Optimal. Leaf size=87 \[ \frac {35 a^4 x}{8}-\frac {8 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
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Rubi [A]
time = 0.06, 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 = {2724, 2718,
2715, 8, 2713} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2724
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 ) \text {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 = 0.27, size = 57, normalized size = 0.66 \begin {gather*} \frac {a^4 (-672 \cos (c+d x)+32 \cos (3 (c+d x))+3 (140 c+140 d x-56 \sin (2 (c+d x))+\sin (4 (c+d x))))}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 111, normalized size = 1.28
method | result | size |
risch | \(\frac {35 a^{4} x}{8}-\frac {7 a^{4} \cos \left (d x +c \right )}{d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}-\frac {7 a^{4} \sin \left (2 d x +2 c \right )}{4 d}\) | \(73\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-4 a^{4} \cos \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-4 a^{4} \cos \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
norman | \(\frac {\frac {35 a^{4} x}{8}-\frac {40 a^{4}}{3 d}-\frac {27 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {35 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {27 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {105 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {35 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {8 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {40 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {136 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 108, normalized size = 1.24 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 70, normalized size = 0.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs.
\(2 (82) = 164\).
time = 0.23, size = 224, normalized size = 2.57 \begin {gather*} \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 {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 72, normalized size = 0.83 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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