Optimal. Leaf size=16 \[ \frac {\sin (a+b x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4380, 2635, 8} \[ \frac {\sin (a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4380
Rubi steps
\begin {align*} \int \frac {\cos ^2(a+b x)-\sin ^2(a+b x)}{\cos ^2(a+b x)+\sin ^2(a+b x)} \, dx &=\int \left (\cos ^2(a+b x)-\sin ^2(a+b x)\right ) \, dx\\ &=\int \cos ^2(a+b x) \, dx-\int \sin ^2(a+b x) \, dx\\ &=\frac {\cos (a+b x) \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 33, normalized size = 2.06 \[ \frac {\sin (2 a) \cos (2 b x)}{2 b}+\frac {\cos (2 a) \sin (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 16, normalized size = 1.00 \[ \frac {\cos \left (b x + a\right ) \sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 14, normalized size = 0.88 \[ \frac {\sin \left (2 \, b x + 2 \, a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 17, normalized size = 1.06 \[ \frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 22, normalized size = 1.38 \[ \frac {\tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 14, normalized size = 0.88 \[ \frac {\sin \left (2\,a+2\,b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 32, normalized size = 2.00 \[ \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{b \sin ^{2}{\left (a + b x \right )} + b \cos ^{2}{\left (a + b x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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