Optimal. Leaf size=202 \[ \frac {A x}{a^3}+\frac {b \left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.32, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4146, 4145,
4004, 3916, 2738, 214} \begin {gather*} \frac {A x}{a^3}+\frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (a^4 (-C)-a^2 b^2 (5 A+2 C)+2 A b^4\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {b \left (-3 a^4 (2 A+C)+5 a^2 A b^2-2 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4145
Rule 4146
Rubi steps
\begin {align*} \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {-2 A \left (a^2-b^2\right )+2 a b (A+C) \sec (c+d x)-\left (A b^2+a^2 C\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {2 A \left (a^2-b^2\right )^2+a b \left (A b^2-a^2 (4 A+3 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (b \left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+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 )}{a^3 \left (a^2-b^2\right )^2 d}\\ &=\frac {A x}{a^3}-\frac {b \left (6 a^4 A-5 a^2 A b^2+2 A b^4+3 a^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^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 = 5.02, size = 642, normalized size = 3.18 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 b \left (-5 a^2 A b^2+2 A b^4+3 a^4 (2 A+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 {\sec (c) \left (2 A \left (a^2-b^2\right )^2 \left (a^2+2 b^2\right ) d x \cos (c)+4 a A b \left (a^2-b^2\right )^2 d x \cos (d x)+4 a^5 A b d x \cos (2 c+d x)-8 a^3 A b^3 d x \cos (2 c+d x)+4 a A b^5 d x \cos (2 c+d x)+a^6 A d x \cos (c+2 d x)-2 a^4 A b^2 d x \cos (c+2 d x)+a^2 A b^4 d x \cos (c+2 d x)+a^6 A d x \cos (3 c+2 d x)-2 a^4 A b^2 d x \cos (3 c+2 d x)+a^2 A b^4 d x \cos (3 c+2 d x)-6 a^4 A b^2 \sin (c)-9 a^2 A b^4 \sin (c)+6 A b^6 \sin (c)-2 a^6 C \sin (c)-5 a^4 b^2 C \sin (c)-2 a^2 b^4 C \sin (c)+17 a^3 A b^3 \sin (d x)-8 a A b^5 \sin (d x)+5 a^5 b C \sin (d x)+4 a^3 b^3 C \sin (d x)-7 a^3 A b^3 \sin (2 c+d x)+4 a A b^5 \sin (2 c+d x)-3 a^5 b C \sin (2 c+d x)+6 a^4 A b^2 \sin (c+2 d x)-3 a^2 A b^4 \sin (c+2 d x)+2 a^6 C \sin (c+2 d x)+a^4 b^2 C \sin (c+2 d x)\right )}{\left (a^2-b^2\right )^2}\right )}{2 a^3 d (A+2 C+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 = 0.28, size = 298, normalized size = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{2} A \,b^{2}+a A \,b^{3}-2 A \,b^{4}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 a^{2} A \,b^{2}-a A \,b^{3}-2 A \,b^{4}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {b \left (6 A \,a^{4}-5 a^{2} A \,b^{2}+2 A \,b^{4}+3 a^{4} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}+\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(298\) |
default | \(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{2} A \,b^{2}+a A \,b^{3}-2 A \,b^{4}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 a^{2} A \,b^{2}-a A \,b^{3}-2 A \,b^{4}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {b \left (6 A \,a^{4}-5 a^{2} A \,b^{2}+2 A \,b^{4}+3 a^{4} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}+\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(298\) |
risch | \(\frac {A x}{a^{3}}-\frac {i \left (-7 A \,a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+4 A a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-3 C \,a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-6 A \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 A \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 A \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-5 C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-17 A \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+8 A a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}-5 C \,a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-4 C \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-6 A \,a^{4} b^{2}+3 a^{2} A \,b^{4}-2 a^{6} C -a^{4} b^{2} C \right )}{a^{3} \left (-a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(996\) |
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. 501 vs.
\(2 (188) = 376\).
time = 2.90, size = 1059, normalized size = 5.24 \begin {gather*} \left [\frac {4 \, {\left (A a^{8} - 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} - A a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (A a^{7} b - 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} - A a b^{7}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (A a^{6} b^{2} - 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} - A b^{8}\right )} d x + {\left (3 \, {\left (2 \, A + C\right )} a^{4} b^{3} - 5 \, A a^{2} b^{5} + 2 \, A b^{7} + {\left (3 \, {\left (2 \, A + C\right )} a^{6} b - 5 \, A a^{4} b^{3} + 2 \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (2 \, A + C\right )} a^{5} b^{2} - 5 \, A a^{3} b^{4} + 2 \, A 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^{7} b + {\left (5 \, A + C\right )} a^{5} b^{3} - {\left (7 \, A + 2 \, C\right )} a^{3} b^{5} + 2 \, A a b^{7} + {\left (2 \, C a^{8} + {\left (6 \, A - C\right )} a^{6} b^{2} - {\left (9 \, A + C\right )} a^{4} b^{4} + 3 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\right )} d\right )}}, \frac {2 \, {\left (A a^{8} - 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} - A a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (A a^{7} b - 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} - A a b^{7}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (A a^{6} b^{2} - 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} - A b^{8}\right )} d x - {\left (3 \, {\left (2 \, A + C\right )} a^{4} b^{3} - 5 \, A a^{2} b^{5} + 2 \, A b^{7} + {\left (3 \, {\left (2 \, A + C\right )} a^{6} b - 5 \, A a^{4} b^{3} + 2 \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (2 \, A + C\right )} a^{5} b^{2} - 5 \, A a^{3} b^{4} + 2 \, A 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^{7} b + {\left (5 \, A + C\right )} a^{5} b^{3} - {\left (7 \, A + 2 \, C\right )} a^{3} b^{5} + 2 \, A a b^{7} + {\left (2 \, C a^{8} + {\left (6 \, A - C\right )} a^{6} b^{2} - {\left (9 \, A + C\right )} a^{4} b^{4} + 3 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\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 {A + C \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. 484 vs.
\(2 (188) = 376\).
time = 0.55, size = 484, normalized size = 2.40 \begin {gather*} -\frac {\frac {{\left (6 \, A a^{4} b + 3 \, C a^{4} b - 5 \, A a^{2} b^{3} + 2 \, A 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^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {{\left (d x + c\right )} A}{a^{3}} + \frac {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 )^{3} + 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A 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 ) - C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, 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 ) + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\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 = 14.24, size = 2500, normalized size = 12.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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