Optimal. Leaf size=133 \[ \frac {a^3 \sin ^6(c+d x)}{6 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^4(c+d x)}{4 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 133, 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} \[ \frac {a^3 \sin ^6(c+d x)}{6 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^4(c+d x)}{4 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^2 (a+x)^5}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^5}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^5+\frac {a^7}{x^2}+\frac {3 a^6}{x}-5 a^4 x-5 a^3 x^2+a^2 x^3+3 a x^4+x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}+\frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 86, normalized size = 0.65 \[ -\frac {a^3 \left (-10 \sin ^6(c+d x)-36 \sin ^5(c+d x)-15 \sin ^4(c+d x)+100 \sin ^3(c+d x)+150 \sin ^2(c+d x)-60 \sin (c+d x)+60 \csc (c+d x)-180 \log (\sin (c+d x))\right )}{60 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 131, normalized size = 0.98 \[ -\frac {144 \, a^{3} \cos \left (d x + c\right )^{6} - 32 \, a^{3} \cos \left (d x + c\right )^{4} - 128 \, a^{3} \cos \left (d x + c\right )^{2} - 720 \, a^{3} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 256 \, a^{3} + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} - 72 \, a^{3} \cos \left (d x + c\right )^{2} + 47 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 120, normalized size = 0.90 \[ \frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, {\left (3 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 147, normalized size = 1.11 \[ -\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{6 d}-\frac {16 a^{3} \sin \left (d x +c \right )}{15 d}-\frac {2 a^{3} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}-\frac {8 a^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15 d}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{4 d}+\frac {3 a^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 107, normalized size = 0.80 \[ \frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3}}{\sin \left (d x + c\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.18, size = 371, normalized size = 2.79 \[ \frac {14\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {10\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {8\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3\,d}-\frac {28\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}+\frac {32\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d}-\frac {32\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3\,d}-\frac {3\,a^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {3\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {46\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {688\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {1064\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {288\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {96\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {3\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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]
________________________________________________________________________________________