Integrand size = 19, antiderivative size = 73 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=a x+\frac {3 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {(8 a+3 b \sec (c+d x)) \tan (c+d x)}{8 d}+\frac {(4 a+3 b \sec (c+d x)) \tan ^3(c+d x)}{12 d} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3966, 3855} \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {\tan ^3(c+d x) (4 a+3 b \sec (c+d x))}{12 d}-\frac {\tan (c+d x) (8 a+3 b \sec (c+d x))}{8 d}+a x+\frac {3 b \text {arctanh}(\sin (c+d x))}{8 d} \]
[In]
[Out]
Rule 3855
Rule 3966
Rubi steps \begin{align*} \text {integral}& = \frac {(4 a+3 b \sec (c+d x)) \tan ^3(c+d x)}{12 d}-\frac {1}{4} \int (4 a+3 b \sec (c+d x)) \tan ^2(c+d x) \, dx \\ & = -\frac {(8 a+3 b \sec (c+d x)) \tan (c+d x)}{8 d}+\frac {(4 a+3 b \sec (c+d x)) \tan ^3(c+d x)}{12 d}+\frac {1}{8} \int (8 a+3 b \sec (c+d x)) \, dx \\ & = a x-\frac {(8 a+3 b \sec (c+d x)) \tan (c+d x)}{8 d}+\frac {(4 a+3 b \sec (c+d x)) \tan ^3(c+d x)}{12 d}+\frac {1}{8} (3 b) \int \sec (c+d x) \, dx \\ & = a x+\frac {3 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {(8 a+3 b \sec (c+d x)) \tan (c+d x)}{8 d}+\frac {(4 a+3 b \sec (c+d x)) \tan ^3(c+d x)}{12 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {a \arctan (\tan (c+d x))}{d}+\frac {3 b \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \tan (c+d x)}{d}+\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \sec (c+d x) \tan ^3(c+d x)}{d} \]
[In]
[Out]
Time = 1.50 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin \left (d x +c \right )^{5}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(104\) |
default | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin \left (d x +c \right )^{5}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(104\) |
parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {b \left (\frac {\sin \left (d x +c \right )^{5}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(109\) |
risch | \(a x +\frac {i \left (15 b \,{\mathrm e}^{7 i \left (d x +c \right )}-48 a \,{\mathrm e}^{6 i \left (d x +c \right )}-9 b \,{\mathrm e}^{5 i \left (d x +c \right )}-96 a \,{\mathrm e}^{4 i \left (d x +c \right )}+9 b \,{\mathrm e}^{3 i \left (d x +c \right )}-80 a \,{\mathrm e}^{2 i \left (d x +c \right )}-15 b \,{\mathrm e}^{i \left (d x +c \right )}-32 a \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(150\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {48 \, a d x \cos \left (d x + c\right )^{4} + 9 \, b \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, b \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (32 \, a \cos \left (d x + c\right )^{3} + 15 \, b \cos \left (d x + c\right )^{2} - 8 \, a \cos \left (d x + c\right ) - 6 \, b\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
[In]
[Out]
\[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \tan ^{4}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a + 3 \, b {\left (\frac {2 \, {\left (5 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (67) = 134\).
Time = 0.93 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.36 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {24 \, {\left (d x + c\right )} a + 9 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 9 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 104 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 33 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 104 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
[In]
[Out]
Time = 15.24 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.66 \[ \int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx=\frac {2\,a\,\mathrm {atan}\left (\frac {64\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^3+9\,a\,b^2}+\frac {9\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^3+9\,a\,b^2}\right )}{d}+\frac {3\,b\,\mathrm {atanh}\left (\frac {27\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (24\,a^2\,b+\frac {27\,b^3}{8}\right )}+\frac {24\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{24\,a^2\,b+\frac {27\,b^3}{8}}\right )}{4\,d}-\frac {\left (\frac {3\,b}{4}-2\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {26\,a}{3}-\frac {11\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {26\,a}{3}-\frac {11\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a+\frac {3\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
[In]
[Out]