Optimal. Leaf size=420 \[ \frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]
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Rubi [A]
time = 3.87, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3079, 3134,
3080, 3855, 2738, 211} \begin {gather*} -\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2738
Rule 3079
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {\left (3 a^2 A-4 A b^2+a b B-3 a (A b-a B) \cos (c+d x)+3 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (6 a^4 A-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \cos (c+d x)+2 b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B-a \left (18 a^4 A b-7 a^2 A b^3+4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B\right ) \cos (c+d x)+3 b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-6 \left (a^2-b^2\right )^3 (4 A b-a B)+3 a b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {(4 A b-a B) \int \sec (c+d x) \, dx}{a^5}+\frac {\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^3 d}\\ &=\frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 3.27, size = 549, normalized size = 1.31 \begin {gather*} \frac {-\frac {48 b \left (-20 a^6 A b+35 a^4 A b^3-28 a^2 A b^5+8 A b^7+8 a^7 B-8 a^5 b^2 B+7 a^3 b^4 B-2 a b^6 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+48 (4 A b-a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 (-4 A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^9 A-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6-120 a A b^8+120 a^6 b^3 B-90 a^4 b^5 B+30 a^2 b^7 B+b \left (72 a^8 A-438 a^6 A b^2+305 a^4 A b^4+28 a^2 A b^6-72 A b^8+144 a^7 b B-50 a^5 b^3 B-7 a^3 b^5 B+18 a b^7 B\right ) \cos (c+d x)+6 a b^2 \left (6 a^6 A-53 a^4 A b^2+57 a^2 A b^4-20 A b^6+20 a^5 b B-15 a^3 b^3 B+5 a b^5 B\right ) \cos (2 (c+d x))+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))+26 a^5 b^4 B \cos (3 (c+d x))-17 a^3 b^6 B \cos (3 (c+d x))+6 a b^8 B \cos (3 (c+d x))\right ) \tan (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{48 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.99, size = 587, normalized size = 1.40
method | result | size |
derivativedivides | \(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +a B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}+\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 A b -a B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\) | \(587\) |
default | \(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +a B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}+\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 A b -a B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\) | \(587\) |
risch | \(\text {Expression too large to display}\) | \(2577\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1662 vs.
\(2 (402) = 804\).
time = 72.79, size = 3393, normalized size = 8.08 \begin {gather*} \text {Too large to display} \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 \cos {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, 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. 996 vs.
\(2 (402) = 804\).
time = 0.51, size = 996, normalized size = 2.37 \begin {gather*} \frac {\frac {3 \, {\left (8 \, B a^{7} b - 20 \, A a^{6} b^{2} - 8 \, B a^{5} b^{3} + 35 \, A a^{4} b^{4} + 7 \, B a^{3} b^{5} - 28 \, A a^{2} b^{6} - 2 \, B a b^{7} + 8 \, A b^{8}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {36 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 116 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 236 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 56 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{3}} + \frac {3 \, {\left (B a - 4 \, A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {3 \, {\left (B a - 4 \, A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} - \frac {6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.11, size = 2500, normalized size = 5.95 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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