Optimal. Leaf size=77 \[ \frac {5 x}{2 a^3}+\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2759, 2761,
2715, 8} \begin {gather*} \frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {5 x}{2 a^3}+\frac {2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2715
Rule 2759
Rule 2761
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int \frac {\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int \cos ^2(c+d x) \, dx}{a^3}\\ &=\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int 1 \, dx}{2 a^3}\\ &=\frac {5 x}{2 a^3}+\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.31, size = 121, normalized size = 1.57 \begin {gather*} -\frac {\cos ^7(c+d x) \left (30 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-22+31 \sin (c+d x)-11 \sin ^2(c+d x)+2 \sin ^3(c+d x)\right )\right )}{6 a^3 d (-1+\sin (c+d x))^4 (1+\sin (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.28, size = 90, normalized size = 1.17
method | result | size |
risch | \(\frac {5 x}{2 a^{3}}+\frac {15 \cos \left (d x +c \right )}{4 a^{3} d}-\frac {\cos \left (3 d x +3 c \right )}{12 a^{3} d}-\frac {3 \sin \left (2 d x +2 c \right )}{4 a^{3} d}\) | \(56\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {11}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(90\) |
default | \(\frac {\frac {2 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {11}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(90\) |
norman | \(\frac {\frac {3055 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {640 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {100 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {673 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {40 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {1969 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {340 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2701 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {1253 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {200 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {289 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {22}{3 a d}+\frac {21 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {500 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3325 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {3197 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {200 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {340 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {101 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}+\frac {640 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {725 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {725 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2435 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {25 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {500 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {1699 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {25 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {169 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {69 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {341 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {100 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 x}{2 a}+\frac {5 x \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {40 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(637\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (71) = 142\).
time = 0.51, size = 184, normalized size = 2.39 \begin {gather*} -\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {9 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 22}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 45, normalized size = 0.58 \begin {gather*} -\frac {2 \, \cos \left (d x + c\right )^{3} - 15 \, d x + 9 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 24 \, \cos \left (d x + c\right )}{6 \, a^{3} 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. 690 vs.
\(2 (71) = 142\).
time = 48.30, size = 690, normalized size = 8.96 \begin {gather*} \begin {cases} \frac {15 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {45 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {45 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {15 d x}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {18 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {36 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {96 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} - \frac {18 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {44}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.83, size = 88, normalized size = 1.14 \begin {gather*} \frac {\frac {15 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.64, size = 57, normalized size = 0.74 \begin {gather*} \frac {5\,x}{2\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________