Optimal. Leaf size=270 \[ -\frac {2 b^2 \left (4 a^2 A b-3 A b^3-3 a^3 B+2 a b^2 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2 A+6 A b^2-4 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]
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Rubi [A]
time = 0.65, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {\left (a^2 A+2 a b B-3 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\left (a^2 A-4 a b B+6 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \tan (c+d x)}{a^3 d \left (a^2-b^2\right )}-\frac {2 b^2 \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}} \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 ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (a^2 A-3 A b^2+2 a b B-a (A b-a B) \cos (c+d x)+2 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-2 \left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right )+a \left (a^2 A+A b^2-2 a b B\right ) \cos (c+d x)+b \left (a^2 A-3 A b^2+2 a b B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (\left (a^2-b^2\right ) \left (a^2 A+6 A b^2-4 a b B\right )+a b \left (a^2 A-3 A b^2+2 a b B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (a^2 A+6 A b^2-4 a b B\right ) \int \sec (c+d x) \, dx}{2 a^4}-\frac {\left (b^2 \left (4 a^2 A b-3 A b^3-3 a^3 B+2 a b^2 B\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 A+6 A b^2-4 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (2 b^2 \left (4 a^2 A b-3 A b^3-3 a^3 B+2 a b^2 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^4 \left (a^2-b^2\right ) d}\\ &=-\frac {2 b^2 \left (4 a^2 A b-3 A b^3-3 a^3 B+2 a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2 A+6 A b^2-4 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 6.29, size = 438, normalized size = 1.62 \begin {gather*} -\frac {2 b^2 \left (-4 a^2 A b+3 A b^3+3 a^3 B-2 a b^2 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{a^4 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d}+\frac {\left (-a^2 A-6 A b^2+4 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {\left (a^2 A+6 A b^2-4 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {A}{4 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {A}{4 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {-2 A b \sin \left (\frac {1}{2} (c+d x)\right )+a B \sin \left (\frac {1}{2} (c+d x)\right )}{a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {-2 A b \sin \left (\frac {1}{2} (c+d x)\right )+a B \sin \left (\frac {1}{2} (c+d x)\right )}{a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{a^3 (a-b) (a+b) d (a+b \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 326, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {-\frac {A}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -4 A b +2 a B}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +6 A \,b^{2}-4 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4}}-\frac {2 b^{2} \left (-\frac {a \left (A b -a B \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 A \,a^{2} b -3 A \,b^{3}-3 a^{3} B +2 B a \,b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}+\frac {A}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -4 A b +2 a B}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -6 A \,b^{2}+4 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4}}}{d}\) | \(326\) |
default | \(\frac {-\frac {A}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -4 A b +2 a B}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +6 A \,b^{2}-4 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4}}-\frac {2 b^{2} \left (-\frac {a \left (A b -a B \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 A \,a^{2} b -3 A \,b^{3}-3 a^{3} B +2 B a \,b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}+\frac {A}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -4 A b +2 a B}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -6 A \,b^{2}+4 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4}}}{d}\) | \(326\) |
risch | \(\text {Expression too large to display}\) | \(1268\) |
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. 630 vs.
\(2 (256) = 512\).
time = 12.30, size = 1329, normalized size = 4.92 \begin {gather*} \left [-\frac {2 \, {\left ({\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} - 2 \, B a b^{5} + 3 \, A b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} - 2 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left ({\left (A a^{6} b - 4 \, B a^{5} b^{2} + 4 \, A a^{4} b^{3} + 8 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} - 4 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (A a^{7} - 4 \, B a^{6} b + 4 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} - 11 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 6 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (A a^{6} b - 4 \, B a^{5} b^{2} + 4 \, A a^{4} b^{3} + 8 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} - 4 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (A a^{7} - 4 \, B a^{6} b + 4 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} - 11 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 6 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (A a^{7} - 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \, {\left (B a^{6} b - 2 \, A a^{5} b^{2} - 3 \, B a^{4} b^{3} + 5 \, A a^{3} b^{4} + 2 \, B a^{2} b^{5} - 3 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, B a^{7} - 3 \, A a^{6} b - 4 \, B a^{5} b^{2} + 6 \, A a^{4} b^{3} + 2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac {4 \, {\left ({\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} - 2 \, B a b^{5} + 3 \, A b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} - 2 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) + {\left ({\left (A a^{6} b - 4 \, B a^{5} b^{2} + 4 \, A a^{4} b^{3} + 8 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} - 4 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (A a^{7} - 4 \, B a^{6} b + 4 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} - 11 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 6 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (A a^{6} b - 4 \, B a^{5} b^{2} + 4 \, A a^{4} b^{3} + 8 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} - 4 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (A a^{7} - 4 \, B a^{6} b + 4 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} - 11 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 6 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a^{7} - 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \, {\left (B a^{6} b - 2 \, A a^{5} b^{2} - 3 \, B a^{4} b^{3} + 5 \, A a^{3} b^{4} + 2 \, B a^{2} b^{5} - 3 \, A a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, B a^{7} - 3 \, A a^{6} b - 4 \, B a^{5} b^{2} + 6 \, A a^{4} b^{3} + 2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right )^{2}\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 \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 378, normalized size = 1.40 \begin {gather*} -\frac {\frac {4 \, {\left (3 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} - 2 \, B a b^{4} + 3 \, 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^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\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 )}} - \frac {{\left (A a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} + \frac {{\left (A a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.28, size = 2500, normalized size = 9.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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