Optimal. Leaf size=87 \[ \frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {8 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8} \]
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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, 2638, 2635, 8, 2633} \[ \frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {8 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2645
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^4 \, dx &=\int \left (a^4+4 a^4 \sin (c+d x)+6 a^4 \sin ^2(c+d x)+4 a^4 \sin ^3(c+d x)+a^4 \sin ^4(c+d x)\right ) \, dx\\ &=a^4 x+a^4 \int \sin ^4(c+d x) \, dx+\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx+\left (6 a^4\right ) \int \sin ^2(c+d x) \, dx\\ &=a^4 x-\frac {4 a^4 \cos (c+d x)}{d}-\frac {3 a^4 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \sin ^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,\cos (c+d x)\right )}{d}\\ &=4 a^4 x-\frac {8 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^4\right ) \int 1 \, dx\\ &=\frac {35 a^4 x}{8}-\frac {8 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 57, normalized size = 0.66 \[ \frac {a^4 (3 (-56 \sin (2 (c+d x))+\sin (4 (c+d x))+140 c+140 d x)-672 \cos (c+d x)+32 \cos (3 (c+d x)))}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 70, normalized size = 0.80 \[ \frac {32 \, a^{4} \cos \left (d x + c\right )^{3} + 105 \, a^{4} d x - 192 \, a^{4} \cos \left (d x + c\right ) + 3 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{3} - 29 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.53, size = 72, normalized size = 0.83 \[ \frac {35}{8} \, a^{4} x + \frac {a^{4} \cos \left (3 \, d x + 3 \, c\right )}{3 \, d} - \frac {7 \, a^{4} \cos \left (d x + c\right )}{d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 111, normalized size = 1.28 \[ \frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-4 a^{4} \cos \left (d x +c \right )+a^{4} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 108, normalized size = 1.24 \[ a^{4} x + \frac {4 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {3 \, {\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{2 \, d} - \frac {4 \, a^{4} \cos \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.59, size = 237, normalized size = 2.72 \[ \frac {35\,a^4\,x}{8}-\frac {\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {27\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {a^4\,\left (105\,c+105\,d\,x\right )}{24}-\frac {a^4\,\left (105\,c+105\,d\,x-320\right )}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{6}-\frac {a^4\,\left (420\,c+420\,d\,x-192\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{6}-\frac {a^4\,\left (420\,c+420\,d\,x-1088\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{4}-\frac {a^4\,\left (630\,c+630\,d\,x-960\right )}{24}\right )+\frac {27\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.89, size = 224, normalized size = 2.57 \[ \begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 a^{4} x \cos ^{2}{\left (c + d x \right )} + a^{4} x - \frac {5 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {8 a^{4} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{4} \cos {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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