Integrand size = 29, antiderivative size = 596 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} (c-i d)^{5/2} f}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} (c+i d)^{5/2} f}-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]
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Time = 3.61 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3650, 3730, 3697, 3696, 95, 214} \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {4 b^2 \left (-2 a^2 d+a b c-b^2 d\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (13 c^2+15 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+6 a^3 b^2 c d^4-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2 (b c-a d)^4 \sqrt {c+d \tan (e+f x)}}-\frac {i \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2} (c-i d)^{5/2}}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2} (c+i d)^{5/2}} \]
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Rule 95
Rule 214
Rule 3650
Rule 3696
Rule 3697
Rule 3730
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} \left (a b c-a^2 d-2 b^2 d\right )+\frac {3}{2} b (b c-a d) \tan (e+f x)+3 b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {4 \int \frac {-\frac {3}{4} \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )-\frac {3}{2} a b (b c-a d)^2 \tan (e+f x)-6 b^2 d \left (a b c-2 a^2 d-b^2 d\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^2} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {8 \int \frac {-\frac {3}{8} \left (3 a^5 c d^3+3 a b^4 c d \left (3 c^2+4 d^2\right )+3 a^3 b^2 c d \left (3 c^2+5 d^2\right )-a^4 b d^2 \left (9 c^2+8 d^2\right )+b^5 \left (3 c^4-14 c^2 d^2-16 d^4\right )-a^2 b^3 \left (3 c^4+35 c^2 d^2+30 d^4\right )\right )-\frac {9}{8} (b c-a d)^3 \left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)+\frac {3}{4} b d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{9 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right )} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {16 \int \frac {\frac {9}{16} (b c-a d)^4 (a c-b c-a d-b d) (a c+b c+a d-b d)-\frac {9}{8} (b c-a d)^4 (b c+a d) (a c-b d) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{9 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2 (c-i d)^2}+\frac {\int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2 (c+i d)^2} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\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 (a-i b)^2 (c-i d)^2 f}+\frac {\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 (a+i b)^2 (c+i d)^2 f} \\ & = -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\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 )}{(a-i b)^2 (c-i d)^2 f}+\frac {\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 )}{(a+i b)^2 (c+i d)^2 f} \\ & = -\frac {i \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 )}{(a-i b)^{5/2} (c-i d)^{5/2} f}+\frac {i \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 )}{(a+i b)^{5/2} (c+i d)^{5/2} f}-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {4 b^2 \left (a b c-2 a^2 d-b^2 d\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {2 d \left (a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (7 c^2+8 d^2\right )+a^2 b^2 d \left (13 c^2+15 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 d \left (3 a^5 c d^4+6 a^3 b^2 c d^4-a^4 b d^3 \left (7 c^2+4 d^2\right )+3 a b^4 c \left (c^4+2 c^2 d^2+2 d^4\right )-b^5 d \left (4 c^4+15 c^2 d^2+8 d^4\right )-a^2 b^3 d \left (7 c^4+28 c^2 d^2+15 d^4\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^4 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \\ \end{align*}
Time = 6.80 (sec) , antiderivative size = 1050, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (-\frac {2 \left (-\frac {3}{2} b^2 \left (a b c-a^2 d-2 b^2 d\right )-a \left (-3 a b^2 d+\frac {3}{2} b^2 (b c-a d)\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (-\frac {2 \left (-c \left (-\frac {3}{2} a b d (b c-a d)^2+6 b^2 c d \left (a b c-2 a^2 d-b^2 d\right )\right )-\frac {3}{4} d^2 \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )\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 (-\frac {9 i (b c-a d)^4 \left (\frac {(a+i b)^2 (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 {(a-i b)^2 (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 )}{8 (-b c+a d) \left (c^2+d^2\right ) f}-\frac {2 \left (d^2 \left (\left (\frac {b c}{2}-\frac {3 a d}{2}\right ) \left (-\frac {3}{2} a b d (b c-a d)^2+6 b^2 c d \left (a b c-2 a^2 d-b^2 d\right )\right )-\frac {3}{4} \left (b d^2-\frac {3}{2} c (-b c+a d)\right ) \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )\right )-c \left (\frac {3}{2} d (-b c+a d) \left (\frac {3}{2} a b c (b c-a d)^2+6 b^2 d^2 \left (a b c-2 a^2 d-b^2 d\right )-\frac {3}{4} d \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )\right )-b c \left (-c \left (-\frac {3}{2} a b d (b c-a d)^2+6 b^2 c d \left (a b c-2 a^2 d-b^2 d\right )\right )-\frac {3}{4} d^2 \left (2 a^3 b c d+6 a b^3 c d-a^4 d^2+b^4 \left (c^2-8 d^2\right )-a^2 b^2 \left (c^2+15 d^2\right )\right )\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{(-b c+a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\right )}{3 (-b c+a d) \left (c^2+d^2\right )}\right )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \]
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Timed out.
\[\int \frac {1}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 73285 vs. \(2 (530) = 1060\).
Time = 141.67 (sec) , antiderivative size = 73285, normalized size of antiderivative = 122.96 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]
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