Optimal. Leaf size=119 \[ -\frac {3 x}{a^3}+\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2844, 3056,
3047, 3102, 12, 2814, 2727} \begin {gather*} \frac {9 \sin (c+d x)}{5 a^3 d}+\frac {3 \sin (c+d x)}{d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {3 x}{a^3}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {3 \sin (c+d x) \cos ^2(c+d x)}{5 a d (a \cos (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2727
Rule 2814
Rule 2844
Rule 3047
Rule 3056
Rule 3102
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) (3 a-6 a \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (18 a^2-27 a^2 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {\int \frac {18 a^2 \cos (c+d x)-27 a^2 \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {\int \frac {45 a^3 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^5}\\ &=\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {3 \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^2}\\ &=-\frac {3 x}{a^3}+\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {3 \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^2}\\ &=-\frac {3 x}{a^3}+\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.56, size = 161, normalized size = 1.35 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )-12 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+96 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+20 \cos ^5\left (\frac {1}{2} (c+d x)\right ) (-3 d x+\sin (c+d x))+\cos \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )-12 \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )\right )}{5 a^3 d (1+\cos (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 85, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(85\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(85\) |
risch | \(-\frac {3 x}{a^{3}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {4 i \left (15 \,{\mathrm e}^{4 i \left (d x +c \right )}+50 \,{\mathrm e}^{3 i \left (d x +c \right )}+70 \,{\mathrm e}^{2 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}+12\right )}{5 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(112\) |
norman | \(\frac {-\frac {3 x}{a}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {45 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}+\frac {591 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {81 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {51 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}-\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 a d}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{20 a d}-\frac {12 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {18 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {12 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{2}}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 137, normalized size = 1.15 \begin {gather*} \frac {\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 126, normalized size = 1.06 \begin {gather*} -\frac {15 \, d x \cos \left (d x + c\right )^{3} + 45 \, d x \cos \left (d x + c\right )^{2} + 45 \, d x \cos \left (d x + c\right ) + 15 \, d x - {\left (5 \, \cos \left (d x + c\right )^{3} + 39 \, \cos \left (d x + c\right )^{2} + 57 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \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. 240 vs.
\(2 (109) = 218\).
time = 2.53, size = 240, normalized size = 2.02 \begin {gather*} \begin {cases} - \frac {60 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} - \frac {60 d x}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} - \frac {9 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {75 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {125 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\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 = 0.47, size = 96, normalized size = 0.81 \begin {gather*} -\frac {\frac {60 \, {\left (d x + c\right )}}{a^{3}} - \frac {40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 85 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.44, size = 113, normalized size = 0.95 \begin {gather*} \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-60\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{20\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________