Integrand size = 29, antiderivative size = 100 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\frac {2 (a c-b d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} f}-\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))} \]
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Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4088, 12, 3916, 2738, 214} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\frac {2 (a c-b d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{f (a-b)^{3/2} (a+b)^{3/2}}-\frac {(b c-a d) \tan (e+f x)}{f \left (a^2-b^2\right ) (a+b \sec (e+f x))} \]
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Rule 12
Rule 214
Rule 2738
Rule 3916
Rule 4088
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))}+\frac {\int \frac {(-a c+b d) \sec (e+f x)}{a+b \sec (e+f x)} \, dx}{-a^2+b^2} \\ & = -\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))}+\frac {(a c-b d) \int \frac {\sec (e+f x)}{a+b \sec (e+f x)} \, dx}{a^2-b^2} \\ & = -\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))}+\frac {(a c-b d) \int \frac {1}{1+\frac {a \cos (e+f x)}{b}} \, dx}{b \left (a^2-b^2\right )} \\ & = -\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))}+\frac {(2 (a c-b d)) \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} (e+f x)\right )\right )}{b \left (a^2-b^2\right ) f} \\ & = \frac {2 (a c-b d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} f}-\frac {(b c-a d) \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\frac {-\frac {2 (a c-b d) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {(-b c+a d) \sin (e+f x)}{(a-b) (a+b) (b+a \cos (e+f x))}}{f} \]
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Time = 0.65 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}+\frac {2 \left (a c -b d \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{f}\) | \(132\) |
default | \(\frac {-\frac {2 \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}+\frac {2 \left (a c -b d \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{f}\) | \(132\) |
risch | \(\frac {2 i \left (a d -b c \right ) \left ({\mathrm e}^{i \left (f x +e \right )} b +a \right )}{a \left (a^{2}-b^{2}\right ) f \left ({\mathrm e}^{2 i \left (f x +e \right )} a +2 \,{\mathrm e}^{i \left (f x +e \right )} b +a \right )}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a c}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) b d}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right ) a c}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right ) b d}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}\) | \(396\) |
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Time = 0.28 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.94 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\left [\frac {{\left (a b c - b^{2} d + {\left (a^{2} c - a b d\right )} \cos \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (f x + e\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + b^{2}}\right ) - 2 \, {\left ({\left (a^{2} b - b^{3}\right )} c - {\left (a^{3} - a b^{2}\right )} d\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} f \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} f\right )}}, \frac {{\left (a b c - b^{2} d + {\left (a^{2} c - a b d\right )} \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (f x + e\right )}\right ) - {\left ({\left (a^{2} b - b^{3}\right )} c - {\left (a^{3} - a b^{2}\right )} d\right )} \sin \left (f x + e\right )}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} f \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} f}\right ] \]
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\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\int \frac {\left (c + d \sec {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}}{\left (a + b \sec {\left (e + f x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.73 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )} {\left (a c - b d\right )}}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )} {\left (a^{2} - b^{2}\right )}}\right )}}{f} \]
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Time = 13.75 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {a-b}}{\sqrt {a+b}}\right )\,\left (a\,c-b\,d\right )}{f\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}+\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,d-b\,c\right )}{f\,\left (a+b\right )\,\left (a-b\right )\,\left (\left (b-a\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a+b\right )} \]
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