Optimal. Leaf size=192 \[ -\frac {b \left (4 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {a \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]
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Rubi [A] time = 0.23, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2754, 12, 2659, 205} \[ -\frac {b \left (4 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {a \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2659
Rule 2754
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {a \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\int \frac {3 b-2 a \cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac {a \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {-10 a b+\left (2 a^2+3 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=\frac {a \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\int \frac {3 b \left (4 a^2+b^2\right )}{a+b \cos (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac {a \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (b \left (4 a^2+b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=\frac {a \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (b \left (4 a^2+b^2\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+b^2\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 \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (2 a^2+3 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2+13 b^2\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 164, normalized size = 0.85 \[ \frac {\frac {\sin (c+d x) \left (6 a^5+10 a^3 b^2+a b^2 \left (2 a^2+13 b^2\right ) \cos ^2(c+d x)-3 b \left (-2 a^4-9 a^2 b^2+b^4\right ) \cos (c+d x)-a b^4\right )}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))^3}-\frac {6 b \left (4 a^2+b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 891, normalized size = 4.64 \[ \left [\frac {3 \, {\left (4 \, a^{5} b + a^{3} b^{3} + {\left (4 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, a^{4} b^{2} + a^{2} b^{4}\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 (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 ) + 2 \, {\left (6 \, a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + a b^{6} + {\left (2 \, a^{5} b^{2} + 11 \, a^{3} b^{4} - 13 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{6} b + 7 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}}, -\frac {3 \, {\left (4 \, a^{5} b + a^{3} b^{3} + {\left (4 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )\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 (6 \, a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + a b^{6} + {\left (2 \, a^{5} b^{2} + 11 \, a^{3} b^{4} - 13 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{6} b + 7 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.95, size = 427, normalized size = 2.22 \[ \frac {\frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 931, normalized size = 4.85 \[ \text {result too large to display} \]
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 = 4.25, size = 382, normalized size = 1.99 \[ \frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^3+2\,a^2\,b+6\,a\,b^2+b^3\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^3+7\,a\,b^2\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^3-2\,a^2\,b+6\,a\,b^2-b^3\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{d\,\left (3\,a\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )+3\,a^2\,b+a^3+b^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )}-\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^2+b^2\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}\,\left (4\,a^2\,b+b^3\right )}\right )\,\left (4\,a^2+b^2\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
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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