Optimal. Leaf size=643 \[ -\frac {(A-i B-C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b c-a d} \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 f}-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2} \]
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Rubi [A]
time = 4.32, antiderivative size = 643, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {3726, 3728,
3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {(c+d \tan (e+f x))^{3/2} \left (-5 a^4 C d+a^3 b B d+a^2 b^2 (3 A d+4 B c-13 C d)-a b^3 (8 A c-9 B d-8 c C)-b^4 (5 A d+4 B c)\right )}{4 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {d \sqrt {c+d \tan (e+f x)} \left (-15 a^4 C d+3 a^3 b B d+a^2 b^2 (d (A-31 C)+4 B c)-a b^3 (8 A c-11 B d-8 c C)-b^4 (7 A d+4 B c+8 C d)\right )}{4 b^3 f \left (a^2+b^2\right )^2}+\frac {\sqrt {b c-a d} \left (-15 a^6 C d^2+3 a^5 b B d^2+a^4 b^2 d (d (A-46 C)+4 B c)+2 a^3 b^3 \left (4 c d (A-C)+B \left (4 c^2+3 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+21 C d^2\right )-a b^5 \left (56 c d (A-C)+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (5 B d+2 c C)-A \left (8 c^2-15 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} f \left (a^2+b^2\right )^3}-\frac {(c-i d)^{5/2} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{5/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3618
Rule 3620
Rule 3715
Rule 3726
Rule 3728
Rule 3734
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} \left (2 (b B-a C) \left (2 b c-\frac {5 a d}{2}\right )+2 A b \left (2 a c+\frac {5 b d}{2}\right )\right )-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac {1}{2} \left (A b^2-a b B+5 a^2 C+4 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{4} \left (b (2 a c+3 b d) \left (5 a^2 C d+b^2 (4 B c+5 A d)+a b (4 A c-4 c C-5 B d)\right )+(2 b c-3 a d) \left (a^2 b B d-5 a^3 C d-A b^2 (4 b c-3 a d)+4 b^3 (c C+B d)+4 a b^2 (B c-2 C d)\right )\right )+2 b^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 {1}{4} d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {\frac {1}{8} \left (-15 a^5 C d^3+3 a^4 b d^2 (5 c C+B d)+a^3 b^2 d^2 (B c+(A-31 C) d)-b^5 c \left (8 A c^2-8 c^2 C-20 B c d-15 A d^2\right )+a^2 b^3 \left (8 A c^3-8 c^3 C-20 B c^2 d-17 A c d^2+47 c C d^2+11 B d^3\right )+a b^4 \left (16 B c^3+40 A c^2 d-40 c^2 C d-31 B c d^2-7 A d^3-8 C d^3\right )\right )+b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)-\frac {1}{8} d \left (15 a^5 C d^2-3 a^4 b d (5 c C+B d)-a^3 b^2 d (B c+(A-31 C) d)+b^5 \left (4 B c^2+9 A c d-24 c C d-8 B d^2\right )-a^2 b^3 \left (4 B c^2+7 A c d+23 c C d+3 B d^2\right )+a b^4 \left (8 A c^2-8 c^2 C-17 B c d-9 A d^2+24 C d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {-b^3 \left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{b^3 \left (a^2+b^2\right )^3}-\frac {\left ((b c-a d) \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{8 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\left ((A-i B-C) (c-i d)^3\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac {\left ((A+i B-C) (c+i d)^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac {\left ((b c-a d) \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b^3 \left (a^2+b^2\right )^3 f}\\ &=-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\left ((A-i B-C) (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (i a+b)^3 f}-\frac {\left ((A+i B-C) (c+i d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (i a-b)^3 f}-\frac {\left ((b c-a d) \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 b^3 \left (a^2+b^2\right )^3 d f}\\ &=\frac {\sqrt {b c-a d} \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 f}-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left ((A-i B-C) (c-i d)^3\right ) \text {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 )}{(a-i b)^3 d f}-\frac {\left ((A+i B-C) (c+i d)^3\right ) \text {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 )}{(a+i b)^3 d f}\\ &=-\frac {(A-i B-C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b c-a d} \left (3 a^5 b B d^2-15 a^6 C d^2+a^4 b^2 d (4 B c+(A-46 C) d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+21 C d^2\right )-a b^5 \left (56 c (A-C) d+B \left (24 c^2-35 d^2\right )\right )-b^6 \left (4 c (2 c C+5 B d)-A \left (8 c^2-15 d^2\right )\right )+2 a^3 b^3 \left (4 c (A-C) d+B \left (4 c^2+3 d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 f}-\frac {d \left (3 a^3 b B d-15 a^4 C d-a b^3 (8 A c-8 c C-11 B d)+a^2 b^2 (4 B c+(A-31 C) d)-b^4 (4 B c+7 A d+8 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (a^3 b B d-5 a^4 C d-b^4 (4 B c+5 A d)-a b^3 (8 A c-8 c C-9 B d)+a^2 b^2 (4 B c+3 A d-13 C d)\right ) (c+d \tan (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(18214\) vs. \(2(643)=1286\).
time = 6.60, size = 18214, normalized size = 28.33 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7282\) vs.
\(2(599)=1198\).
time = 0.78, size = 7283, normalized size = 11.33
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(7283\) |
default | \(\text {Expression too large to display}\) | \(7283\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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(-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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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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