Integrand size = 28, antiderivative size = 141 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\left (a^2+b^2\right )^2}{a b^4 d (b+a \cot (c+d x))}-\frac {4 a \left (a^2+b^2\right ) \log (b+a \cot (c+d x))}{b^5 d}-\frac {4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac {\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}-\frac {a \tan ^2(c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \]
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Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3167, 908} \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}-\frac {4 a \left (a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}+\frac {\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right )^2}{a b^4 d (a \cot (c+d x)+b)}-\frac {a \tan ^2(c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \]
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Rule 908
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4 (b+a x)^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{b^2 x^4}-\frac {2 a}{b^3 x^3}+\frac {3 a^2+2 b^2}{b^4 x^2}-\frac {4 a \left (a^2+b^2\right )}{b^5 x}+\frac {\left (a^2+b^2\right )^2}{b^4 (b+a x)^2}+\frac {4 a^2 \left (a^2+b^2\right )}{b^5 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {\left (a^2+b^2\right )^2}{a b^4 d (b+a \cot (c+d x))}-\frac {4 a \left (a^2+b^2\right ) \log (b+a \cot (c+d x))}{b^5 d}-\frac {4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac {\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}-\frac {a \tan ^2(c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \\ \end{align*}
Time = 6.02 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\frac {b^4 \sec ^4(c+d x)}{3 (a+b \tan (c+d x))}+\frac {4}{3} \left (-a \left (\left (a^2+b^2\right ) \log (a+b \tan (c+d x))-a b \tan (c+d x)+\frac {1}{2} b^2 \tan ^2(c+d x)\right )+\left (a^2+b^2\right ) \left (-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)-\frac {a^2+b^2}{a+b \tan (c+d x)}\right )\right )}{b^5 d} \]
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Time = 1.82 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-a b \tan \left (d x +c \right )^{2}+3 \tan \left (d x +c \right ) a^{2}+2 \tan \left (d x +c \right ) b^{2}}{b^{4}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{b^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 a \left (a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5}}}{d}\) | \(114\) |
default | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-a b \tan \left (d x +c \right )^{2}+3 \tan \left (d x +c \right ) a^{2}+2 \tan \left (d x +c \right ) b^{2}}{b^{4}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{b^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 a \left (a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5}}}{d}\) | \(114\) |
risch | \(-\frac {8 i \left (-2 i a \,b^{2}+3 a^{2} b +2 b^{3}-3 i a^{3}-3 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-5 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-9 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-9 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) b^{4} d}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{5} d}+\frac {4 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{5} d}-\frac {4 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{3} d}\) | \(342\) |
norman | \(\frac {-\frac {2 \left (24 a^{2}+16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 b^{3} d}+\frac {4 \left (2 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b^{3} d}+\frac {4 \left (2 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{b^{3} d}-\frac {2 \left (36 a^{4}+44 a^{2} b^{2}+9 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d \,b^{4}}+\frac {2 \left (36 a^{4}+44 a^{2} b^{2}+9 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d \,b^{4}}+\frac {2 \left (4 a^{4}+4 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{4} d a}-\frac {2 \left (4 a^{4}+4 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{b^{4} d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}+\frac {4 a \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5} d}+\frac {4 a \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5} d}-\frac {4 a \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{5} d}\) | \(400\) |
parallelrisch | \(\frac {-16 \left (a^{2}+b^{2}\right ) a \left (a \cos \left (2 d x +2 c \right )+\frac {3 a}{4}+\frac {b \sin \left (2 d x +2 c \right )}{2}+\frac {a \cos \left (4 d x +4 c \right )}{4}+\frac {b \sin \left (4 d x +4 c \right )}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )+16 \left (a^{2}+b^{2}\right ) a \left (a \cos \left (2 d x +2 c \right )+\frac {3 a}{4}+\frac {b \sin \left (2 d x +2 c \right )}{2}+\frac {a \cos \left (4 d x +4 c \right )}{4}+\frac {b \sin \left (4 d x +4 c \right )}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 \left (a^{2}+b^{2}\right ) a \left (a \cos \left (2 d x +2 c \right )+\frac {3 a}{4}+\frac {b \sin \left (2 d x +2 c \right )}{2}+\frac {a \cos \left (4 d x +4 c \right )}{4}+\frac {b \sin \left (4 d x +4 c \right )}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {16 \left (-3 a^{4}-3 a^{2} b^{2}-b^{4}\right ) \cos \left (2 d x +2 c \right )}{3}+\frac {2 \left (-6 a^{4}-9 a^{2} b^{2}-4 b^{4}\right ) \cos \left (4 d x +4 c \right )}{3}-\frac {4 a \,b^{3} \sin \left (2 d x +2 c \right )}{3}+\frac {2 a \,b^{3} \sin \left (4 d x +4 c \right )}{3}-12 a^{4}-10 a^{2} b^{2}}{b^{5} d \left (2 b \sin \left (2 d x +2 c \right )+a \cos \left (4 d x +4 c \right )+4 a \cos \left (2 d x +2 c \right )+b \sin \left (4 d x +4 c \right )+3 a \right )}\) | \(400\) |
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (139) = 278\).
Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.99 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {4 \, {\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 6 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) + 2 \, {\left (a b^{3} \cos \left (d x + c\right ) - 2 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (a b^{5} d \cos \left (d x + c\right )^{4} + b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}}{b^{6} \tan \left (d x + c\right ) + a b^{5}} - \frac {b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{2} + 2 \, b^{2}\right )} \tan \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{3} + a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{5}}}{3 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (a^{3} + a b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} - \frac {b^{4} \tan \left (d x + c\right )^{3} - 3 \, a b^{3} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} \tan \left (d x + c\right ) + 6 \, b^{4} \tan \left (d x + c\right )}{b^{6}} - \frac {3 \, {\left (4 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 3 \, a^{4} + 2 \, a^{2} b^{2} - b^{4}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )} b^{5}}}{3 \, d} \]
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Time = 26.12 (sec) , antiderivative size = 1132, normalized size of antiderivative = 8.03 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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