Optimal. Leaf size=100 \[ -\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^4(c+d x)}{4 a d}+\frac {2 \csc ^3(c+d x)}{3 a d}-\frac {\csc ^2(c+d x)}{a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^4(c+d x)}{4 a d}+\frac {2 \csc ^3(c+d x)}{3 a d}-\frac {\csc ^2(c+d x)}{a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^6 (a-x)^3 (a+x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}-\frac {a^4}{x^5}-\frac {2 a^3}{x^4}+\frac {2 a^2}{x^3}+\frac {a}{x^2}-\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{a d}+\frac {2 \csc ^3(c+d x)}{3 a d}+\frac {\csc ^4(c+d x)}{4 a d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {\log (\sin (c+d x))}{a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 68, normalized size = 0.68 \[ -\frac {12 \csc ^5(c+d x)-15 \csc ^4(c+d x)-40 \csc ^3(c+d x)+60 \csc ^2(c+d x)+60 \csc (c+d x)+60 \log (\sin (c+d x))}{60 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 118, normalized size = 1.18 \[ -\frac {60 \, \cos \left (d x + c\right )^{4} + 60 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 80 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (4 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 32}{60 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 82, normalized size = 0.82 \[ -\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {137 \, \sin \left (d x + c\right )^{5} - 60 \, \sin \left (d x + c\right )^{4} - 60 \, \sin \left (d x + c\right )^{3} + 40 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 12}{a \sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.48, size = 97, normalized size = 0.97 \[ -\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{5 a d \sin \left (d x +c \right )^{5}}-\frac {1}{a d \sin \left (d x +c \right )^{2}}+\frac {1}{4 a d \sin \left (d x +c \right )^{4}}+\frac {2}{3 a d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 70, normalized size = 0.70 \[ -\frac {\frac {60 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac {60 \, \sin \left (d x + c\right )^{4} + 60 \, \sin \left (d x + c\right )^{3} - 40 \, \sin \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) + 12}{a \sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.24, size = 204, normalized size = 2.04 \[ \frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,a\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a\,d}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {1}{5}\right )}{32\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________