Optimal. Leaf size=242 \[ \frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-3 a b^4 (A+2 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.68, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4175, 4165,
4083, 3855, 3916, 2738, 214} \begin {gather*} \frac {a \tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\left (-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-3 a b^4 (A+2 C)+2 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {\tan (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4165
Rule 4175
Rubi steps
\begin {align*} \int \frac {\sec ^2(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 {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) \left (-2 b \left (A b^2-a (b B-a C)\right )+\left (a^2 b B-2 b^3 B-a^3 C+a b^2 (A+2 C)\right ) \sec (c+d x)+2 b \left (a^2-b^2\right ) C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \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 (-b^2 \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right )-2 b \left (a^2-b^2\right )^2 C \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {C \int \sec (c+d x) \, dx}{b^3}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-3 a b^4 (A+2 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-3 a b^4 (A+2 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-3 a b^4 (A+2 C)\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^4 \left (a^2-b^2\right )^2 d}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {\left (3 a A b^4-a^2 b^3 B-2 b^5 B+2 a^5 C-5 a^3 b^2 C+6 a 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^3 (a+b)^{5/2} d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \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 [C] Result contains complex when optimal does not.
time = 6.13, size = 514, normalized size = 2.12 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-4 C (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 C (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 \left (-a^2 b^3 B-2 b^5 B+2 a^5 C-5 a^3 b^2 C+3 a b^4 (A+2 C)\right ) \text {ArcTan}\left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^2 (i \cos (c)+\sin (c))}{\left (a^2-b^2\right )^{5/2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {b \left (a \sec (c) \left (b \left (4 A b^4+a^3 b B-10 a b^3 B-7 a^4 C+a^2 b^2 (5 A+16 C)\right ) \sin (d x)+a \left (b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sin (2 c+d x)+\left (A b^4-3 a b^3 B-2 a^4 C+a^2 b^2 (2 A+5 C)\right ) \sin (c+2 d x)\right )\right )+\left (a^2+2 b^2\right ) \left (-A b^4+3 a b^3 B+2 a^4 C-a^2 b^2 (2 A+5 C)\right ) \tan (c)\right )}{a \left (a^2-b^2\right )^2}\right )}{2 b^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 1.28, size = 364, normalized size = 1.50
method | result | size |
derivativedivides | \(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{2}+a A \,b^{3}+2 A \,b^{4}-a^{2} b^{2} B -4 a \,b^{3} B -2 a^{4} C +a^{3} b C +6 C \,a^{2} b^{2}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (2 a^{2} A \,b^{2}-a A \,b^{3}+2 A \,b^{4}+a^{2} b^{2} B -4 a \,b^{3} B -2 a^{4} C -a^{3} b C +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (3 a A \,b^{4}-a^{2} b^{3} B -2 b^{5} B +2 a^{5} C -5 a^{3} b^{2} C +6 C a \,b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}}{d}\) | \(364\) |
default | \(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{2}+a A \,b^{3}+2 A \,b^{4}-a^{2} b^{2} B -4 a \,b^{3} B -2 a^{4} C +a^{3} b C +6 C \,a^{2} b^{2}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (2 a^{2} A \,b^{2}-a A \,b^{3}+2 A \,b^{4}+a^{2} b^{2} B -4 a \,b^{3} B -2 a^{4} C -a^{3} b C +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (3 a A \,b^{4}-a^{2} b^{3} B -2 b^{5} B +2 a^{5} C -5 a^{3} b^{2} C +6 C a \,b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}}{d}\) | \(364\) |
risch | \(\text {Expression too large to display}\) | \(1489\) |
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. 721 vs.
\(2 (232) = 464\).
time = 53.99, size = 1501, normalized size = 6.20 \begin {gather*} \left [-\frac {{\left (2 \, C a^{5} b^{2} - 5 \, C a^{3} b^{4} - B a^{2} b^{5} + 3 \, {\left (A + 2 \, C\right )} a b^{6} - 2 \, B b^{7} + {\left (2 \, C a^{7} - 5 \, C a^{5} b^{2} - B a^{4} b^{3} + 3 \, {\left (A + 2 \, C\right )} a^{3} b^{4} - 2 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{6} b - 5 \, C a^{4} b^{3} - B a^{3} b^{4} + 3 \, {\left (A + 2 \, C\right )} a^{2} b^{5} - 2 \, B a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} - {\left (A + 9 \, C\right )} a^{4} b^{4} + 5 \, B a^{3} b^{5} - {\left (A - 6 \, C\right )} a^{2} b^{6} - 4 \, B a b^{7} + 2 \, A b^{8} + {\left (2 \, C a^{7} b - {\left (2 \, A + 7 \, C\right )} a^{5} b^{3} + 3 \, B a^{4} b^{4} + {\left (A + 5 \, C\right )} a^{3} b^{5} - 3 \, B a^{2} b^{6} + A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{3} - 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} - a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} d\right )}}, -\frac {{\left (2 \, C a^{5} b^{2} - 5 \, C a^{3} b^{4} - B a^{2} b^{5} + 3 \, {\left (A + 2 \, C\right )} a b^{6} - 2 \, B b^{7} + {\left (2 \, C a^{7} - 5 \, C a^{5} b^{2} - B a^{4} b^{3} + 3 \, {\left (A + 2 \, C\right )} a^{3} b^{4} - 2 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{6} b - 5 \, C a^{4} b^{3} - B a^{3} b^{4} + 3 \, {\left (A + 2 \, C\right )} a^{2} b^{5} - 2 \, B a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} - {\left (A + 9 \, C\right )} a^{4} b^{4} + 5 \, B a^{3} b^{5} - {\left (A - 6 \, C\right )} a^{2} b^{6} - 4 \, B a b^{7} + 2 \, A b^{8} + {\left (2 \, C a^{7} b - {\left (2 \, A + 7 \, C\right )} a^{5} b^{3} + 3 \, B a^{4} b^{4} + {\left (A + 5 \, C\right )} a^{3} b^{5} - 3 \, B a^{2} b^{6} + A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b^{3} - 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} - a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} d\right )}}\right ] \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 ^{2}{\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. 632 vs.
\(2 (232) = 464\).
time = 0.59, size = 632, normalized size = 2.61 \begin {gather*} -\frac {\frac {{\left (2 \, C a^{5} - 5 \, C a^{3} b^{2} - B a^{2} b^{3} + 3 \, A a b^{4} + 6 \, C a b^{4} - 2 \, B b^{5}\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^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} + \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}} - \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{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 )}^{2}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.91, size = 2500, normalized size = 10.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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