Optimal. Leaf size=61 \[ -\frac {(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 a+3 b)-\frac {b \sin ^3(c+d x) \cos (c+d x)}{4 d} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ -\frac {(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 a+3 b)-\frac {b \sin ^3(c+d x) \cos (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3014
Rubi steps
\begin {align*} \int \sin ^2(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {b \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} (4 a+3 b) \int \sin ^2(c+d x) \, dx\\ &=-\frac {(4 a+3 b) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} (4 a+3 b) \int 1 \, dx\\ &=\frac {1}{8} (4 a+3 b) x-\frac {(4 a+3 b) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 45, normalized size = 0.74 \[ \frac {4 (4 a+3 b) (c+d x)-8 (a+b) \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 50, normalized size = 0.82 \[ \frac {{\left (4 \, a + 3 \, b\right )} d x + {\left (2 \, b \cos \left (d x + c\right )^{3} - {\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 43, normalized size = 0.70 \[ \frac {1}{8} \, {\left (4 \, a + 3 \, b\right )} x + \frac {b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {{\left (a + b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 65, normalized size = 1.07 \[ \frac {b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 74, normalized size = 1.21 \[ \frac {{\left (d x + c\right )} {\left (4 \, a + 3 \, b\right )} - \frac {{\left (4 \, a + 5 \, b\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, a + 3 \, b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.55, size = 68, normalized size = 1.11 \[ x\,\left (\frac {a}{2}+\frac {3\,b}{8}\right )-\frac {\left (\frac {a}{2}+\frac {5\,b}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {a}{2}+\frac {3\,b}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 158, normalized size = 2.59 \[ \begin {cases} \frac {a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a x \cos ^{2}{\left (c + d x \right )}}{2} - \frac {a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 b x \cos ^{4}{\left (c + d x \right )}}{8} - \frac {5 b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {3 b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\relax (c )}\right ) \sin ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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