Optimal. Leaf size=95 \[ -\frac {\sin ^4(c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{a d}-\frac {2 \sin (c+d x)}{a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {\sin ^4(c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{a d}-\frac {2 \sin (c+d x)}{a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^3 (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a^3+\frac {a^5}{x^2}-\frac {a^4}{x}+2 a^2 x+a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}-\frac {2 \sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{a d}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^4(c+d x)}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 66, normalized size = 0.69 \[ -\frac {3 \sin ^4(c+d x)-4 \sin ^3(c+d x)-12 \sin ^2(c+d x)+24 \sin (c+d x)+12 \csc (c+d x)+12 \log (\sin (c+d x))}{12 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 85, normalized size = 0.89 \[ \frac {32 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (8 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{2} - 11\right )} \sin \left (d x + c\right ) - 96 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 256}{96 \, a d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 95, normalized size = 1.00 \[ -\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {12 \, {\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4} - 4 \, a^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{3} \sin \left (d x + c\right )^{2} + 24 \, a^{3} \sin \left (d x + c\right )}{a^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 94, normalized size = 0.99 \[ -\frac {\sin ^{4}\left (d x +c \right )}{4 d a}+\frac {\sin ^{3}\left (d x +c \right )}{3 d a}+\frac {\sin ^{2}\left (d x +c \right )}{a d}-\frac {2 \sin \left (d x +c \right )}{a d}-\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 74, normalized size = 0.78 \[ -\frac {\frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )}{a} + \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac {12}{a \sin \left (d x + c\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.36, size = 272, normalized size = 2.86 \[ \frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a\,d}-\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a\,d}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a\,d}-\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{a\,d}+\frac {\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{a\,d}-\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a\,d}+\frac {20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,a\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,a\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3\,a\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {9\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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]
________________________________________________________________________________________