Optimal. Leaf size=105 \[ -\frac {a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2950, 2709, 3767, 8, 3768, 3770} \[ -\frac {a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2709
Rule 2950
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^6(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\left (a^3 A^3\right ) \int \frac {\cot ^6(c+d x)}{(A-A \sin (c+d x))^2} \, dx\\ &=\frac {a^3 \int \left (-A^4 \csc ^2(c+d x)-2 A^4 \csc ^3(c+d x)+2 A^4 \csc ^5(c+d x)+A^4 \csc ^6(c+d x)\right ) \, dx}{A^3}\\ &=-\left (\left (a^3 A\right ) \int \csc ^2(c+d x) \, dx\right )+\left (a^3 A\right ) \int \csc ^6(c+d x) \, dx-\left (2 a^3 A\right ) \int \csc ^3(c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^5(c+d x) \, dx\\ &=\frac {a^3 A \cot (c+d x) \csc (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}-\left (a^3 A\right ) \int \csc (c+d x) \, dx+\frac {1}{2} \left (3 a^3 A\right ) \int \csc ^3(c+d x) \, dx+\frac {\left (a^3 A\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (a^3 A\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {a^3 A \cot ^5(c+d x)}{5 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac {1}{4} \left (3 a^3 A\right ) \int \csc (c+d x) \, dx\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {a^3 A \cot ^5(c+d x)}{5 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.07, size = 268, normalized size = 2.55 \[ a^3 A \left (-\frac {7 \tan \left (\frac {1}{2} (c+d x)\right )}{30 d}+\frac {7 \cot \left (\frac {1}{2} (c+d x)\right )}{30 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{16 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{16 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}+\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{160 d}-\frac {19 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{480 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{160 d}+\frac {19 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{480 d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 201, normalized size = 1.91 \[ \frac {56 \, A a^{3} \cos \left (d x + c\right )^{5} - 80 \, A a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left (A a^{3} \cos \left (d x + c\right )^{3} + A a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 174, normalized size = 1.66 \[ \frac {3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 90 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {274 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 90 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 25 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.72, size = 132, normalized size = 1.26 \[ \frac {7 a^{3} A \cot \left (d x +c \right )}{15 d}+\frac {a^{3} A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{4 d}-\frac {a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{4 d}-\frac {a^{3} A \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} A \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{5 d}-\frac {4 a^{3} A \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 175, normalized size = 1.67 \[ \frac {15 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 60 \, A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {120 \, A a^{3}}{\tan \left (d x + c\right )} - \frac {8 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} A a^{3}}{\tan \left (d x + c\right )^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.16, size = 244, normalized size = 2.32 \[ -\frac {A\,a^3\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-25\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+90\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-90\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+25\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}{480\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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]
________________________________________________________________________________________