Optimal. Leaf size=129 \[ -\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a b x}{8} \]
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Rubi [A] time = 0.18, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2862, 2669, 2635, 8} \[ -\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a b x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2862
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx &=-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{7} \int \cos ^4(c+d x) (2 b+2 a \sin (c+d x)) (a+b \sin (c+d x)) \, dx\\ &=-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{42} \int \cos ^4(c+d x) \left (14 a b+2 \left (a^2+6 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{3} (a b) \int \cos ^4(c+d x) \, dx\\ &=-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{4} (a b) \int \cos ^2(c+d x) \, dx\\ &=-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{8} (a b) \int 1 \, dx\\ &=\frac {a b x}{8}-\frac {\left (a^2+6 b^2\right ) \cos ^5(c+d x)}{105 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 132, normalized size = 1.02 \[ \frac {-105 \left (8 a^2+3 b^2\right ) \cos (c+d x)-105 \left (4 a^2+b^2\right ) \cos (3 (c+d x))-84 a^2 \cos (5 (c+d x))+210 a b \sin (2 (c+d x))-210 a b \sin (4 (c+d x))-70 a b \sin (6 (c+d x))+840 a b c+840 a b d x+21 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 85, normalized size = 0.66 \[ \frac {120 \, b^{2} \cos \left (d x + c\right )^{7} - 168 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, a b d x - 35 \, {\left (8 \, a b \cos \left (d x + c\right )^{5} - 2 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 141, normalized size = 1.09 \[ \frac {1}{8} \, a b x + \frac {b^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} - \frac {{\left (4 \, a^{2} - b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 105, normalized size = 0.81 \[ \frac {-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 81, normalized size = 0.63 \[ -\frac {672 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 96 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} b^{2}}{3360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.01, size = 256, normalized size = 1.98 \[ \frac {a\,b\,x}{8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^2-4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (4\,a^2+4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {4\,a^2}{5}+\frac {4\,b^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,a^2+8\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {22\,a^2}{5}-\frac {8\,b^2}{5}\right )+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {2\,a^2}{5}+\frac {4\,b^2}{35}-\frac {11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {31\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}-\frac {31\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.77, size = 223, normalized size = 1.73 \[ \begin {cases} - \frac {a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {a b x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a b x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {a b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 b^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \sin {\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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