Optimal. Leaf size=144 \[ \frac {\left (3 a^2-b^2\right ) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}{4 b^5 d}+\frac {(a+b \sin (c+d x))^8}{8 b^5 d}-\frac {4 a (a+b \sin (c+d x))^7}{7 b^5 d} \]
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Rubi [A] time = 0.13, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 697} \[ \frac {\left (3 a^2-b^2\right ) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}{4 b^5 d}+\frac {(a+b \sin (c+d x))^8}{8 b^5 d}-\frac {4 a (a+b \sin (c+d x))^7}{7 b^5 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^3 \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^3-4 \left (a^3-a b^2\right ) (a+x)^4+2 \left (3 a^2-b^2\right ) (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}{4 b^5 d}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac {\left (3 a^2-b^2\right ) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac {4 a (a+b \sin (c+d x))^7}{7 b^5 d}+\frac {(a+b \sin (c+d x))^8}{8 b^5 d}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 120, normalized size = 0.83 \[ \frac {\frac {1}{3} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^6+\frac {1}{4} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4+\frac {1}{8} (a+b \sin (c+d x))^8-\frac {4}{7} a (a+b \sin (c+d x))^7-\frac {4}{5} a (a-b) (a+b) (a+b \sin (c+d x))^5}{b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 117, normalized size = 0.81 \[ \frac {105 \, b^{3} \cos \left (d x + c\right )^{8} - 140 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{6} - 8 \, {\left (45 \, a b^{2} \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - 56 \, a^{3} - 24 \, a b^{2} - 4 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 185, normalized size = 1.28 \[ \frac {b^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {3 \, a b^{2} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (6 \, a^{2} b - b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (24 \, a^{2} b + b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, {\left (10 \, a^{2} b + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, a^{3} - 9 \, a b^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (20 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (8 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 135, normalized size = 0.94 \[ \frac {b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+3 a \,b^{2} \left (-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a^{2} b \left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 144, normalized size = 1.00 \[ \frac {105 \, b^{3} \sin \left (d x + c\right )^{8} + 360 \, a b^{2} \sin \left (d x + c\right )^{7} + 140 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{6} + 168 \, {\left (a^{3} - 6 \, a b^{2}\right )} \sin \left (d x + c\right )^{5} + 1260 \, a^{2} b \sin \left (d x + c\right )^{2} - 210 \, {\left (6 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{4} + 840 \, a^{3} \sin \left (d x + c\right ) - 280 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{3}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 141, normalized size = 0.98 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (a\,b^2-\frac {2\,a^3}{3}\right )-{\sin \left (c+d\,x\right )}^5\,\left (\frac {6\,a\,b^2}{5}-\frac {a^3}{5}\right )+{\sin \left (c+d\,x\right )}^6\,\left (\frac {a^2\,b}{2}-\frac {b^3}{3}\right )-{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,a^2\,b}{2}-\frac {b^3}{4}\right )+a^3\,\sin \left (c+d\,x\right )+\frac {b^3\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {3\,a^2\,b\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {3\,a\,b^2\,{\sin \left (c+d\,x\right )}^7}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.92, size = 202, normalized size = 1.40 \[ \begin {cases} \frac {8 a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{2} b \cos ^{6}{\left (c + d x \right )}}{2 d} + \frac {8 a b^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {4 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {b^{3} \cos ^{8}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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