Integrand size = 28, antiderivative size = 151 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {6 a^2 b^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3169, 2717, 2718, 2672, 327, 212, 2670, 14, 294} \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {6 a^2 b^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}-\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d} \]
[In]
[Out]
Rule 14
Rule 212
Rule 294
Rule 327
Rule 2670
Rule 2672
Rule 2717
Rule 2718
Rule 3169
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \cos (c+d x)+4 a^3 b \sin (c+d x)+6 a^2 b^2 \sin (c+d x) \tan (c+d x)+4 a b^3 \sin (c+d x) \tan ^2(c+d x)+b^4 \sin (c+d x) \tan ^3(c+d x)\right ) \, dx \\ & = a^4 \int \cos (c+d x) \, dx+\left (4 a^3 b\right ) \int \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sin (c+d x) \tan (c+d x) \, dx+\left (4 a b^3\right ) \int \sin (c+d x) \tan ^2(c+d x) \, dx+b^4 \int \sin (c+d x) \tan ^3(c+d x) \, dx \\ & = -\frac {4 a^3 b \cos (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {4 a^3 b \cos (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {6 a^2 b^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {6 a^2 b^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d} \\ \end{align*}
Time = 3.55 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.77 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {16 a b^3-16 a b \left (a^2-b^2\right ) \cos (c+d x)-24 a^2 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 b^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a^2 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 b^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^4}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+32 a b^3 \sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-\frac {b^4}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+4 a^4 \sin (c+d x)-24 a^2 b^2 \sin (c+d x)+4 b^4 \sin (c+d x)}{4 d} \]
[In]
[Out]
Time = 1.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\sin \left (d x +c \right ) a^{4}-4 \cos \left (d x +c \right ) a^{3} b +6 a^{2} b^{2} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(157\) |
default | \(\frac {\sin \left (d x +c \right ) a^{4}-4 \cos \left (d x +c \right ) a^{3} b +6 a^{2} b^{2} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(157\) |
parts | \(\frac {a^{4} \sin \left (d x +c \right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {4 a^{3} b \cos \left (d x +c \right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {6 a^{2} b^{2} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(168\) |
parallelrisch | \(\frac {-12 \left (a +\frac {b}{2}\right ) b^{2} \left (a -\frac {b}{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 \left (a +\frac {b}{2}\right ) b^{2} \left (a -\frac {b}{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-8 a^{3} b +16 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (-4 a^{3} b +4 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (a^{4}-6 a^{2} b^{2}+3 b^{4}\right ) \sin \left (d x +c \right )-12 \cos \left (d x +c \right ) a^{3} b +28 \cos \left (d x +c \right ) a \,b^{3}-8 a^{3} b +16 a \,b^{3}}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(227\) |
risch | \(-\frac {2 \,{\mathrm e}^{i \left (d x +c \right )} a^{3} b}{d}+\frac {2 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{3}}{d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 d}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b^{2}}{d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 d}-\frac {2 \,{\mathrm e}^{-i \left (d x +c \right )} a^{3} b}{d}+\frac {2 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{3}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b^{2}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 d}+\frac {b^{3} {\mathrm e}^{i \left (d x +c \right )} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )} a +i b +8 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {6 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{d}-\frac {3 b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d}-\frac {6 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}+\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(356\) |
norman | \(\frac {\frac {8 a^{3} b +16 a \,b^{3}}{d}+\frac {16 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {\left (2 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (2 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {\left (2 a^{4}-12 a^{2} b^{2}+7 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {\left (2 a^{4}-12 a^{2} b^{2}+7 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {24 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {2 \left (2 a^{4}-12 a^{2} b^{2}-3 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {2 \left (2 a^{4}-12 a^{2} b^{2}-3 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 \left (12 a^{3} b +8 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}+\frac {2 \left (12 a^{3} b +16 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {\left (24 a^{3} b +16 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {3 b^{2} \left (4 a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {3 b^{2} \left (4 a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(444\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {16 \, a b^{3} \cos \left (d x + c\right ) - 16 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (b^{4} + 2 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
[In]
[Out]
Timed out. \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.94 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 16 \, a b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 12 \, a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 16 \, a^{3} b \cos \left (d x + c\right ) - 4 \, a^{4} \sin \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {4 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{3} b + 4 \, a b^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 24.83 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.46 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^4-24\,a^2\,b^2+2\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^4-12\,a^2\,b^2+3\,b^4\right )-16\,a\,b^3+8\,a^3\,b-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^4-12\,a^2\,b^2+3\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (16\,a\,b^3-16\,a^3\,b\right )+8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,b^4-12\,a^2\,b^2\right )}{d} \]
[In]
[Out]