Optimal. Leaf size=87 \[ -\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8} \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2645, 2637, 2635, 8, 2633} \[ -\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \, dx &=\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\\ &=a^4 x+a^4 \int \cos ^4(c+d x) \, dx+\left (4 a^4\right ) \int \cos (c+d x) \, dx+\left (4 a^4\right ) \int \cos ^3(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=a^4 x+\frac {4 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \cos ^2(c+d x) \, dx+\left (3 a^4\right ) \int 1 \, dx-\frac {\left (4 a^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=4 a^4 x+\frac {8 a^4 \sin (c+d x)}{d}+\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {1}{8} \left (3 a^4\right ) \int 1 \, dx\\ &=\frac {35 a^4 x}{8}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 56, normalized size = 0.64 \[ \frac {a^4 (672 \sin (c+d x)+168 \sin (2 (c+d x))+32 \sin (3 (c+d x))+3 \sin (4 (c+d x))+420 c+420 d x)}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 63, normalized size = 0.72 \[ \frac {105 \, a^{4} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} + 32 \, a^{4} \cos \left (d x + c\right )^{2} + 81 \, a^{4} \cos \left (d x + c\right ) + 160 \, a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 72, normalized size = 0.83 \[ \frac {35}{8} \, a^{4} x + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{4} \sin \left (3 \, d x + 3 \, c\right )}{3 \, d} + \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {7 \, a^{4} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 111, normalized size = 1.28 \[ \frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} \sin \left (d x +c \right )+a^{4} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 106, normalized size = 1.22 \[ a^{4} x - \frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{2 \, d} + \frac {4 \, a^{4} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 89, normalized size = 1.02 \[ \frac {35\,a^4\,x}{8}+\frac {\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {385\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {511\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}+\frac {93\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.08, size = 224, normalized size = 2.57 \[ \begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 a^{4} x \cos ^{2}{\left (c + d x \right )} + a^{4} x + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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