Optimal. Leaf size=106 \[ \frac {5}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {4 \csc (c+d x)}{a^4 d}+\frac {9 \log (\sin (c+d x))}{a^4 d}-\frac {9 \log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 106, 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 = {2707, 77} \[ \frac {5}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {4 \csc (c+d x)}{a^4 d}+\frac {9 \log (\sin (c+d x))}{a^4 d}-\frac {9 \log (\sin (c+d x)+1)}{a^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{x^3 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^3}-\frac {4}{a^3 x^2}+\frac {9}{a^4 x}-\frac {2}{a^2 (a+x)^3}-\frac {5}{a^3 (a+x)^2}-\frac {9}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {4 \csc (c+d x)}{a^4 d}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {9 \log (\sin (c+d x))}{a^4 d}-\frac {9 \log (1+\sin (c+d x))}{a^4 d}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {5}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.80, size = 73, normalized size = 0.69 \[ \frac {\frac {10}{\sin (c+d x)+1}+\frac {2}{(\sin (c+d x)+1)^2}-\csc ^2(c+d x)+8 \csc (c+d x)+18 \log (\sin (c+d x))-18 \log (\sin (c+d x)+1)}{2 a^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 196, normalized size = 1.85 \[ -\frac {27 \, \cos \left (d x + c\right )^{2} - 18 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 18 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 26}{2 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 3 \, a^{4} d \cos \left (d x + c\right )^{2} + 2 \, a^{4} d - 2 \, {\left (a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.18, size = 185, normalized size = 1.75 \[ -\frac {\frac {144 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {108 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {4 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 272 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 402 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 272 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 101, normalized size = 0.95 \[ -\frac {1}{2 a^{4} d \sin \left (d x +c \right )^{2}}+\frac {4}{a^{4} d \sin \left (d x +c \right )}+\frac {9 \ln \left (\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {1}{a^{4} d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{a^{4} d \left (1+\sin \left (d x +c \right )\right )}-\frac {9 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 103, normalized size = 0.97 \[ \frac {\frac {18 \, \sin \left (d x + c\right )^{3} + 27 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 1}{a^{4} \sin \left (d x + c\right )^{4} + 2 \, a^{4} \sin \left (d x + c\right )^{3} + a^{4} \sin \left (d x + c\right )^{2}} - \frac {18 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {18 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.61, size = 228, normalized size = 2.15 \[ \frac {9\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^4\,d}-\frac {48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {129\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{2}}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {18\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________