Optimal. Leaf size=67 \[ \frac {(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac {(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac {(a \sin (c+d x)+a)^5}{5 a d} \]
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Rubi [A] time = 0.08, antiderivative size = 67, 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 = {2833, 12, 43} \[ \frac {(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac {(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac {(a \sin (c+d x)+a)^5}{5 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 (a+x)^4-2 a (a+x)^5+(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {(a+a \sin (c+d x))^5}{5 a d}-\frac {(a+a \sin (c+d x))^6}{3 a^2 d}+\frac {(a+a \sin (c+d x))^7}{7 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 80, normalized size = 1.19 \[ -\frac {a^4 (-7245 \sin (c+d x)+3395 \sin (3 (c+d x))-609 \sin (5 (c+d x))+15 \sin (7 (c+d x))+5460 \cos (2 (c+d x))-1680 \cos (4 (c+d x))+140 \cos (6 (c+d x))-630)}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 97, normalized size = 1.45 \[ -\frac {70 \, a^{4} \cos \left (d x + c\right )^{6} - 315 \, a^{4} \cos \left (d x + c\right )^{4} + 420 \, a^{4} \cos \left (d x + c\right )^{2} + {\left (15 \, a^{4} \cos \left (d x + c\right )^{6} - 171 \, a^{4} \cos \left (d x + c\right )^{4} + 332 \, a^{4} \cos \left (d x + c\right )^{2} - 176 \, a^{4}\right )} \sin \left (d x + c\right )}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 71, normalized size = 1.06 \[ \frac {15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 70, normalized size = 1.04 \[ \frac {\frac {a^{4} \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {2 a^{4} \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {6 a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+a^{4} \left (\sin ^{4}\left (d x +c \right )\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 1.06 \[ \frac {15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.41, size = 69, normalized size = 1.03 \[ \frac {\frac {a^4\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {2\,a^4\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {6\,a^4\,{\sin \left (c+d\,x\right )}^5}{5}+a^4\,{\sin \left (c+d\,x\right )}^4+\frac {a^4\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.24, size = 95, normalized size = 1.42 \[ \begin {cases} \frac {a^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac {2 a^{4} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {6 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{4} \sin ^{4}{\left (c + d x \right )}}{d} + \frac {a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{4} \sin ^{2}{\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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