Optimal. Leaf size=679 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{5/2} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{5/2} f}-\frac {b^{3/2} \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{7/2} f}-\frac {d \left (2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )+A \left (2 a^2 d^2+b^2 \left (3 c^2+5 d^2\right )\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]
[Out]
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Rubi [A]
time = 3.47, antiderivative size = 678, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3730, 3734,
3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right )^2 (b c-a d)^3 \sqrt {c+d \tan (e+f x)}}-\frac {b^{3/2} \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{7/2}}-\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 (c-i d)^{5/2}}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 (c+i d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 214
Rule 3618
Rule 3620
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (5 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+3 a d)\right )+(A b-a B-b C) (b c-a d) \tan (e+f x)+\frac {5}{2} \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{4} \left (2 a^3 d^2 (A c-c C+B d)-a^2 b (4 A+C) d \left (c^2+d^2\right )+b^3 (2 B c-5 A d) \left (c^2+d^2\right )-a b^2 \left (2 c^3 C-B c^2 d+4 c C d^2-3 B d^3-2 A \left (c^3+2 c d^2\right )\right )\right )+\frac {3}{2} (b c-a d)^2 (A b c-a B c-b c C+a A d+b B d-a C d) \tan (e+f x)+\frac {3}{4} b d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {4 \int \frac {-\frac {3}{8} \left (b^4 (2 B c-5 A d) \left (c^2+d^2\right )^2+2 a^4 d^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-2 a^3 b d^2 \left (3 c^3 C-4 B c^2 d+c C d^2-2 B d^3-A c \left (3 c^2+d^2\right )\right )+a^2 b^2 d \left (c^4 C+4 c^2 C d^2-4 B c d^3-C d^4-2 A \left (3 c^4+7 c^2 d^2+2 d^4\right )\right )-a b^3 \left (2 c^5 C+B c^4 d+10 c^3 C d^2-6 B c^2 d^3+4 c C d^4-3 B d^5-2 A \left (c^5+5 c^3 d^2+2 c d^4\right )\right )\right )+\frac {3}{4} (b c-a d)^3 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )-b \left (c^2 C-2 B c d-C d^2\right )\right ) \tan (e+f x)+\frac {3}{8} b d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2}\\ &=-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {4 \int \frac {\frac {3}{4} (b c-a d)^3 \left (a^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-\frac {3}{4} (b c-a d)^3 \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)}{\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 )^2}+\frac {\left (b^2 \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)^3}\\ &=-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(A-i B-C) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2 (c-i d)^2}+\frac {(A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2 (c+i d)^2}+\frac {\left (b^2 \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^3 f}\\ &=-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(i A+B-i C) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 (c-i d)^2 f}-\frac {(i (A+i B-C)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 (c+i d)^2 f}+\frac {\left (b^2 \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\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 )}{\left (a^2+b^2\right )^2 d (b c-a d)^3 f}\\ &=-\frac {b^{3/2} \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{7/2} f}-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {(i (i A+B-i C)) \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)^2 (c-i d)^2 d f}-\frac {(A+i B-C) \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)^2 (c+i d)^2 d f}\\ &=-\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{5/2} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{5/2} f}-\frac {b^{3/2} \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{7/2} f}-\frac {d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}\\ \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. \(6052\) vs. \(2(679)=1358\).
time = 6.28, size = 6052, normalized size = 8.91 \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. \(15714\) vs.
\(2(639)=1278\).
time = 0.75, size = 15715, normalized size = 23.14
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(15715\) |
default | \(\text {Expression too large to display}\) | \(15715\) |
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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