Integrand size = 12, antiderivative size = 97 \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\frac {a x}{a^2+b^2}+\frac {2 a c \text {arctanh}\left (\frac {b-(a-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (a^2+b^2\right ) \sqrt {a^2+b^2-c^2}}+\frac {b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2} \]
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Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3238, 3217, 3203, 632, 212} \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\frac {2 a c \text {arctanh}\left (\frac {b-(a-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (a^2+b^2\right ) \sqrt {a^2+b^2-c^2}}+\frac {b \log (a \cos (x)+b \sin (x)+c)}{a^2+b^2}+\frac {a x}{a^2+b^2} \]
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Rule 212
Rule 632
Rule 3203
Rule 3217
Rule 3238
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x)}{c+a \cos (x)+b \sin (x)} \, dx \\ & = \frac {a x}{a^2+b^2}+\frac {b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2}-\frac {(a c) \int \frac {1}{c+a \cos (x)+b \sin (x)} \, dx}{a^2+b^2} \\ & = \frac {a x}{a^2+b^2}+\frac {b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2}-\frac {(2 a c) \text {Subst}\left (\int \frac {1}{a+c+2 b x+(-a+c) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = \frac {a x}{a^2+b^2}+\frac {b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2}+\frac {(4 a c) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2-c^2\right )-x^2} \, dx,x,2 b+2 (-a+c) \tan \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = \frac {a x}{a^2+b^2}+\frac {2 a c \text {arctanh}\left (\frac {b-(a-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (a^2+b^2\right ) \sqrt {a^2+b^2-c^2}}+\frac {b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\frac {a x+\frac {2 a c \text {arctanh}\left (\frac {b+(-a+c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}}+b \log (c+a \cos (x)+b \sin (x))}{a^2+b^2} \]
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Time = 0.77 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.76
method | result | size |
default | \(\frac {\frac {2 \left (a b -c b \right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -c \tan \left (\frac {x}{2}\right )^{2}-2 b \tan \left (\frac {x}{2}\right )-a -c \right )}{2 a -2 c}+\frac {2 \left (a c -b^{2}+\frac {\left (a b -c b \right ) b}{a -c}\right ) \arctan \left (\frac {2 \left (a -c \right ) \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {-a^{2}-b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}-b^{2}+c^{2}}}}{a^{2}+b^{2}}+\frac {-b \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )+2 a \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}+b^{2}}\) | \(171\) |
risch | \(\text {Expression too large to display}\) | \(1314\) |
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (93) = 186\).
Time = 0.32 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.70 \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\left [\frac {\sqrt {a^{2} + b^{2} - c^{2}} a c \log \left (\frac {2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4} - {\left (a^{2} - b^{2}\right )} c^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} c \cos \left (x\right ) - {\left (a^{4} - b^{4} - 2 \, {\left (a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (x\right )^{2} + 2 \, {\left ({\left (a^{2} b + b^{3}\right )} c - {\left (a^{3} b + a b^{3} - 2 \, a b c^{2}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) + 2 \, {\left (2 \, a b c \cos \left (x\right )^{2} - a b c + {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) - {\left (a^{3} + a b^{2} + {\left (a^{2} - b^{2}\right )} c \cos \left (x\right )\right )} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2} - c^{2}}}{2 \, a c \cos \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + b c\right )} \sin \left (x\right )}\right ) + 2 \, {\left (a^{3} + a b^{2} - a c^{2}\right )} x + {\left (a^{2} b + b^{3} - b c^{2}\right )} \log \left (2 \, a c \cos \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + b c\right )} \sin \left (x\right )\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} - {\left (a^{2} + b^{2}\right )} c^{2}\right )}}, -\frac {2 \, \sqrt {-a^{2} - b^{2} + c^{2}} a c \arctan \left (\frac {{\left (a c \cos \left (x\right ) + b c \sin \left (x\right ) + a^{2} + b^{2}\right )} \sqrt {-a^{2} - b^{2} + c^{2}}}{{\left (a^{2} b + b^{3} - b c^{2}\right )} \cos \left (x\right ) - {\left (a^{3} + a b^{2} - a c^{2}\right )} \sin \left (x\right )}\right ) - 2 \, {\left (a^{3} + a b^{2} - a c^{2}\right )} x - {\left (a^{2} b + b^{3} - b c^{2}\right )} \log \left (2 \, a c \cos \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + b c\right )} \sin \left (x\right )\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} - {\left (a^{2} + b^{2}\right )} c^{2}\right )}}\right ] \]
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\[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\int \frac {1}{a + b \tan {\left (x \right )} + c \sec {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.63 \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=-\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, c\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - c \tan \left (\frac {1}{2} \, x\right ) - b}{\sqrt {-a^{2} - b^{2} + c^{2}}}\right )\right )} a c}{{\left (a^{2} + b^{2}\right )} \sqrt {-a^{2} - b^{2} + c^{2}}} + \frac {a x}{a^{2} + b^{2}} + \frac {b \log \left (-a \tan \left (\frac {1}{2} \, x\right )^{2} + c \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a + c\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}{a^{2} + b^{2}} \]
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Time = 42.59 (sec) , antiderivative size = 988, normalized size of antiderivative = 10.19 \[ \int \frac {1}{a+c \sec (x)+b \tan (x)} \, dx=\frac {\ln \left (32\,a\,c-32\,c^2+32\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )+\frac {\left (32\,a^2\,b-32\,b\,c^2+32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )\,\left (-a^2+2\,a\,c+3\,b^2-2\,c^2\right )-\frac {\left (a^2\,b-b\,c^2+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (32\,a^4-64\,a^3\,c-64\,a^2\,b^2+32\,a^2\,c^2-32\,b^2\,c^2+96\,a\,b^2\,c+32\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )\,\left (4\,a^2-4\,c\,a+b^2\right )+\frac {32\,\left (a-c\right )\,\left (a^2\,b-b\,c^2+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (3\,a^3\,b-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^3\,c+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,b^2+a^2\,b\,c+2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,c^2+3\,a\,b^3-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a\,b^2\,c-4\,a\,b\,c^2+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^4+b^3\,c-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^2\,c^2\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (a^2\,b-b\,c^2+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (b\,\left (a^2-c^2\right )+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )}{c^2\,\left (a^2+b^2-c^2\right )+{\left (a^2+b^2-c^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )}{b+a\,1{}\mathrm {i}}+\frac {\ln \left (32\,a\,c-32\,c^2+32\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )+\frac {\left (32\,a^2\,b-32\,b\,c^2+32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )\,\left (-a^2+2\,a\,c+3\,b^2-2\,c^2\right )-\frac {\left (a^2\,b-b\,c^2+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (32\,a^4-64\,a^3\,c-64\,a^2\,b^2+32\,a^2\,c^2-32\,b^2\,c^2+96\,a\,b^2\,c+32\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a-c\right )\,\left (4\,a^2-4\,c\,a+b^2\right )+\frac {32\,\left (a-c\right )\,\left (a^2\,b-b\,c^2+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (3\,a^3\,b-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^3\,c+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,b^2+a^2\,b\,c+2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,c^2+3\,a\,b^3-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,a\,b^2\,c-4\,a\,b\,c^2+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^4+b^3\,c-2\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^2\,c^2\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (a^2\,b-b\,c^2+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (a^2+b^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (b\,\left (a^2-c^2\right )+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )}{c^2\,\left (a^2+b^2-c^2\right )+{\left (a^2+b^2-c^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a+b\,1{}\mathrm {i}} \]
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