Optimal. Leaf size=465 \[ \frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
[Out]
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Rubi [A]
time = 3.53, antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4183, 4177,
4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\tan (c+d x) \sec (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{2 b^3 d \left (a^2-b^2\right )^2}+\frac {\tan (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{2 b^4 d \left (a^2-b^2\right )^2}-\frac {a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4167
Rule 4177
Rule 4183
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec ^3(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec ^2(c+d x) \left (2 \left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)-2 \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-2 a \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 b \left (a^3 b B-4 a b^3 B-2 a^4 C+b^4 (2 A+C)+a^2 b^2 (A+4 C)\right ) \sec (c+d x)+2 \left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-2 a b \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 \left (a^2-b^2\right )^2 \left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}+\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \int \sec (c+d x) \, dx}{2 b^5}\\ &=\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^2}\\ &=\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^6 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (2 a^4 A b^2-5 a^2 A b^4+6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+12 a^6 C-29 a^4 b^2 C+20 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1124\) vs. \(2(465)=930\).
time = 6.47, size = 1124, normalized size = 2.42 \begin {gather*} \frac {2 a \left (2 a^4 A b^2-5 a^2 A b^4+6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+12 a^6 C-29 a^4 b^2 C+20 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^5 \sqrt {a^2-b^2} \left (-a^2+b^2\right )^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {\left (-2 A b^2+6 a b B-12 a^2 C-b^2 C\right ) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-6 a^4 A b^3 \sin (c+d x)+12 a^2 A b^5 \sin (c+d x)+18 a^5 b^2 B \sin (c+d x)-32 a^3 b^4 B \sin (c+d x)+8 a b^6 B \sin (c+d x)-36 a^6 b C \sin (c+d x)+72 a^4 b^3 C \sin (c+d x)-38 a^2 b^5 C \sin (c+d x)+8 b^7 C \sin (c+d x)-4 a^5 A b^2 \sin (2 (c+d x))+10 a^3 A b^4 \sin (2 (c+d x))+12 a^6 b B \sin (2 (c+d x))-14 a^4 b^3 B \sin (2 (c+d x))-12 a^2 b^5 B \sin (2 (c+d x))+8 b^7 B \sin (2 (c+d x))-24 a^7 C \sin (2 (c+d x))+26 a^5 b^2 C \sin (2 (c+d x))+20 a^3 b^4 C \sin (2 (c+d x))-16 a b^6 C \sin (2 (c+d x))-6 a^4 A b^3 \sin (3 (c+d x))+12 a^2 A b^5 \sin (3 (c+d x))+18 a^5 b^2 B \sin (3 (c+d x))-32 a^3 b^4 B \sin (3 (c+d x))+8 a b^6 B \sin (3 (c+d x))-36 a^6 b C \sin (3 (c+d x))+64 a^4 b^3 C \sin (3 (c+d x))-22 a^2 b^5 C \sin (3 (c+d x))-2 a^5 A b^2 \sin (4 (c+d x))+5 a^3 A b^4 \sin (4 (c+d x))+6 a^6 b B \sin (4 (c+d x))-11 a^4 b^3 B \sin (4 (c+d x))+2 a^2 b^5 B \sin (4 (c+d x))-12 a^7 C \sin (4 (c+d x))+21 a^5 b^2 C \sin (4 (c+d x))-6 a^3 b^4 C \sin (4 (c+d x))\right )}{8 b^4 \left (-a^2+b^2\right )^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.37, size = 554, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b B -6 a C -C b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-2 A \,b^{2}+6 a b B -12 a^{2} C -b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b B -6 a C -C b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (2 A \,b^{2}-6 a b B +12 a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {2 a \left (\frac {\frac {\left (2 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}-4 a^{3} b B +a^{2} b^{2} B +8 a \,b^{3} B +6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}-4 a^{3} b B -a^{2} b^{2} B +8 a \,b^{3} B +6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}-6 a^{5} b B +15 a^{3} b^{3} B -12 a \,b^{5} B +12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}}{d}\) | \(554\) |
default | \(\frac {\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b B -6 a C -C b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-2 A \,b^{2}+6 a b B -12 a^{2} C -b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b B -6 a C -C b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (2 A \,b^{2}-6 a b B +12 a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {2 a \left (\frac {\frac {\left (2 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}-4 a^{3} b B +a^{2} b^{2} B +8 a \,b^{3} B +6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}-4 a^{3} b B -a^{2} b^{2} B +8 a \,b^{3} B +6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}-6 a^{5} b B +15 a^{3} b^{3} B -12 a \,b^{5} B +12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}}{d}\) | \(554\) |
risch | \(\text {Expression too large to display}\) | \(2997\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1740 vs.
\(2 (446) = 892\).
time = 0.63, size = 1740, normalized size = 3.74 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 21.04, size = 2500, normalized size = 5.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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