Optimal. Leaf size=67 \[ \frac {4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (a+a \sin (c+d x))^6}{3 a^4 d}+\frac {(a+a \sin (c+d x))^7}{7 a^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45}
\begin {gather*} \frac {(a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac {4 (a \sin (c+d x)+a)^5}{5 a^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\text {Subst}\left (\int (a-x)^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^2 (a+x)^4-4 a (a+x)^5+(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (a+a \sin (c+d x))^6}{3 a^4 d}+\frac {(a+a \sin (c+d x))^7}{7 a^5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 58, normalized size = 0.87 \begin {gather*} -\frac {a^2 \cos ^6(c+d x) (1+\sin (c+d x))^2 \left (29-40 \sin (c+d x)+15 \sin ^2(c+d x)\right )}{105 d (-1+\sin (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.21, size = 99, normalized size = 1.48
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) a^{2}}{3}+\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(99\) |
default | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) a^{2}}{3}+\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(99\) |
risch | \(\frac {45 a^{2} \sin \left (d x +c \right )}{64 d}-\frac {a^{2} \sin \left (7 d x +7 c \right )}{448 d}-\frac {a^{2} \cos \left (6 d x +6 c \right )}{96 d}+\frac {a^{2} \sin \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \cos \left (4 d x +4 c \right )}{16 d}+\frac {19 a^{2} \sin \left (3 d x +3 c \right )}{192 d}-\frac {5 a^{2} \cos \left (2 d x +2 c \right )}{32 d}\) | \(118\) |
norman | \(\frac {\frac {4 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {28 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {194 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {1032 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {194 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {28 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {40 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(263\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 95, normalized size = 1.42 \begin {gather*} \frac {15 \, a^{2} \sin \left (d x + c\right )^{7} + 35 \, a^{2} \sin \left (d x + c\right )^{6} - 21 \, a^{2} \sin \left (d x + c\right )^{5} - 105 \, a^{2} \sin \left (d x + c\right )^{4} - 35 \, a^{2} \sin \left (d x + c\right )^{3} + 105 \, a^{2} \sin \left (d x + c\right )^{2} + 105 \, a^{2} \sin \left (d x + c\right )}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 71, normalized size = 1.06 \begin {gather*} -\frac {35 \, a^{2} \cos \left (d x + c\right )^{6} + {\left (15 \, a^{2} \cos \left (d x + c\right )^{6} - 24 \, a^{2} \cos \left (d x + c\right )^{4} - 32 \, a^{2} \cos \left (d x + c\right )^{2} - 64 \, a^{2}\right )} \sin \left (d x + c\right )}{105 \, 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. 158 vs.
\(2 (58) = 116\).
time = 0.66, size = 158, normalized size = 2.36 \begin {gather*} \begin {cases} \frac {8 a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {8 a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} + \frac {4 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{2} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.28, size = 117, normalized size = 1.75 \begin {gather*} -\frac {a^{2} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \cos \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac {5 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} - \frac {a^{2} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {19 \, a^{2} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {45 \, a^{2} \sin \left (d x + c\right )}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.54, size = 92, normalized size = 1.37 \begin {gather*} \frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a^2\,{\sin \left (c+d\,x\right )}^6}{3}-\frac {a^2\,{\sin \left (c+d\,x\right )}^5}{5}-a^2\,{\sin \left (c+d\,x\right )}^4-\frac {a^2\,{\sin \left (c+d\,x\right )}^3}{3}+a^2\,{\sin \left (c+d\,x\right )}^2+a^2\,\sin \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________