Optimal. Leaf size=369 \[ \frac {\left (12 A b^2+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 1.15, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4186, 4185,
4189, 4004, 3916, 2738, 214} \begin {gather*} \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {x \left (a^2 (A+2 C)+12 A b^2\right )}{2 a^5}-\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \sin (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}+\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}-\frac {b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 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 4185
Rule 4186
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (2 A b^2-a^2 (A-C)\right )+2 a b (A+C) \sec (c+d x)-3 \left (A b^2+a^2 C\right ) \sec ^2(c+d x)\right )}{(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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+a b \left (A b^2-a^2 (4 A+3 C)\right ) \sec (c+d x)+2 \left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (2 b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right )+2 a \left (2 A b^4-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \sec (c+d x)-2 b \left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {2 \left (a^2-b^2\right )^2 \left (12 A b^2+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b \left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 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^5 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (12 A b^2+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6+6 a^6 C-5 a^4 b^2 C+2 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^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos (c+d x) \sin (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 [A]
time = 2.56, size = 256, normalized size = 0.69 \begin {gather*} \frac {2 \left (12 A b^2+a^2 (A+2 C)\right ) (c+d x)+\frac {4 b \left (12 A b^6+5 a^4 b^2 (4 A-C)+6 a^6 C+a^2 b^4 (-29 A+2 C)\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 )^{5/2}}-12 a A b \sin (c+d x)-\frac {2 a b^3 \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}+\frac {2 a b^2 \left (-7 A b^4+a^2 b^2 (10 A-3 C)+6 a^4 C\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+a^2 A \sin (2 (c+d x))}{4 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 404, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {2 b \left (\frac {-\frac {\left (10 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}+6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-3 a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-3 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+12 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(404\) |
default | \(\frac {\frac {2 b \left (\frac {-\frac {\left (10 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}+6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-3 a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-3 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+12 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(404\) |
risch | \(\text {Expression too large to display}\) | \(1472\) |
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. 748 vs.
\(2 (350) = 700\).
time = 4.32, size = 1554, normalized size = 4.21 \begin {gather*} \left [\frac {2 \, {\left ({\left (A + 2 \, C\right )} a^{10} + 3 \, {\left (3 \, A - 2 \, C\right )} a^{8} b^{2} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{6} b^{4} + {\left (35 \, A - 2 \, C\right )} a^{4} b^{6} - 12 \, A a^{2} b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (A + 2 \, C\right )} a^{9} b + 3 \, {\left (3 \, A - 2 \, C\right )} a^{7} b^{3} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{5} b^{5} + {\left (35 \, A - 2 \, C\right )} a^{3} b^{7} - 12 \, A a b^{9}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left ({\left (A + 2 \, C\right )} a^{8} b^{2} + 3 \, {\left (3 \, A - 2 \, C\right )} a^{6} b^{4} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{4} b^{6} + {\left (35 \, A - 2 \, C\right )} a^{2} b^{8} - 12 \, A b^{10}\right )} d x + {\left (6 \, C a^{6} b^{3} + 5 \, {\left (4 \, A - C\right )} a^{4} b^{5} - {\left (29 \, A - 2 \, C\right )} a^{2} b^{7} + 12 \, A b^{9} + {\left (6 \, C a^{8} b + 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} - {\left (29 \, A - 2 \, C\right )} a^{4} b^{5} + 12 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, C a^{7} b^{2} + 5 \, {\left (4 \, A - C\right )} a^{5} b^{4} - {\left (29 \, A - 2 \, C\right )} a^{3} b^{6} + 12 \, A a b^{8}\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 ({\left (6 \, A - 5 \, C\right )} a^{7} b^{3} - {\left (27 \, A - 7 \, C\right )} a^{5} b^{5} + {\left (33 \, A - 2 \, C\right )} a^{3} b^{7} - 12 \, A a b^{9} - {\left (A a^{10} - 3 \, A a^{8} b^{2} + 3 \, A a^{6} b^{4} - A a^{4} b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A a^{9} b - 3 \, A a^{7} b^{3} + 3 \, A a^{5} b^{5} - A a^{3} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (11 \, A - 6 \, C\right )} a^{8} b^{2} - {\left (43 \, A - 9 \, C\right )} a^{6} b^{4} + {\left (50 \, A - 3 \, C\right )} a^{4} b^{6} - 18 \, A a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{13} - 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} - a^{7} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} b^{8}\right )} d\right )}}, \frac {{\left ({\left (A + 2 \, C\right )} a^{10} + 3 \, {\left (3 \, A - 2 \, C\right )} a^{8} b^{2} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{6} b^{4} + {\left (35 \, A - 2 \, C\right )} a^{4} b^{6} - 12 \, A a^{2} b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (A + 2 \, C\right )} a^{9} b + 3 \, {\left (3 \, A - 2 \, C\right )} a^{7} b^{3} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{5} b^{5} + {\left (35 \, A - 2 \, C\right )} a^{3} b^{7} - 12 \, A a b^{9}\right )} d x \cos \left (d x + c\right ) + {\left ({\left (A + 2 \, C\right )} a^{8} b^{2} + 3 \, {\left (3 \, A - 2 \, C\right )} a^{6} b^{4} - 3 \, {\left (11 \, A - 2 \, C\right )} a^{4} b^{6} + {\left (35 \, A - 2 \, C\right )} a^{2} b^{8} - 12 \, A b^{10}\right )} d x - {\left (6 \, C a^{6} b^{3} + 5 \, {\left (4 \, A - C\right )} a^{4} b^{5} - {\left (29 \, A - 2 \, C\right )} a^{2} b^{7} + 12 \, A b^{9} + {\left (6 \, C a^{8} b + 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} - {\left (29 \, A - 2 \, C\right )} a^{4} b^{5} + 12 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, C a^{7} b^{2} + 5 \, {\left (4 \, A - C\right )} a^{5} b^{4} - {\left (29 \, A - 2 \, C\right )} a^{3} b^{6} + 12 \, A a b^{8}\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 ({\left (6 \, A - 5 \, C\right )} a^{7} b^{3} - {\left (27 \, A - 7 \, C\right )} a^{5} b^{5} + {\left (33 \, A - 2 \, C\right )} a^{3} b^{7} - 12 \, A a b^{9} - {\left (A a^{10} - 3 \, A a^{8} b^{2} + 3 \, A a^{6} b^{4} - A a^{4} b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A a^{9} b - 3 \, A a^{7} b^{3} + 3 \, A a^{5} b^{5} - A a^{3} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (11 \, A - 6 \, C\right )} a^{8} b^{2} - {\left (43 \, A - 9 \, C\right )} a^{6} b^{4} + {\left (50 \, A - 3 \, C\right )} a^{4} b^{6} - 18 \, A a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{13} - 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} - a^{7} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} 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 {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{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. 2485 vs.
\(2 (350) = 700\).
time = 0.81, size = 2485, normalized size = 6.73 \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 = 17.74, size = 2500, normalized size = 6.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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