Optimal. Leaf size=30 \[ a x-\frac {b \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b x}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2635, 8} \[ a x-\frac {b \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(e+f x)\right ) \, dx &=a x+b \int \sin ^2(e+f x) \, dx\\ &=a x-\frac {b \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} b \int 1 \, dx\\ &=a x+\frac {b x}{2}-\frac {b \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.10 \[ a x+\frac {b (e+f x)}{2 f}-\frac {b \sin (2 (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 0.97 \[ \frac {{\left (2 \, a + b\right )} f x - b \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 26, normalized size = 0.87 \[ \frac {1}{4} \, b {\left (2 \, x - \frac {\sin \left (2 \, f x + 2 \, e\right )}{f}\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 32, normalized size = 1.07 \[ a x +\frac {b \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 29, normalized size = 0.97 \[ a x + \frac {{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.65, size = 27, normalized size = 0.90 \[ -\frac {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{4}-f\,x\,\left (a+\frac {b}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 51, normalized size = 1.70 \[ a x + b \left (\begin {cases} \frac {x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {\sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \sin ^{2}{\relax (e )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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