Integrand size = 28, antiderivative size = 153 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{2 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {\left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac {\sec ^3(c+d x)}{3 b d}-\frac {a \sec (c+d x) \tan (c+d x)}{2 b^2 d} \]
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Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3183, 3853, 3855, 3153, 212} \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {\left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}-\frac {a \text {arctanh}(\sin (c+d x))}{2 b^2 d}-\frac {a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac {\sec ^3(c+d x)}{3 b d} \]
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Rule 212
Rule 3153
Rule 3183
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x)}{3 b d}-\frac {a \int \sec ^3(c+d x) \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^2} \\ & = \frac {\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac {\sec ^3(c+d x)}{3 b d}-\frac {a \sec (c+d x) \tan (c+d x)}{2 b^2 d}-\frac {a \int \sec (c+d x) \, dx}{2 b^2}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \sec (c+d x) \, dx}{b^4}+\frac {\left (a^2+b^2\right )^2 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4} \\ & = -\frac {a \text {arctanh}(\sin (c+d x))}{2 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac {\sec ^3(c+d x)}{3 b d}-\frac {a \sec (c+d x) \tan (c+d x)}{2 b^2 d}-\frac {\left (a^2+b^2\right )^2 \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^4 d} \\ & = -\frac {a \text {arctanh}(\sin (c+d x))}{2 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {\left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac {\sec ^3(c+d x)}{3 b d}-\frac {a \sec (c+d x) \tan (c+d x)}{2 b^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(321\) vs. \(2(153)=306\).
Time = 2.15 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.10 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {48 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )+\sec ^3(c+d x) \left (12 a^2 b+20 b^3+12 b \left (a^2+b^2\right ) \cos (2 (c+d x))+6 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 a \left (2 a^2+3 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-6 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 a b^2 \sin (2 (c+d x))\right )}{24 b^4 d} \]
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Time = 1.00 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.76
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +3 b^{2}}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a +b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -3 b^{2}}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {2 \left (-a^{4}-2 a^{2} b^{2}-b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4} \sqrt {a^{2}+b^{2}}}}{d}\) | \(269\) |
default | \(\frac {-\frac {1}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +3 b^{2}}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a +b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -3 b^{2}}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {2 \left (-a^{4}-2 a^{2} b^{2}-b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4} \sqrt {a^{2}+b^{2}}}}{d}\) | \(269\) |
risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (3 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+20 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 i b a +6 a^{2}+6 b^{2}\right )}{3 d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{4} d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 b^{2} d}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (i a -b \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\right )}{d \,b^{4}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (i a -b \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\right )}{d \,b^{4}}-\frac {a^{3} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{4} d}-\frac {3 a \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 b^{2} d}\) | \(347\) |
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Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.69 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {6 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, b^{3} + 12 \, {\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, b^{4} d \cos \left (d x + c\right )^{3}} \]
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\[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (143) = 286\).
Time = 0.32 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.36 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (6 \, a^{2} + 8 \, b^{2} - \frac {3 \, a b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a b \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {12 \, {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{b^{3} - \frac {3 \, b^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, b^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {b^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{4}} + \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{4}} - \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}}}{6 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.82 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {\frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} + \frac {2 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{2} + 8 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} b^{3}}}{6 \, d} \]
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Time = 24.30 (sec) , antiderivative size = 724, normalized size of antiderivative = 4.73 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {b^3\,\left (\cos \left (c+d\,x\right )+\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {\cos \left (3\,c+3\,d\,x\right )}{3}+\frac {5}{6}\right )-b^2\,\left (\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{4}+\frac {9\,a\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}\right )+b\,\left (\frac {3\,a^2\,\cos \left (c+d\,x\right )}{4}+\frac {a^2}{2}+\frac {a^2\,\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {a^2\,\cos \left (3\,c+3\,d\,x\right )}{4}\right )+\frac {\mathrm {atanh}\left (\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{2}-\frac {3\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-\frac {a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {3\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{2}}{b^4\,d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
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