Optimal. Leaf size=77 \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^4}{4 b^3 d}-\frac {(a+b \sin (c+d x))^6}{6 b^3 d}+\frac {2 a (a+b \sin (c+d x))^5}{5 b^3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 77, 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} \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^4}{4 b^3 d}-\frac {(a+b \sin (c+d x))^6}{6 b^3 d}+\frac {2 a (a+b \sin (c+d x))^5}{5 b^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^3(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 ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^3+2 a (a+x)^4-(a+x)^5\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^4}{4 b^3 d}+\frac {2 a (a+b \sin (c+d x))^5}{5 b^3 d}-\frac {(a+b \sin (c+d x))^6}{6 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 56, normalized size = 0.73 \[ \frac {(a+b \sin (c+d x))^4 \left (-a^2+4 a b \sin (c+d x)+5 b^2 \cos (2 (c+d x))+10 b^2\right )}{60 b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 95, normalized size = 1.23 \[ \frac {10 \, b^{3} \cos \left (d x + c\right )^{6} - 15 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (9 \, a b^{2} \cos \left (d x + c\right )^{4} - 10 \, a^{3} - 6 \, a b^{2} - {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 112, normalized size = 1.45 \[ -\frac {10 \, b^{3} \sin \left (d x + c\right )^{6} + 36 \, a b^{2} \sin \left (d x + c\right )^{5} + 45 \, a^{2} b \sin \left (d x + c\right )^{4} - 15 \, b^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 60 \, a b^{2} \sin \left (d x + c\right )^{3} - 90 \, a^{2} b \sin \left (d x + c\right )^{2} - 60 \, a^{3} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 115, normalized size = 1.49 \[ \frac {b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-\frac {3 a^{2} b \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 100, normalized size = 1.30 \[ -\frac {10 \, b^{3} \sin \left (d x + c\right )^{6} + 36 \, a b^{2} \sin \left (d x + c\right )^{5} - 90 \, a^{2} b \sin \left (d x + c\right )^{2} + 15 \, {\left (3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{4} - 60 \, a^{3} \sin \left (d x + c\right ) + 20 \, {\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{3}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.13, size = 98, normalized size = 1.27 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (a\,b^2-\frac {a^3}{3}\right )-{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,a^2\,b}{4}-\frac {b^3}{4}\right )+a^3\,\sin \left (c+d\,x\right )-\frac {b^3\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {3\,a^2\,b\,{\sin \left (c+d\,x\right )}^2}{2}-\frac {3\,a\,b^2\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.87, size = 151, normalized size = 1.96 \[ \begin {cases} \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {2 a b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{3} \cos ^{6}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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