Integrand size = 29, antiderivative size = 470 \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i (a-i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{5/2} f}+\frac {i (a+i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{5/2} f}-\frac {b^{7/2} (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f} \]
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Time = 7.55 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3646, 3726, 3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 f \left (c^2+d^2\right )^2}-\frac {b^{7/2} (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {i (a-i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{5/2}}+\frac {i (a+i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{5/2}}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}} \]
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Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3646
Rule 3726
Rule 3728
Rule 3736
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(a+b \tan (e+f x))^{3/2} \left (\frac {1}{2} \left (5 b^3 c^2+3 a^3 c d-13 a b^2 c d+11 a^2 b d^2\right )+\frac {3}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)-\frac {1}{2} b \left (2 a d (2 b c-a d)-b^2 \left (5 c^2+3 d^2\right )\right ) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 d \left (c^2+d^2\right )} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {\sqrt {a+b \tan (e+f x)} \left (\frac {3}{4} \left (12 a^3 b c d^3-2 a^2 b^2 d^2 \left (5 c^2-7 d^2\right )+a^4 d^2 \left (c^2-d^2\right )-4 a b^3 c d \left (c^2+6 d^2\right )+b^4 \left (5 c^4+11 c^2 d^2\right )\right )+\frac {3}{2} d^2 \left (a^2 c-b^2 c+2 a b d\right ) \left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)+\frac {3}{4} b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}+\frac {4 \int \frac {\frac {3}{8} \left (20 a^4 b c d^4-40 a^2 b^3 c d^4+2 a^5 d^3 \left (c^2-d^2\right )-20 a^3 b^2 d^3 \left (c^2-d^2\right )+a b^4 d \left (9 c^4+28 c^2 d^2-d^4\right )-b^5 c \left (5 c^4+10 c^2 d^2+d^4\right )\right )-\frac {3}{4} d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right ) \tan (e+f x)-\frac {3}{8} b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 d^3 \left (c^2+d^2\right )^2} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}+\frac {4 \text {Subst}\left (\int \frac {\frac {3}{8} \left (20 a^4 b c d^4-40 a^2 b^3 c d^4+2 a^5 d^3 \left (c^2-d^2\right )-20 a^3 b^2 d^3 \left (c^2-d^2\right )+a b^4 d \left (9 c^4+28 c^2 d^2-d^4\right )-b^5 c \left (5 c^4+10 c^2 d^2+d^4\right )\right )-\frac {3}{4} d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right ) x-\frac {3}{8} b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 d^3 \left (c^2+d^2\right )^2 f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}+\frac {4 \text {Subst}\left (\int \left (-\frac {3 b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 \left (d^3 \left (10 a^4 b c d-20 a^2 b^3 c d+2 b^5 c d+a^5 \left (c^2-d^2\right )-10 a^3 b^2 \left (c^2-d^2\right )+5 a b^4 \left (c^2-d^2\right )\right )-d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right ) x\right )}{4 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{3 d^3 \left (c^2+d^2\right )^2 f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (b^4 (5 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 d^3 f}+\frac {\text {Subst}\left (\int \frac {d^3 \left (10 a^4 b c d-20 a^2 b^3 c d+2 b^5 c d+a^5 \left (c^2-d^2\right )-10 a^3 b^2 \left (c^2-d^2\right )+5 a b^4 \left (c^2-d^2\right )\right )-d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^3 \left (c^2+d^2\right )^2 f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (b^3 (5 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{d^3 f}+\frac {\text {Subst}\left (\int \left (\frac {i d^3 \left (10 a^4 b c d-20 a^2 b^3 c d+2 b^5 c d+a^5 \left (c^2-d^2\right )-10 a^3 b^2 \left (c^2-d^2\right )+5 a b^4 \left (c^2-d^2\right )\right )+d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {i d^3 \left (10 a^4 b c d-20 a^2 b^3 c d+2 b^5 c d+a^5 \left (c^2-d^2\right )-10 a^3 b^2 \left (c^2-d^2\right )+5 a b^4 \left (c^2-d^2\right )\right )-d^3 \left (2 a^5 c d-20 a^3 b^2 c d+10 a b^4 c d-5 a^4 b \left (c^2-d^2\right )+10 a^2 b^3 \left (c^2-d^2\right )-b^5 \left (c^2-d^2\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d^3 \left (c^2+d^2\right )^2 f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}+\frac {(i a+b)^5 \text {Subst}\left (\int \frac {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 {(i a-b)^5 \text {Subst}\left (\int \frac {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 {\left (b^3 (5 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{d^3 f} \\ & = -\frac {b^{7/2} (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f}+\frac {(i a+b)^5 \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i 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)^5 \text {Subst}\left (\int \frac {1}{a+i b-(c+i 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-i b)^{9/2} \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 )}{(c-i d)^{5/2} f}+\frac {i (a+i b)^{9/2} \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 )}{(c+i d)^{5/2} f}-\frac {b^{7/2} (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 9.28 (sec) , antiderivative size = 2261, normalized size of antiderivative = 4.81 \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Result too large to show} \]
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Timed out.
\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 47668 vs. \(2 (398) = 796\).
Time = 236.60 (sec) , antiderivative size = 95363, normalized size of antiderivative = 202.90 \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{9/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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