Optimal. Leaf size=65 \[ \frac {3 a x}{8}-\frac {b \cos ^4(c+d x)}{4 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d} \]
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Rubi [A]
time = 0.03, 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 = {3567, 2715, 8}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3567
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 = 0.11, size = 62, normalized size = 0.95 \begin {gather*} \frac {3 a (c+d x)}{8 d}-\frac {b \cos ^4(c+d x)}{4 d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \sin (4 (c+d x))}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 52, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {b \left (\cos ^{4}\left (d x +c \right )\right )}{4}}{d}\) | \(52\) |
default | \(\frac {a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {b \left (\cos ^{4}\left (d x +c \right )\right )}{4}}{d}\) | \(52\) |
risch | \(\frac {3 a x}{8}-\frac {b \cos \left (4 d x +4 c \right )}{32 d}+\frac {a \sin \left (4 d x +4 c \right )}{32 d}-\frac {b \cos \left (2 d x +2 c \right )}{8 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 61, normalized size = 0.94 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 51, normalized size = 0.78 \begin {gather*} -\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} \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*} \int \left (a + b \tan {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs.
\(2 (57) = 114\).
time = 0.64, size = 426, normalized size = 6.55 \begin {gather*} \frac {12 \, a d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 24 \, a d x \tan \left (d x\right )^{4} \tan \left (c\right )^{2} + 24 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{4} - 5 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 20 \, a \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 20 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{4} + 12 \, a d x \tan \left (d x\right )^{4} + 48 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 6 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{2} + 32 \, b \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 12 \, a d x \tan \left (c\right )^{4} + 6 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{4} - 12 \, a \tan \left (d x\right )^{4} \tan \left (c\right ) + 24 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 24 \, a \tan \left (d x\right )^{2} \tan \left (c\right )^{3} - 12 \, a \tan \left (d x\right ) \tan \left (c\right )^{4} + 24 \, a d x \tan \left (d x\right )^{2} + 3 \, b \tan \left (d x\right )^{4} + 24 \, a d x \tan \left (c\right )^{2} - 36 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 3 \, b \tan \left (c\right )^{4} + 12 \, a \tan \left (d x\right )^{3} - 24 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) - 24 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 12 \, a \tan \left (c\right )^{3} + 12 \, a d x + 6 \, b \tan \left (d x\right )^{2} + 32 \, b \tan \left (d x\right ) \tan \left (c\right ) + 6 \, b \tan \left (c\right )^{2} + 20 \, a \tan \left (d x\right ) + 20 \, a \tan \left (c\right ) - 5 \, b}{32 \, {\left (d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 2 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{2} + 2 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{4} + d \tan \left (d x\right )^{4} + 4 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + d \tan \left (c\right )^{4} + 2 \, d \tan \left (d x\right )^{2} + 2 \, d \tan \left (c\right )^{2} + d\right )}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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