Optimal. Leaf size=129 \[ \frac {11 a^2 x}{16}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2836, 2715, 8,
2713} \begin {gather*} \frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {11 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {11 a^2 x}{16} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 2836
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx &=\int \left (a^2 \cos ^4(c+d x)+2 a^2 \cos ^5(c+d x)+a^2 \cos ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^4(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^5(c+d x) \, dx\\ &=\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{4} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{6} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {1}{8} \left (3 a^2\right ) \int 1 \, dx+\frac {1}{8} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^2 x}{8}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {1}{16} \left (5 a^2\right ) \int 1 \, dx\\ &=\frac {11 a^2 x}{16}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 73, normalized size = 0.57 \begin {gather*} \frac {a^2 (660 d x+1200 \sin (c+d x)+465 \sin (2 (c+d x))+200 \sin (3 (c+d x))+75 \sin (4 (c+d x))+24 \sin (5 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 121, normalized size = 0.94
method | result | size |
risch | \(\frac {11 a^{2} x}{16}+\frac {5 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} \sin \left (5 d x +5 c \right )}{40 d}+\frac {5 a^{2} \sin \left (4 d x +4 c \right )}{64 d}+\frac {5 a^{2} \sin \left (3 d x +3 c \right )}{24 d}+\frac {31 a^{2} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(121\) |
default | \(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(121\) |
norman | \(\frac {\frac {11 a^{2} x}{16}+\frac {53 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {87 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {501 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {331 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {187 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {11 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {33 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {165 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {55 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {165 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {33 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {11 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 121, normalized size = 0.94 \begin {gather*} \frac {128 \, {\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^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 30 \, {\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^{2}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 89, normalized size = 0.69 \begin {gather*} \frac {165 \, a^{2} d x + {\left (40 \, a^{2} \cos \left (d x + c\right )^{5} + 96 \, a^{2} \cos \left (d x + c\right )^{4} + 110 \, a^{2} \cos \left (d x + c\right )^{3} + 128 \, a^{2} \cos \left (d x + c\right )^{2} + 165 \, a^{2} \cos \left (d x + c\right ) + 256 \, a^{2}\right )} \sin \left (d x + c\right )}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (122) = 244\).
time = 0.45, size = 343, normalized size = 2.66 \begin {gather*} \begin {cases} \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {16 a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {8 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {2 a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{2} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.48, size = 106, normalized size = 0.82 \begin {gather*} \frac {11}{16} \, a^{2} x + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{40 \, d} + \frac {5 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {5 \, a^{2} \sin \left (3 \, d x + 3 \, c\right )}{24 \, d} + \frac {31 \, a^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {5 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.86, size = 121, normalized size = 0.94 \begin {gather*} \frac {11\,a^2\,x}{16}+\frac {\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {187\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {331\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {501\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {53\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________