Optimal. Leaf size=100 \[ -\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-a^3 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-a^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3473
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc (c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)+a^3 \cot ^2(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^2(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} a^3 \int \csc ^3(c+d x) \, dx-a^3 \int 1 \, dx-\frac {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-a^3 x+\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{8} a^3 \int \csc (c+d x) \, dx\\ &=-a^3 x+\frac {13 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.56, size = 133, normalized size = 1.33 \[ \frac {a^3 \left (-22 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+22 \sec ^2\left (\frac {1}{2} (c+d x)\right )-\left ((4 \sin (c+d x)+1) \csc ^4\left (\frac {1}{2} (c+d x)\right )\right )-8 \left (13 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-13 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+8 c+8 d x\right )\right )}{64 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.52, size = 190, normalized size = 1.90 \[ -\frac {16 \, a^{3} d x \cos \left (d x + c\right )^{4} - 32 \, a^{3} d x \cos \left (d x + c\right )^{2} - 22 \, a^{3} \cos \left (d x + c\right )^{3} + 16 \, a^{3} d x + 16 \, a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 26 \, a^{3} \cos \left (d x + c\right ) - 13 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 174, normalized size = 1.74 \[ \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 192 \, {\left (d x + c\right )} a^{3} - 312 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {650 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 141, normalized size = 1.41 \[ -a^{3} x -\frac {a^{3} \cot \left (d x +c \right )}{d}-\frac {a^{3} c}{d}-\frac {13 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {13 a^{3} \cos \left (d x +c \right )}{8 d}-\frac {13 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{3}}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 147, normalized size = 1.47 \[ -\frac {16 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.89, size = 237, normalized size = 2.37 \[ \frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+13\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{13\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {13\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
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]
________________________________________________________________________________________