Integrand size = 29, antiderivative size = 132 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^2 b \cot (c+d x)}{d}+\frac {3 a^3 \sec (c+d x)}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2991, 3852, 8, 2702, 327, 213, 2700, 14, 294} \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \sec (c+d x)}{2 d}-\frac {a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a^2 b \tan (c+d x)}{d}-\frac {3 a^2 b \cot (c+d x)}{d}-\frac {3 a b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d} \]
[In]
[Out]
Rule 8
Rule 14
Rule 213
Rule 294
Rule 327
Rule 2700
Rule 2702
Rule 2991
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \int \left (b^3 \sec ^2(c+d x)+3 a b^2 \csc (c+d x) \sec ^2(c+d x)+3 a^2 b \csc ^2(c+d x) \sec ^2(c+d x)+a^3 \csc ^3(c+d x) \sec ^2(c+d x)\right ) \, dx \\ & = a^3 \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+\left (3 a b^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx+b^3 \int \sec ^2(c+d x) \, dx \\ & = \frac {a^3 \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {b^3 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {3 a b^2 \sec (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \tan (c+d x)}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {3 a b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^2 b \cot (c+d x)}{d}+\frac {3 a^3 \sec (c+d x)}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^2 b \cot (c+d x)}{d}+\frac {3 a^3 \sec (c+d x)}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(132)=264\).
Time = 1.14 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.02 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\csc ^4(c+d x) \left (2 a^3+12 a b^2-6 \left (a^3+2 a b^2\right ) \cos (2 (c+d x))+3 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 a \left (a^2+2 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-3 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^2 b \sin (c+d x)+6 b^3 \sin (c+d x)-12 a^2 b \sin (3 (c+d x))-2 b^3 \sin (3 (c+d x))\right )}{2 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
[In]
[Out]
Time = 0.91 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \tan \left (d x +c \right )}{d}\) | \(130\) |
default | \(\frac {a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \tan \left (d x +c \right )}{d}\) | \(130\) |
parallelrisch | \(\frac {12 a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+12 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +\left (-72 a^{2} b -16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (a^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 a b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-18 a^{2}-48 b^{2}\right )}{8 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 d}\) | \(162\) |
risch | \(\frac {i \left (-3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-12 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{2} b +2 b^{3}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}\) | \(264\) |
norman | \(\frac {\frac {a^{3}}{8 d}+\frac {a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (3 a^{3}+12 a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {\left (21 a^{3}+48 a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (45 a^{3}+144 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \left (13 a^{2}+36 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {3 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {3 a^{2} b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {3 b \left (7 a^{2}+2 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 b \left (7 a^{2}+2 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \left (9 a^{2}+4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {b \left (9 a^{2}+4 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 a \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(359\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.48 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {4 \, a^{3} + 12 \, a b^{2} - 6 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (3 \, a^{2} b + b^{3} - {\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, a b^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{2} b {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + 4 \, b^{3} \tan \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.36 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, {\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {16 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} + 3 \, a b^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {18 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
[In]
[Out]
Time = 11.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.26 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{2}+3\,a\,b^2\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {17\,a^3}{2}+24\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (30\,a^2\,b+8\,b^3\right )-\frac {a^3}{2}-6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
[In]
[Out]