Optimal. Leaf size=43 \[ \frac {a x}{2}-\frac {b \cos ^2(c+d x)}{2 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 (c+d x)}{2 d}+\frac {a x}{2}-\frac {b \cos ^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3567
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {b \cos ^2(c+d x)}{2 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac {b \cos ^2(c+d x)}{2 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a \int 1 \, dx\\ &=\frac {a x}{2}-\frac {b \cos ^2(c+d x)}{2 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 1.07 \begin {gather*} \frac {a (c+d x)}{2 d}-\frac {b \cos ^2(c+d x)}{2 d}+\frac {a \sin (2 (c+d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 41, normalized size = 0.95
method | result | size |
risch | \(\frac {a x}{2}-\frac {b \cos \left (2 d x +2 c \right )}{4 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(36\) |
derivativedivides | \(\frac {-\frac {b \left (\cos ^{2}\left (d x +c \right )\right )}{2}+a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(41\) |
default | \(\frac {-\frac {b \left (\cos ^{2}\left (d x +c \right )\right )}{2}+a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 38, normalized size = 0.88 \begin {gather*} \frac {{\left (d x + c\right )} a + \frac {a \tan \left (d x + c\right ) - b}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 35, normalized size = 0.81 \begin {gather*} \frac {a d x - b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, 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 ^{2}{\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. 146 vs.
\(2 (37) = 74\).
time = 0.51, size = 146, normalized size = 3.40 \begin {gather*} \frac {2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, a d x \tan \left (d x\right )^{2} + 2 \, a d x \tan \left (c\right )^{2} - b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) - 2 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, a d x + b \tan \left (d x\right )^{2} + 4 \, b \tan \left (d x\right ) \tan \left (c\right ) + b \tan \left (c\right )^{2} + 2 \, a \tan \left (d x\right ) + 2 \, a \tan \left (c\right ) - b}{4 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + d \tan \left (d x\right )^{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 = 3.68, size = 31, normalized size = 0.72 \begin {gather*} \frac {a\,x}{2}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b}{2}-\frac {a\,\mathrm {tan}\left (c+d\,x\right )}{2}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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