Optimal. Leaf size=30 \[ a x+\frac {b x}{2}-\frac {b \cos (c+d x) \sin (c+d x)}{2 d} \]
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Rubi [A]
time = 0.01, 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 = {2715, 8}
\begin {gather*} a x-\frac {b \sin (c+d x) \cos (c+d x)}{2 d}+\frac {b x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(c+d x)\right ) \, dx &=a x+b \int \sin ^2(c+d x) \, dx\\ &=a x-\frac {b \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} b \int 1 \, dx\\ &=a x+\frac {b x}{2}-\frac {b \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.10 \begin {gather*} a x+\frac {b (c+d x)}{2 d}-\frac {b \sin (2 (c+d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 32, normalized size = 1.07
method | result | size |
risch | \(a x +\frac {b x}{2}-\frac {b \sin \left (2 d x +2 c \right )}{4 d}\) | \(24\) |
default | \(a x +\frac {b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(32\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(37\) |
norman | \(\frac {\left (a +\frac {b}{2}\right ) x +\frac {b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\left (a +\frac {b}{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a +b \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 29, normalized size = 0.97 \begin {gather*} a x + \frac {{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} b}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 29, normalized size = 0.97 \begin {gather*} \frac {{\left (2 \, a + b\right )} d x - b \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 [A]
time = 0.07, size = 51, normalized size = 1.70 \begin {gather*} a x + b \left (\begin {cases} \frac {x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {x \cos ^{2}{\left (c + d x \right )}}{2} - \frac {\sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sin ^{2}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, b {\left (2 \, x - \frac {\sin \left (2 \, d x + 2 \, c\right )}{d}\right )} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.40, size = 27, normalized size = 0.90 \begin {gather*} -\frac {\frac {b\,\sin \left (2\,c+2\,d\,x\right )}{4}-d\,x\,\left (a+\frac {b}{2}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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