ) + 7 \, {\left (A + i \, B\right )} a^{2} \cos \left (5 \, f x + 5 \, e\right ) - 5 \, {\left (-i \, A - B\right )} a^{2} \sin \left (9 \, f x + 9 \, e\right ) - 2 \, {\left (-6 i \, A + B\right )} a^{2} \sin \left (7 \, f x + 7 \, e\right ) - 7 \, {\left (-i \, A + B\right )} a^{2} \sin \left (5 \, f x + 5 \, e\right )\right )} \sqrt {a} \sqrt {c}}{-4900 \, {\left (i \, c^{4} \cos \left (2 \, f x + 2 \, e\right ) - c^{4} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{4}\right )} f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 9.96 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}-7\,B\,\cos \left (4\,e+4\,f\,x\right )+5\,B\,\cos \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (4\,e+4\,f\,x\right )-5\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}\right )}{70\,c^3\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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