Optimal. Leaf size=77 \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^3}{3 b^3 d}-\frac {(a+b \sin (c+d x))^5}{5 b^3 d}+\frac {a (a+b \sin (c+d x))^4}{2 b^3 d} \]
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Rubi [A] time = 0.07, 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} \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^3}{3 b^3 d}-\frac {(a+b \sin (c+d x))^5}{5 b^3 d}+\frac {a (a+b \sin (c+d x))^4}{2 b^3 d} \]
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))^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^2 \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)^2+2 a (a+x)^3-(a+x)^4\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))^3}{3 b^3 d}+\frac {a (a+b \sin (c+d x))^4}{2 b^3 d}-\frac {(a+b \sin (c+d x))^5}{5 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 56, normalized size = 0.73 \[ \frac {(a+b \sin (c+d x))^3 \left (-a^2+3 a b \sin (c+d x)+3 b^2 \cos (2 (c+d x))+7 b^2\right )}{30 b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 69, normalized size = 0.90 \[ -\frac {15 \, a b \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 10 \, a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 80, normalized size = 1.04 \[ -\frac {6 \, b^{2} \sin \left (d x + c\right )^{5} + 15 \, a b \sin \left (d x + c\right )^{4} + 10 \, a^{2} \sin \left (d x + c\right )^{3} - 10 \, b^{2} \sin \left (d x + c\right )^{3} - 30 \, a b \sin \left (d x + c\right )^{2} - 30 \, a^{2} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 78, normalized size = 1.01 \[ \frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-\frac {a b \left (\cos ^{4}\left (d x +c \right )\right )}{2}+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 73, normalized size = 0.95 \[ -\frac {6 \, b^{2} \sin \left (d x + c\right )^{5} + 15 \, a b \sin \left (d x + c\right )^{4} - 30 \, a b \sin \left (d x + c\right )^{2} + 10 \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{3} - 30 \, a^{2} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 74, normalized size = 0.96 \[ -\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {a^2}{3}-\frac {b^2}{3}\right )-a^2\,\sin \left (c+d\,x\right )+\frac {b^2\,{\sin \left (c+d\,x\right )}^5}{5}-a\,b\,{\sin \left (c+d\,x\right )}^2+\frac {a\,b\,{\sin \left (c+d\,x\right )}^4}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.16, size = 107, normalized size = 1.39 \[ \begin {cases} \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a b \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac {2 b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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