Integrand size = 49, antiderivative size = 379 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(B+i (A-C)) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b} (c-i d)^{5/2} f}+\frac {(i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} (c+i d)^{5/2} f}+\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]
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Time = 2.06 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {3730, 3697, 3696, 95, 214} \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(B+i (A-C)) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b} (c-i d)^{5/2}}+\frac {(i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b} (c+i d)^{5/2}}+\frac {2 \left (A d^2-B c d+c^2 C\right ) \sqrt {a+b \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}+\frac {2 \sqrt {a+b \tan (e+f x)} \left (b \left (4 c^2 d^2 (2 A-C)+2 A d^4-5 B c^3 d+B c d^3+2 c^4 C\right )-3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{3 f \left (c^2+d^2\right )^2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}} \]
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Rule 95
Rule 214
Rule 3696
Rule 3697
Rule 3730
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {\frac {1}{2} \left (2 A b d^2+3 A c (b c-a d)-(b c-3 a d) (c C-B d)\right )+\frac {3}{2} (b c-a d) (B c-(A-C) d) \tan (e+f x)+b \left (c^2 C-B c d+A d^2\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{3 (b c-a d) \left (c^2+d^2\right )} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {-\frac {3}{4} (b c-a d)^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-\frac {3}{4} (b c-a d)^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 (b c-a d)^2 \left (c^2+d^2\right )^2} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(A-i B-C) \int \frac {1+i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {(A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)^2} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(A-i B-C) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c-i d)^2 f}+\frac {(A+i B-C) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c+i d)^2 f} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(A-i B-C) \text {Subst}\left (\int \frac {1}{i a+b-(i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^2 f}+\frac {(A+i B-C) \text {Subst}\left (\int \frac {1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^2 f} \\ & = -\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b} (c-i d)^{5/2} f}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} (c+i d)^{5/2} f}+\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )-3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \\ \end{align*}
Time = 6.00 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.06 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\frac {3 (b c-a d)^2 \left (\frac {(i A+B-i C) (c+i d)^2 \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+i b} \sqrt {-c+i d}}+\frac {i (A+i B-C) (c-i d)^2 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} \sqrt {c+i d}}\right )+\frac {2 (b c-a d) \left (c^2+d^2\right ) \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (2 c^4 C-5 B c^3 d+4 c^2 (2 A-C) d^2+B c d^3+2 A d^4\right )+3 a d^2 \left (2 c (-A+C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{3 (b c-a d)^2 \left (c^2+d^2\right )^2 f} \]
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Timed out.
\[\int \frac {A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}}{\sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\sqrt {a + b \tan {\left (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]
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