Optimal. Leaf size=94 \[ -\frac {4 \sin ^3(c+d x)}{3 a d}+\frac {4 \sin (c+d x)}{a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {3 x}{2 a} \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2767, 2748, 2635, 8, 2633} \[ -\frac {4 \sin ^3(c+d x)}{3 a d}+\frac {4 \sin (c+d x)}{a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {3 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2767
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {\int \cos ^2(c+d x) (3 a-4 a \cos (c+d x)) \, dx}{a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {3 \int \cos ^2(c+d x) \, dx}{a}+\frac {4 \int \cos ^3(c+d x) \, dx}{a}\\ &=-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {3 \int 1 \, dx}{2 a}-\frac {4 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac {3 x}{2 a}+\frac {4 \sin (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {4 \sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 143, normalized size = 1.52 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (21 \sin \left (c+\frac {d x}{2}\right )+18 \sin \left (c+\frac {3 d x}{2}\right )+18 \sin \left (2 c+\frac {3 d x}{2}\right )-2 \sin \left (2 c+\frac {5 d x}{2}\right )-2 \sin \left (3 c+\frac {5 d x}{2}\right )+\sin \left (3 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {7 d x}{2}\right )-36 d x \cos \left (c+\frac {d x}{2}\right )+69 \sin \left (\frac {d x}{2}\right )-36 d x \cos \left (\frac {d x}{2}\right )\right )}{48 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 70, normalized size = 0.74 \[ -\frac {9 \, d x \cos \left (d x + c\right ) + 9 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 7 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 88, normalized size = 0.94 \[ -\frac {\frac {9 \, {\left (d x + c\right )}}{a} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 136, normalized size = 1.45 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 176, normalized size = 1.87 \[ \frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 70, normalized size = 0.74 \[ \frac {\frac {15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{12}+\frac {\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{24}}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {3\,x}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.90, size = 570, normalized size = 6.06 \[ \begin {cases} - \frac {9 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {27 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {27 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {9 d x}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {6 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {48 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {50 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {24 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\relax (c )}}{a \cos {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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