Optimal. Leaf size=138 \[ \frac {\left (a^2-2 b^2\right ) \sin ^6(c+d x)}{6 d}-\frac {\left (2 a^2-b^2\right ) \sin ^4(c+d x)}{4 d}+\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a b \sin ^7(c+d x)}{7 d}-\frac {4 a b \sin ^5(c+d x)}{5 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {b^2 \sin ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.13, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2837, 12, 772} \[ \frac {\left (a^2-2 b^2\right ) \sin ^6(c+d x)}{6 d}-\frac {\left (2 a^2-b^2\right ) \sin ^4(c+d x)}{4 d}+\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a b \sin ^7(c+d x)}{7 d}-\frac {4 a b \sin ^5(c+d x)}{5 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {b^2 \sin ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+x)^2 \left (b^2-x^2\right )^2}{b} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int x (a+x)^2 \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 b^4 x+2 a b^4 x^2+b^2 \left (-2 a^2+b^2\right ) x^3-4 a b^2 x^4+\left (a^2-2 b^2\right ) x^5+2 a x^6+x^7\right ) \, dx,x,b \sin (c+d x)\right )}{b^6 d}\\ &=\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}-\frac {\left (2 a^2-b^2\right ) \sin ^4(c+d x)}{4 d}-\frac {4 a b \sin ^5(c+d x)}{5 d}+\frac {\left (a^2-2 b^2\right ) \sin ^6(c+d x)}{6 d}+\frac {2 a b \sin ^7(c+d x)}{7 d}+\frac {b^2 \sin ^8(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 138, normalized size = 1.00 \[ -\frac {840 \left (10 a^2+3 b^2\right ) \cos (2 (c+d x))+420 \left (8 a^2+b^2\right ) \cos (4 (c+d x))+560 a^2 \cos (6 (c+d x))-16800 a b \sin (c+d x)+1120 a b \sin (3 (c+d x))+2016 a b \sin (5 (c+d x))+480 a b \sin (7 (c+d x))-280 b^2 \cos (6 (c+d x))-105 b^2 \cos (8 (c+d x))-2590 b^2}{107520 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 85, normalized size = 0.62 \[ \frac {105 \, b^{2} \cos \left (d x + c\right )^{8} - 140 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{6} - 16 \, {\left (15 \, a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} - 4 \, a b \cos \left (d x + c\right )^{2} - 8 \, a b\right )} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 152, normalized size = 1.10 \[ \frac {b^{2} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a b \sin \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {3 \, a b \sin \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a b \sin \left (3 \, d x + 3 \, c\right )}{96 \, d} + \frac {5 \, a b \sin \left (d x + c\right )}{32 \, d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (8 \, a^{2} + b^{2}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (10 \, a^{2} + 3 \, b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 101, normalized size = 0.73 \[ \frac {-\frac {a^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{6}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 108, normalized size = 0.78 \[ \frac {105 \, b^{2} \sin \left (d x + c\right )^{8} + 240 \, a b \sin \left (d x + c\right )^{7} - 672 \, a b \sin \left (d x + c\right )^{5} + 140 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{3} - 210 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{4} + 420 \, a^{2} \sin \left (d x + c\right )^{2}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.47, size = 108, normalized size = 0.78 \[ \frac {{\sin \left (c+d\,x\right )}^6\,\left (\frac {a^2}{6}-\frac {b^2}{3}\right )-{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2}{2}-\frac {b^2}{4}\right )+\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {b^2\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {2\,a\,b\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {4\,a\,b\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {2\,a\,b\,{\sin \left (c+d\,x\right )}^7}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.70, size = 163, normalized size = 1.18 \[ \begin {cases} - \frac {a^{2} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {16 a b \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {8 a b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {2 a b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \sin {\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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