Integrand size = 21, antiderivative size = 217 \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]
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Time = 0.74 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2881, 3134, 3080, 3855, 2738, 211} \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {\left (a^2-3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 d \left (a^2-b^2\right )} \]
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Rule 211
Rule 2738
Rule 2881
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (a^2-3 b^2-a b \cos (c+d x)+2 b^2 \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )} \\ & = \frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-2 b \left (2 a^2-3 b^2\right )+a \left (a^2+b^2\right ) \cos (c+d x)+b \left (a^2-3 b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )} \\ & = -\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (a^4+5 a^2 b^2-6 b^4+a b \left (a^2-3 b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )} \\ & = -\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (b^3 \left (4 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )}+\frac {\left (a^2+6 b^2\right ) \int \sec (c+d x) \, dx}{2 a^4} \\ & = \frac {\left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (2 b^3 \left (4 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right ) d} \\ & = -\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {b \left (2 a^2-3 b^2\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \\ \end{align*}
Time = 3.81 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {8 b^3 \left (-4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-2 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-12 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a b^4 \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-8 a b \tan (c+d x)}{4 a^4 d} \]
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Time = 1.83 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}-6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4}}-\frac {2 b^{3} \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {1}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4}}}{d}\) | \(273\) |
default | \(\frac {\frac {1}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}-6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4}}-\frac {2 b^{3} \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {1}{2 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4}}}{d}\) | \(273\) |
risch | \(-\frac {i \left (a^{3} {\mathrm e}^{5 i \left (d x +c \right )} b -3 b^{3} a \,{\mathrm e}^{5 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+8 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+10 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+7 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-9 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a +4 a^{2} b^{2}-6 b^{4}\right )}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (a^{2}-b^{2}\right ) \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{2} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{a^{4} d}\) | \(689\) |
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Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (204) = 408\).
Time = 0.71 (sec) , antiderivative size = 899, normalized size of antiderivative = 4.14 \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {2 \, {\left ({\left (4 \, a^{2} b^{4} - 3 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left ({\left (a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} - 2 \, {\left (2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right )^{2}\right )}}, -\frac {4 \, {\left ({\left (4 \, a^{2} b^{4} - 3 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left ({\left (a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} - 2 \, {\left (2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \]
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\[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.35 \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}} + \frac {4 \, {\left (4 \, a^{2} b^{3} - 3 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {{\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {{\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}}{2 \, d} \]
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Time = 21.33 (sec) , antiderivative size = 3699, normalized size of antiderivative = 17.05 \[ \int \frac {\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]
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