Optimal. Leaf size=261 \[ -\frac {b \left (a^2 (4 A+3 C)+b^2 (A+2 C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {a \left (2 a^4 C+a^2 b^2 (2 A-5 C)+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2} \]
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Rubi [A] time = 0.57, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3032, 3021, 2754, 12, 2659, 205} \[ -\frac {b \left (a^2 (4 A+3 C)+b^2 (A+2 C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a \left (a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2659
Rule 2754
Rule 3021
Rule 3032
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\int \frac {3 b \left (A b^2+a^2 C\right )-a \left (2 A b^2-\left (a^2-3 b^2\right ) C\right ) \cos (c+d x)-3 b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {2 a b^2 \left (a^2 C-b^2 (5 A+6 C)\right )+b \left (a^2 b^2 (2 A-3 C)+2 a^4 C+3 b^4 (A+2 C)\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\int \frac {3 b^4 \left (b^2 (A+2 C)+a^2 (4 A+3 C)\right )}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )^3}\\ &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (b \left (b^2 (A+2 C)+a^2 (4 A+3 C)\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (b \left (b^2 (A+2 C)+a^2 (4 A+3 C)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {b \left (4 a^2 A+A b^2+3 a^2 C+2 b^2 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 224, normalized size = 0.86 \[ \frac {\frac {24 b \left (a^2 (4 A+3 C)+b^2 (A+2 C)\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+\frac {2 \sin (c+d x) \left (6 b \left (a^4 (2 A+C)+9 a^2 b^2 (A+C)-A b^4\right ) \cos (c+d x)+a \left (2 a^4 (6 A+5 C)+a^2 b^2 (22 A+17 C)+\left (2 a^4 C+a^2 b^2 (2 A-5 C)+b^4 (13 A+18 C)\right ) \cos (2 (c+d x))+b^4 (11 A+18 C)\right )\right )}{(a+b \cos (c+d x))^3}}{24 d \left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 1103, normalized size = 4.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 689, normalized size = 2.64 \[ \frac {\frac {3 \, {\left (4 \, A a^{2} b + 3 \, C a^{2} b + A b^{3} + 2 \, C b^{3}\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} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, A a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{4} b \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 ) + 12 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, 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 )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1727, normalized size = 6.62 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 491, normalized size = 1.88 \[ \frac {\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A\,a^3+C\,a^3+7\,A\,a\,b^2+9\,C\,a\,b^2\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,A\,a^3+A\,b^3+2\,C\,a^3+6\,A\,a\,b^2+2\,A\,a^2\,b+6\,C\,a\,b^2+3\,C\,a^2\,b\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A\,a^3-A\,b^3+2\,C\,a^3+6\,A\,a\,b^2-2\,A\,a^2\,b+6\,C\,a\,b^2-3\,C\,a^2\,b\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{d\,\left (3\,a\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )+3\,a^2\,b+a^3+b^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )}-\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\,\left (4\,A\,a^2+A\,b^2+3\,C\,a^2+2\,C\,b^2\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}\,\left (A\,b^3+2\,C\,b^3+4\,A\,a^2\,b+3\,C\,a^2\,b\right )}\right )\,\left (4\,A\,a^2+A\,b^2+3\,C\,a^2+2\,C\,b^2\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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