Optimal. Leaf size=118 \[ \frac {15 x}{8 a}-\frac {4 \sin (c+d x)}{a d}+\frac {15 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {4 \sin ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3904, 3872,
2715, 8, 2713} \begin {gather*} \frac {4 \sin ^3(c+d x)}{3 a d}-\frac {4 \sin (c+d x)}{a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {15 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac {15 x}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 3904
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {\int \cos ^4(c+d x) (-5 a+4 a \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {4 \int \cos ^3(c+d x) \, dx}{a}+\frac {5 \int \cos ^4(c+d x) \, dx}{a}\\ &=\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {15 \int \cos ^2(c+d x) \, dx}{4 a}+\frac {4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac {4 \sin (c+d x)}{a d}+\frac {15 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {4 \sin ^3(c+d x)}{3 a d}+\frac {15 \int 1 \, dx}{8 a}\\ &=\frac {15 x}{8 a}-\frac {4 \sin (c+d x)}{a d}+\frac {15 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {4 \sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 173, normalized size = 1.47 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (360 d x \cos \left (\frac {d x}{2}\right )+360 d x \cos \left (c+\frac {d x}{2}\right )-552 \sin \left (\frac {d x}{2}\right )-168 \sin \left (c+\frac {d x}{2}\right )-120 \sin \left (c+\frac {3 d x}{2}\right )-120 \sin \left (2 c+\frac {3 d x}{2}\right )+40 \sin \left (2 c+\frac {5 d x}{2}\right )+40 \sin \left (3 c+\frac {5 d x}{2}\right )-5 \sin \left (3 c+\frac {7 d x}{2}\right )-5 \sin \left (4 c+\frac {7 d x}{2}\right )+3 \sin \left (4 c+\frac {9 d x}{2}\right )+3 \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{384 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 100, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {115 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {109 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(100\) |
default | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {115 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {109 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(100\) |
risch | \(\frac {15 x}{8 a}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )}}{8 a d}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a d}-\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (4 d x +4 c \right )}{32 a d}-\frac {\sin \left (3 d x +3 c \right )}{12 a d}+\frac {\sin \left (2 d x +2 c \right )}{2 a d}\) | \(117\) |
norman | \(\frac {\frac {15 x}{8 a}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {157 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {187 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {41 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {15 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {45 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {15 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 217, normalized size = 1.84 \begin {gather*} -\frac {\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 79, normalized size = 0.67 \begin {gather*} \frac {45 \, d x \cos \left (d x + c\right ) + 45 \, d x + {\left (6 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 13 \, \cos \left (d x + c\right )^{2} - 19 \, \cos \left (d x + c\right ) - 64\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\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*} \frac {\int \frac {\cos ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 101, normalized size = 0.86 \begin {gather*} \frac {\frac {45 \, {\left (d x + c\right )}}{a} - \frac {24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 115 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, \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 )}^{4} a}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.45, size = 98, normalized size = 0.83 \begin {gather*} \frac {15\,x}{8\,a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d}-\frac {\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}+\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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