Optimal. Leaf size=102 \[ \frac {a^4 \sin ^5(c+d x)}{5 d}-\frac {8 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac {7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^4 x}{2} \]
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Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {16 a^4 \sin ^3(c+d x)}{15 d}+\frac {32 a^4 \sin (c+d x)}{5 d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{5 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{10 d}+\frac {7 a^4 x}{2}+\frac {\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rule 2751
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx &=\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {4}{5} \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {4}{5} \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac {4 a^4 x}{5}+\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \left (4 a^4\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (16 a^4\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (16 a^4\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (24 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {4 a^4 x}{5}+\frac {16 a^4 \sin (c+d x)}{5 d}+\frac {12 a^4 \cos (c+d x) \sin (c+d x)}{5 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{5} \left (12 a^4\right ) \int 1 \, dx-\frac {\left (16 a^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {16 a^4 x}{5}+\frac {32 a^4 \sin (c+d x)}{5 d}+\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {16 a^4 \sin ^3(c+d x)}{15 d}+\frac {1}{10} \left (3 a^4\right ) \int 1 \, dx\\ &=\frac {7 a^4 x}{2}+\frac {32 a^4 \sin (c+d x)}{5 d}+\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac {(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {16 a^4 \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 63, normalized size = 0.62 \[ \frac {a^4 (1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x))+840 d x)}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 76, normalized size = 0.75 \[ \frac {105 \, a^{4} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{4} + 30 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 105 \, a^{4} \cos \left (d x + c\right ) + 166 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 89, normalized size = 0.87 \[ \frac {7}{2} \, a^{4} x + \frac {a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac {29 \, a^{4} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {2 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {49 \, a^{4} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 133, normalized size = 1.30 \[ \frac {\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 128, normalized size = 1.25 \[ \frac {8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 105, normalized size = 1.03 \[ \frac {7\,a^4\,x}{2}+\frac {7\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {98\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {896\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {158\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.17, size = 280, normalized size = 2.75 \[ \begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right )^{4} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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