Optimal. Leaf size=598 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} (c-i d)^{3/2} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} (c+i d)^{3/2} f}-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 2.36, antiderivative size = 598, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {3730, 3697,
3696, 95, 214} \begin {gather*} -\frac {2 \left (A b^2-a (b B-a C)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \sqrt {a+b \tan (e+f x)} \left (a^4 (-d) \left (d^2 (3 A+5 C)-3 B c d+8 c^2 C\right )+8 a^3 b B d \left (c^2+d^2\right )-a^2 b^2 \left (11 A c^2 d+17 A d^3+3 B c^3-3 B c d^2+5 c^2 C d-C d^3\right )+2 a b^3 \left (c^2+d^2\right ) (3 A c+B d-3 c C)-b^4 \left (d \left (5 A c^2+8 A d^2+3 c^2 C\right )-3 B \left (c^3+2 c d^2\right )\right )\right )}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-4 a^4 C d+7 a^3 b B d-a^2 b^2 (2 d (5 A-C)+3 B c)+a b^3 (6 A c+B d-6 c C)+b^4 (3 B c-4 A d)\right )}{3 f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {(i A+B-i C) \tanh ^{-1}\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)^{3/2}}-\frac {(B-i (A-C)) \tanh ^{-1}\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)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 214
Rule 3696
Rule 3697
Rule 3730
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (4 A b^2 d-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right )+\frac {3}{2} (A b-a B-b C) (b c-a d) \tan (e+f x)+2 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} \left (-\left (a b c-a^2 d-2 b^2 d\right ) \left (a^2 (3 A+C) d-b^2 (3 B c-4 A d)-a b (3 A c-3 c C+B d)\right )+(b c+a d) \left (3 b^3 c C-7 a^2 b B d+4 a^3 C d-A b^2 (3 b c-7 a d)+3 a b^2 (B c-C d)\right )\right )+\frac {3}{4} \left (a^2 B-b^2 B-2 a b (A-C)\right ) (b c-a d)^2 \tan (e+f x)-\frac {1}{2} d \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+10 A d-2 C d)\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\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 \sqrt {c+d \tan (e+f x)}}+\frac {8 \int \frac {\frac {3}{8} (b c-a d)^3 \left (a^2 (A c-c C+B d)-b^2 (A c-c C+B d)+2 a b (B c-(A-C) d)\right )-\frac {3}{8} (b c-a d)^3 \left (2 a b (A c-c C+B d)-a^2 (B c-(A-C) d)+b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\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 \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 (a-i b)^2 (c-i d)}+\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 (a+i b)^2 (c+i d)}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\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 \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 (a-i b)^2 (c-i d) 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 (a+i b)^2 (c+i d) f}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\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 \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 )}{(a-i b)^2 (c-i d) 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 )}{(a+i b)^2 (c+i d) f}\\ &=-\frac {(i A+B-i C) \tanh ^{-1}\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)^{3/2} f}-\frac {(B-i (A-C)) \tanh ^{-1}\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)^{3/2} f}-\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (7 a^3 b B d-4 a^4 C d+b^4 (3 B c-4 A d)+a b^3 (6 A c-6 c C+B d)-a^2 b^2 (3 B c+2 (5 A-C) d)\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 d \left (8 a^3 b B d \left (c^2+d^2\right )+2 a b^3 (3 A c-3 c C+B d) \left (c^2+d^2\right )-a^4 d \left (8 c^2 C-3 B c d+(3 A+5 C) d^2\right )-a^2 b^2 \left (3 B c^3+11 A c^2 d+5 c^2 C d-3 B c d^2+17 A d^3-C d^3\right )-b^4 \left (d \left (5 A c^2+3 c^2 C+8 A d^2\right )-3 B \left (c^3+2 c d^2\right )\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 \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 6.62, size = 902, normalized size = 1.51 \begin {gather*} -\frac {2 \left (A b^2-a (b B-a C)\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {2 \left (-a \left (-2 a \left (A b^2-a (b B-a C)\right ) d+\frac {3}{2} b (A b-a B-b C) (b c-a d)\right )+\frac {1}{2} b^2 \left (4 A b^2 d-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {3 (b c-a d)^3 \left (\frac {(a+i b)^2 (i A+B-i C) (c+i d) \tanh ^{-1}\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 (A+i B-C) (i c+d) \tanh ^{-1}\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 )}{4 (-b c+a d) \left (c^2+d^2\right ) f}-\frac {2 \left (d^2 \left (\left (-\frac {b c}{2}-\frac {a d}{2}\right ) \left (-2 a \left (A b^2-a (b B-a C)\right ) d+\frac {3}{2} b (A b-a B-b C) (b c-a d)\right )+\frac {1}{2} \left (b^2 d-\frac {1}{2} a (b c-a d)\right ) \left (4 A b^2 d-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right )\right )-c \left (\frac {1}{2} d (b c-a d) \left (-2 b \left (A b^2-a (b B-a C)\right ) d-\frac {3}{2} a (A b-a B-b C) (b c-a d)+\frac {1}{2} b \left (4 A b^2 d-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right )\right )-c d \left (-a \left (-2 a \left (A b^2-a (b B-a C)\right ) d+\frac {3}{2} b (A b-a B-b C) (b c-a d)\right )+\frac {1}{2} b^2 \left (4 A b^2 d-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\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 )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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