Integrand size = 32, antiderivative size = 444 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} (c-i d)^{5/2} f}-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \]
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Time = 2.04 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3640, 3677, 3679, 12, 3625, 214} \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f (c-i d)^{5/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 f (c-i d)^2 (c+i d)^5 \sqrt {c+d \tan (e+f x)}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 f (-d+i c)^3 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {-21 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \]
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Rule 12
Rule 214
Rule 3625
Rule 3640
Rule 3677
Rule 3679
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a (5 i c-13 d)-4 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx}{5 a^2 (i c-d)} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {3}{4} a^2 \left (5 c^2+20 i c d-47 d^2\right )-\frac {3}{2} a^2 (5 c+21 i d) d \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx}{15 a^4 (c+i d)^2} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (\frac {3}{8} a^3 \left (5 i c^3-35 c^2 d-135 i c d^2+361 d^3\right )+\frac {3}{2} a^3 d \left (5 i c^2-30 c d-89 i d^2\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx}{15 a^6 (i c-d)^3} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {2 \int \frac {\sqrt {a+i a \tan (e+f x)} \left (\frac {3}{16} a^4 \left (15 i c^4-90 c^3 d-260 i c^2 d^2+502 c d^3-707 i d^4\right )+\frac {3}{8} a^4 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{45 a^7 (i c-d)^3 \left (c^2+d^2\right )} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}}-\frac {4 \int \frac {45 a^5 (i c-d)^5 \sqrt {a+i a \tan (e+f x)}}{32 \sqrt {c+d \tan (e+f x)}} \, dx}{45 a^8 (i c-d)^3 \left (c^2+d^2\right )^2} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^3 (c-i d)^2} \\ & = -\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}}-\frac {i \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{4 a (c-i d)^2 f} \\ & = -\frac {i \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} (c-i d)^{5/2} f}-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \\ \end{align*}
Time = 7.44 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {a}{5 (i a c-a d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {-\frac {1}{2} a^2 (5 i c-13 d)+4 a^2 d}{3 (i a c-a d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {-\frac {3}{2} i a^3 (5 c+21 i d) d-\frac {3}{4} a^3 \left (5 c^2+20 i c d-47 d^2\right )}{(i a c-a d) f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 \left (-\frac {3}{2} a^3 c d \left (5 i c^2-30 c d-89 i d^2\right )+\frac {3}{8} a^3 d \left (5 i c^3-35 c^2 d-135 i c d^2+361 d^3\right )\right ) \sqrt {a+i a \tan (e+f x)}}{3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {45 i a^5 (i c-d)^5 \arctan \left (\frac {\sqrt {-a c+i a d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {2} \sqrt {-a c+i a d} \left (c^2+d^2\right ) f}-\frac {2 \left (-\frac {3}{8} a^4 c d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right )+\frac {3}{16} a^4 d \left (15 i c^4-90 c^3 d-260 i c^2 d^2+502 c d^3-707 i d^4\right )\right ) \sqrt {a+i a \tan (e+f x)}}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\right )}{3 a \left (c^2+d^2\right )}}{a (i a c-a d)}}{3 a (i a c-a d)}}{5 a (i a c-a d)} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 10144 vs. \(2 (374 ) = 748\).
Time = 1.44 (sec) , antiderivative size = 10145, normalized size of antiderivative = 22.85
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(10145\) |
default | \(\text {Expression too large to display}\) | \(10145\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1774 vs. \(2 (352) = 704\).
Time = 0.39 (sec) , antiderivative size = 1774, normalized size of antiderivative = 4.00 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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