Integrand size = 19, antiderivative size = 184 \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
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Time = 0.40 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3918, 4088, 12, 3916, 2738, 214} \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {5 a b \tan (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {b \tan (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3} \]
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Rule 12
Rule 214
Rule 2738
Rule 3916
Rule 3918
Rule 4088
Rubi steps \begin{align*} \text {integral}& = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\sec (c+d x) (-3 a+2 b \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )} \\ & = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) \left (2 \left (3 a^2+2 b^2\right )-5 a b \sec (c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2} \\ & = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int -\frac {3 a \left (2 a^2+3 b^2\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3} \\ & = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3} \\ & = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b \left (a^2-b^2\right )^3} \\ & = -\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b \left (a^2-b^2\right )^3 d} \\ & = \frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b \tan (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {5 a b \tan (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \tan (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=-\frac {\frac {12 a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {b \left (18 a^4+17 a^2 b^2+10 b^4+6 a b \left (9 a^2+b^2\right ) \cos (c+d x)+\left (18 a^4-5 a^2 b^2+2 b^4\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{(b+a \cos (c+d x))^3}}{12 (a-b)^3 (a+b)^3 d} \]
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Time = 0.70 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (6 a^{2}+3 a b +2 b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (9 a^{2}+b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-3 a b +2 b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(284\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (6 a^{2}+3 a b +2 b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (9 a^{2}+b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-3 a b +2 b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(284\) |
risch | \(\frac {i b \left (27 a^{6} b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 a^{4} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+6 a^{2} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+18 a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+81 a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-36 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+12 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+108 a^{6} b \,{\mathrm e}^{3 i \left (d x +c \right )}+42 a^{4} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-8 a^{2} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+8 b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+36 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+120 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+12 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+81 a^{6} b \,{\mathrm e}^{i \left (d x +c \right )}-12 a^{4} b^{3} {\mathrm e}^{i \left (d x +c \right )}+6 a^{2} b^{5} {\mathrm e}^{i \left (d x +c \right )}+18 a^{7}-5 a^{5} b^{2}+2 b^{4} a^{3}\right )}{3 a^{3} \left (-a^{2}+b^{2}\right )^{3} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {3 a \,b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}\) | \(692\) |
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Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (169) = 338\).
Time = 0.33 (sec) , antiderivative size = 905, normalized size of antiderivative = 4.92 \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\left [-\frac {3 \, {\left (2 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (2 \, a^{6} + 3 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 2 \, {\left (11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7} + {\left (18 \, a^{6} b - 23 \, a^{4} b^{3} + 7 \, a^{2} b^{5} - 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d\right )}}, \frac {3 \, {\left (2 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (2 \, a^{6} + 3 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7} + {\left (18 \, a^{6} b - 23 \, a^{4} b^{3} + 7 \, a^{2} b^{5} - 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d\right )}}\right ] \]
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\[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
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Exception generated. \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (169) = 338\).
Time = 0.36 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.19 \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=-\frac {\frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {18 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}}}{3 \, d} \]
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Time = 17.86 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.05 \[ \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+3\,b^2\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,\left (2\,a^3+3\,a\,b^2\right )\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}}\right )\,\left (2\,a^2+3\,b^2\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+3\,a\,b^2+2\,b^3\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,a^2\,b+b^3\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2\,b-3\,a\,b^2+2\,b^3\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )+3\,a\,b^2+3\,a^2\,b+a^3+b^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )} \]
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