Integrand size = 21, antiderivative size = 73 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {13 a^4 x}{2}+\frac {4 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \tan (c+d x)}{d} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2836, 2717, 2715, 8, 3855, 3852} \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {4 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \tan (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {13 a^4 x}{2} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2717
Rule 2836
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (6 a^4+4 a^4 \cos (c+d x)+a^4 \cos ^2(c+d x)+4 a^4 \sec (c+d x)+a^4 \sec ^2(c+d x)\right ) \, dx \\ & = 6 a^4 x+a^4 \int \cos ^2(c+d x) \, dx+a^4 \int \sec ^2(c+d x) \, dx+\left (4 a^4\right ) \int \cos (c+d x) \, dx+\left (4 a^4\right ) \int \sec (c+d x) \, dx \\ & = 6 a^4 x+\frac {4 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^4 \int 1 \, dx-\frac {a^4 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {13 a^4 x}{2}+\frac {4 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(241\) vs. \(2(73)=146\).
Time = 3.00 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.30 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {1}{64} a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (26 x-\frac {16 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {16 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {16 \cos (d x) \sin (c)}{d}+\frac {\cos (2 d x) \sin (2 c)}{d}+\frac {16 \cos (c) \sin (d x)}{d}+\frac {\cos (2 c) \sin (2 d x)}{d}+\frac {4 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
[In]
[Out]
Time = 2.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} \sin \left (d x +c \right )+6 a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} \tan \left (d x +c \right )}{d}\) | \(82\) |
default | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} \sin \left (d x +c \right )+6 a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} \tan \left (d x +c \right )}{d}\) | \(82\) |
parts | \(\frac {a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{4} \left (d x +c \right )}{d}+\frac {4 a^{4} \sin \left (d x +c \right )}{d}\) | \(93\) |
parallelrisch | \(\frac {a^{4} \left (52 d x \cos \left (d x +c \right )-32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+9 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )+16 \sin \left (2 d x +2 c \right )\right )}{8 d \cos \left (d x +c \right )}\) | \(96\) |
risch | \(\frac {13 a^{4} x}{2}-\frac {i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {2 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(144\) |
norman | \(\frac {-\frac {13 a^{4} x}{2}-\frac {11 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {24 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {39 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-13 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {39 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {4 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {4 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(258\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.44 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {13 \, a^{4} d x \cos \left (d x + c\right ) + 4 \, a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} \cos \left (d x + c\right ) + 2 \, a^{4}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
[In]
[Out]
\[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=a^{4} \left (\int 4 \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 24 \, {\left (d x + c\right )} a^{4} + 8 \, a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{4} \sin \left (d x + c\right ) + 4 \, a^{4} \tan \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {13 \, {\left (d x + c\right )} a^{4} + 8 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (7 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 14.57 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx=\frac {13\,a^4\,x}{2}+\frac {8\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {-5\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+11\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
[In]
[Out]