Optimal. Leaf size=65 \[ \frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8}-\frac {b \cos ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3486, 2635, 8} \[ \frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8}-\frac {b \cos ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3486
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {b \cos ^4(c+d x)}{4 d}+a \int \cos ^4(c+d x) \, dx\\ &=-\frac {b \cos ^4(c+d x)}{4 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {b \cos ^4(c+d x)}{4 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{8}-\frac {b \cos ^4(c+d x)}{4 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 62, normalized size = 0.95 \[ \frac {3 a (c+d x)}{8 d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \sin (4 (c+d x))}{32 d}-\frac {b \cos ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 51, normalized size = 0.78 \[ -\frac {2 \, b \cos \left (d x + c\right )^{4} - 3 \, a d x - {\left (2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 426, normalized size = 6.55 \[ \frac {12 \, a d x \tan \left (d x\right )^{4} \tan \relax (c)^{4} + 24 \, a d x \tan \left (d x\right )^{4} \tan \relax (c)^{2} + 24 \, a d x \tan \left (d x\right )^{2} \tan \relax (c)^{4} - 5 \, b \tan \left (d x\right )^{4} \tan \relax (c)^{4} - 20 \, a \tan \left (d x\right )^{4} \tan \relax (c)^{3} - 20 \, a \tan \left (d x\right )^{3} \tan \relax (c)^{4} + 12 \, a d x \tan \left (d x\right )^{4} + 48 \, a d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 6 \, b \tan \left (d x\right )^{4} \tan \relax (c)^{2} + 32 \, b \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 12 \, a d x \tan \relax (c)^{4} + 6 \, b \tan \left (d x\right )^{2} \tan \relax (c)^{4} - 12 \, a \tan \left (d x\right )^{4} \tan \relax (c) + 24 \, a \tan \left (d x\right )^{3} \tan \relax (c)^{2} + 24 \, a \tan \left (d x\right )^{2} \tan \relax (c)^{3} - 12 \, a \tan \left (d x\right ) \tan \relax (c)^{4} + 24 \, a d x \tan \left (d x\right )^{2} + 3 \, b \tan \left (d x\right )^{4} + 24 \, a d x \tan \relax (c)^{2} - 36 \, b \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 3 \, b \tan \relax (c)^{4} + 12 \, a \tan \left (d x\right )^{3} - 24 \, a \tan \left (d x\right )^{2} \tan \relax (c) - 24 \, a \tan \left (d x\right ) \tan \relax (c)^{2} + 12 \, a \tan \relax (c)^{3} + 12 \, a d x + 6 \, b \tan \left (d x\right )^{2} + 32 \, b \tan \left (d x\right ) \tan \relax (c) + 6 \, b \tan \relax (c)^{2} + 20 \, a \tan \left (d x\right ) + 20 \, a \tan \relax (c) - 5 \, b}{32 \, {\left (d \tan \left (d x\right )^{4} \tan \relax (c)^{4} + 2 \, d \tan \left (d x\right )^{4} \tan \relax (c)^{2} + 2 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{4} + d \tan \left (d x\right )^{4} + 4 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + d \tan \relax (c)^{4} + 2 \, d \tan \left (d x\right )^{2} + 2 \, d \tan \relax (c)^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 52, normalized size = 0.80 \[ \frac {-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) b}{4}+a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 61, normalized size = 0.94 \[ \frac {3 \, {\left (d x + c\right )} a + \frac {3 \, a \tan \left (d x + c\right )^{3} + 5 \, a \tan \left (d x + c\right ) - 2 \, b}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 41, normalized size = 0.63 \[ \frac {3\,a\,x}{8}+\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {3\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{8}+\frac {5\,a\,\mathrm {tan}\left (c+d\,x\right )}{8}-\frac {b}{4}\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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