Optimal. Leaf size=89 \[ -\frac {(6 a+5 b) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {(6 a+5 b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x (6 a+5 b)-\frac {b \sin ^5(c+d x) \cos (c+d x)}{6 d} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ -\frac {(6 a+5 b) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {(6 a+5 b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x (6 a+5 b)-\frac {b \sin ^5(c+d x) \cos (c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3014
Rubi steps
\begin {align*} \int \sin ^4(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {b \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{6} (6 a+5 b) \int \sin ^4(c+d x) \, dx\\ &=-\frac {(6 a+5 b) \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{8} (6 a+5 b) \int \sin ^2(c+d x) \, dx\\ &=-\frac {(6 a+5 b) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {(6 a+5 b) \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{16} (6 a+5 b) \int 1 \, dx\\ &=\frac {1}{16} (6 a+5 b) x-\frac {(6 a+5 b) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {(6 a+5 b) \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 70, normalized size = 0.79 \[ \frac {-3 (16 a+15 b) \sin (2 (c+d x))+(6 a+9 b) \sin (4 (c+d x))+72 a c+72 a d x-b \sin (6 (c+d x))+60 b c+60 b d x}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 69, normalized size = 0.78 \[ \frac {3 \, {\left (6 \, a + 5 \, b\right )} d x - {\left (8 \, b \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a + 13 \, b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (10 \, a + 11 \, b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 68, normalized size = 0.76 \[ \frac {1}{16} \, {\left (6 \, a + 5 \, b\right )} x - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (2 \, a + 3 \, b\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {{\left (16 \, a + 15 \, b\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 86, normalized size = 0.97 \[ \frac {b \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 104, normalized size = 1.17 \[ \frac {3 \, {\left (d x + c\right )} {\left (6 \, a + 5 \, b\right )} - \frac {3 \, {\left (10 \, a + 11 \, b\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (6 \, a + 5 \, b\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (6 \, a + 5 \, b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.94, size = 92, normalized size = 1.03 \[ x\,\left (\frac {3\,a}{8}+\frac {5\,b}{16}\right )-\frac {\left (\frac {5\,a}{8}+\frac {11\,b}{16}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (a+\frac {5\,b}{6}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {3\,a}{8}+\frac {5\,b}{16}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6+3\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.37, size = 258, normalized size = 2.90 \[ \begin {cases} \frac {3 a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cos ^{4}{\left (c + d x \right )}}{8} - \frac {5 a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b x \cos ^{6}{\left (c + d x \right )}}{16} - \frac {11 b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\relax (c )}\right ) \sin ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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