Optimal. Leaf size=585 \[ \frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{3 d^4 f \left (c^2+d^2\right )^2}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d^3 f \left (c^2+d^2\right )^2}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(-b+i a)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}-\frac {(a-i b)^3 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}} \]
[Out]
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Rubi [A] time = 2.97, antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3645, 3637, 3630, 3539, 3537, 63, 208} \[ \frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-17 B c^2 d^3-8 B c^4 d-3 B d^5+16 c^5 C\right )\right )}{3 d^4 f \left (c^2+d^2\right )^2}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d^3 f \left (c^2+d^2\right )^2}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+4 c^2 C d^2+2 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {(-b+i a)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}-\frac {(a-i b)^3 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 3537
Rule 3539
Rule 3630
Rule 3637
Rule 3645
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{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))^2 \left (\frac {3}{2} \left (A d (a c+2 b d)+\frac {2}{3} \left (3 b c-\frac {3 a d}{2}\right ) (c C-B d)\right )+\frac {3}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac {3}{2} b \left (2 c^2 C-B c d+(A+C) d^2\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 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {(a+b \tan (e+f x)) \left (-\frac {3}{4} \left ((4 b c-a d) \left (a d^2 (B c-(A-C) d)-b (2 c C-B d) \left (c^2+d^2\right )\right )-d (a c+4 b d) \left (a d (A c-c C+B d)+2 b \left (c^2 C-B c d+A d^2\right )\right )\right )-\frac {3}{4} d^2 \left (2 a b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac {3}{4} b \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\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 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}-\frac {8 \int \frac {-\frac {3}{8} \left (6 a b^2 d \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )-2 b^3 c \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )-3 a^3 d^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+15 a^2 b d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )+\frac {9}{8} d^3 \left (3 a^2 b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-3 a b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)-\frac {3}{8} b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{9 d^3 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}-\frac {8 \int \frac {\frac {9}{8} d^3 \left (a^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a^2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )+b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )+\frac {9}{8} d^3 \left (3 a^2 b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-3 a b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{9 d^3 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}+\frac {\left ((a-i b)^3 (A-i B-C)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {\left ((a+i b)^3 (A+i B-C)\right ) \int \frac {1-i \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 ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}+\frac {\left (i (a-i b)^3 (A-i B-C)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d)^2 f}-\frac {\left (i (a+i b)^3 (A+i B-C)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d)^2 f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}-\frac {\left ((a-i b)^3 (A-i B-C)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(c-i d)^2 d f}-\frac {\left ((a+i b)^3 (A+i B-C)\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(c+i d)^2 d f}\\ &=-\frac {(a-i b)^3 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(i a-b)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 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-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f}\\ \end {align*}
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Mathematica [C] time = 6.91, size = 670, normalized size = 1.15 \[ \frac {2 C (a+b \tan (e+f x))^3}{3 d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {3 (-2 a C d-b B d+2 b c C) (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {3 (a+b \tan (e+f x)) \left (b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-b B d+2 b c C)\right )}{2 d f (c+d \tan (e+f x))^{3/2}}-\frac {3 \left (-\frac {2 \left (16 a^3 C d^3+9 a^2 b B d^3-48 a^2 b c C d^2-18 a b^2 B c d^2+48 a b^2 c^2 C d-2 A b^3 c d^2+8 b^3 B c^2 d+b^3 B d^3-16 b^3 c^3 C+2 b^3 c C d^2\right )}{3 d (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (\frac {\left (\frac {3}{2} c d^4 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right )+\frac {3}{2} d^5 \left (-\left (a^3 (A-C)\right )+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right )\right ) \left (\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{3 (-d+i c) (c+d \tan (e+f x))^{3/2}}-\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (d+i c) (c+d \tan (e+f x))^{3/2}}\right )}{d}-\frac {3}{2} d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{(-d+i c) \sqrt {c+d \tan (e+f x)}}-\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{(d+i c) \sqrt {c+d \tan (e+f x)}}\right )\right )}{3 d}\right )}{4 d f}\right )}{d}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 85156, normalized size = 145.57 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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