Integrand size = 31, antiderivative size = 110 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{2} a^3 (6 A+7 B) x+\frac {a^3 (3 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 B \sin (c+d x)}{2 d}-\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3054, 3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {a^3 (3 A+B) \text {arctanh}(\sin (c+d x))}{d}-\frac {(2 A-B) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}+\frac {1}{2} a^3 x (6 A+7 B)+\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
[In]
[Out]
Rule 2814
Rule 3047
Rule 3054
Rule 3055
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+a \cos (c+d x))^2 (a (3 A+B)-a (2 A-B) \cos (c+d x)) \sec (c+d x) \, dx \\ & = -\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int (a+a \cos (c+d x)) \left (2 a^2 (3 A+B)+5 a^2 B \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (2 a^3 (3 A+B)+\left (5 a^3 B+2 a^3 (3 A+B)\right ) \cos (c+d x)+5 a^3 B \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {5 a^3 B \sin (c+d x)}{2 d}-\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (2 a^3 (3 A+B)+a^3 (6 A+7 B) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (6 A+7 B) x+\frac {5 a^3 B \sin (c+d x)}{2 d}-\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\left (a^3 (3 A+B)\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (6 A+7 B) x+\frac {a^3 (3 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 B \sin (c+d x)}{2 d}-\frac {(2 A-B) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^2 \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(272\) vs. \(2(110)=220\).
Time = 2.64 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.47 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{32} a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 (6 A+7 B) x-\frac {4 (3 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (3 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (A+3 B) \cos (d x) \sin (c)}{d}+\frac {B \cos (2 d x) \sin (2 c)}{d}+\frac {4 (A+3 B) \cos (c) \sin (d x)}{d}+\frac {B \cos (2 c) \sin (2 d x)}{d}+\frac {4 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
[In]
[Out]
Time = 2.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {A \,a^{3} \tan \left (d x +c \right )}{d}+\frac {\left (A \,a^{3}+3 B \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}\right ) \left (d x +c \right )}{d}+\frac {B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(120\) |
parallelrisch | \(-\frac {3 \left (\cos \left (d x +c \right ) \left (A +\frac {B}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\cos \left (d x +c \right ) \left (A +\frac {B}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {A}{6}-\frac {B}{2}\right ) \sin \left (2 d x +2 c \right )-\frac {B \sin \left (3 d x +3 c \right )}{24}-x \left (A +\frac {7 B}{6}\right ) d \cos \left (d x +c \right )-\frac {\sin \left (d x +c \right ) \left (A +\frac {B}{8}\right )}{3}\right ) a^{3}}{d \cos \left (d x +c \right )}\) | \(124\) |
derivativedivides | \(\frac {A \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 B \,a^{3} \sin \left (d x +c \right )+3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \left (d x +c \right )+A \,a^{3} \tan \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(128\) |
default | \(\frac {A \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 B \,a^{3} \sin \left (d x +c \right )+3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \left (d x +c \right )+A \,a^{3} \tan \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(128\) |
risch | \(3 a^{3} A x +\frac {7 a^{3} B x}{2}-\frac {i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,a^{3}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{3}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3}}{2 d}+\frac {i B \,a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i A \,a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}\) | \(240\) |
norman | \(\frac {\left (-\frac {7}{2} B \,a^{3}-3 A \,a^{3}\right ) x +\left (-\frac {21}{2} B \,a^{3}-9 A \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7}{2} B \,a^{3}+3 A \,a^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{2} B \,a^{3}+9 A \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 A \,a^{3}-7 B \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{3}+7 B \,a^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 B \,a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \left (A -3 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a^{3} \left (A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3} \left (4 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{3} \left (6 A -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{3} \left (3 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{3} \left (3 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(344\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {{\left (6 \, A + 7 \, B\right )} a^{3} d x \cos \left (d x + c\right ) + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (B a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 2 \, A a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
[In]
[Out]
\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a^{3} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.27 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {12 \, {\left (d x + c\right )} A a^{3} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 6 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a^{3} \sin \left (d x + c\right ) + 12 \, B a^{3} \sin \left (d x + c\right ) + 4 \, A a^{3} \tan \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.75 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=-\frac {\frac {4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - {\left (6 \, A a^{3} + 7 \, B a^{3}\right )} {\left (d x + c\right )} - 2 \, {\left (3 \, A a^{3} + B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 2 \, {\left (3 \, A a^{3} + B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.79 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {6\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
[In]
[Out]