Optimal. Leaf size=51 \[ \frac {B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac {(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac {B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac {(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^3 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((A-B) (a+x)^3+\frac {B (a+x)^4}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {(A-B) (a+a \sin (c+d x))^4}{4 a d}+\frac {B (a+a \sin (c+d x))^5}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 36, normalized size = 0.71 \[ \frac {a^3 (\sin (c+d x)+1)^4 (5 A+4 B \sin (c+d x)-B)}{20 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 94, normalized size = 1.84 \[ \frac {5 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 40 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 4 \, {\left (B a^{3} \cos \left (d x + c\right )^{4} - {\left (5 \, A + 7 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, A + 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 116, normalized size = 2.27 \[ \frac {4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \, A a^{3} \sin \left (d x + c\right )^{4} + 15 \, B a^{3} \sin \left (d x + c\right )^{4} + 20 \, A a^{3} \sin \left (d x + c\right )^{3} + 20 \, B a^{3} \sin \left (d x + c\right )^{3} + 30 \, A a^{3} \sin \left (d x + c\right )^{2} + 10 \, B a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 98, normalized size = 1.92 \[ \frac {\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (a^{3} A +3 B \,a^{3}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 a^{3} A +3 B \,a^{3}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a^{3} A \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 84, normalized size = 1.65 \[ \frac {4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{4} + 20 \, {\left (A + B\right )} a^{3} \sin \left (d x + c\right )^{3} + 10 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.07, size = 81, normalized size = 1.59 \[ \frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^2\,\left (3\,A+B\right )}{2}+\frac {a^3\,{\sin \left (c+d\,x\right )}^4\,\left (A+3\,B\right )}{4}+\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}+A\,a^3\,\sin \left (c+d\,x\right )+a^3\,{\sin \left (c+d\,x\right )}^3\,\left (A+B\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.02, size = 151, normalized size = 2.96 \[ \begin {cases} \frac {A a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {A a^{3} \sin {\left (c + d x \right )}}{d} - \frac {3 A a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac {B a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 B a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac {B a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{3} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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