Integrand size = 31, antiderivative size = 95 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=-\frac {a C \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {2 \left (A b^2+a^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b} d}+\frac {C \tan (c+d x)}{b d} \]
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Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4168, 4083, 3855, 3916, 2738, 214} \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=\frac {2 \left (a^2 C+A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^2 d \sqrt {a-b} \sqrt {a+b}}-\frac {a C \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {C \tan (c+d x)}{b d} \]
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4168
Rubi steps \begin{align*} \text {integral}& = \frac {C \tan (c+d x)}{b d}+\frac {\int \frac {\sec (c+d x) (A b-a C \sec (c+d x))}{a+b \sec (c+d x)} \, dx}{b} \\ & = \frac {C \tan (c+d x)}{b d}-\frac {(a C) \int \sec (c+d x) \, dx}{b^2}+\left (A+\frac {a^2 C}{b^2}\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx \\ & = -\frac {a C \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {C \tan (c+d x)}{b d}+\frac {\left (A b^2+a^2 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^3} \\ & = -\frac {a C \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {C \tan (c+d x)}{b d}+\frac {\left (2 \left (A b^2+a^2 C\right )\right ) \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} (c+d x)\right )\right )}{b^3 d} \\ & = -\frac {a C \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {2 \left (A b^2+a^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b} d}+\frac {C \tan (c+d x)}{b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.82 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.48 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=\frac {2 \cos (c+d x) (b+a \cos (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \left (a C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-a C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 i \left (A b^2+a^2 C\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (\cos (c)-i \sin (c))}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {b C \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b C \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{b^2 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))} \]
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Time = 0.35 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {-\frac {C}{b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}-\frac {C}{b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {2 \left (-A \,b^{2}-C \,a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(137\) |
default | \(\frac {-\frac {C}{b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}-\frac {C}{b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {2 \left (-A \,b^{2}-C \,a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(137\) |
risch | \(\frac {2 i C}{d b \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C \,a^{2}}{\sqrt {a^{2}-b^{2}}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C \,a^{2}}{\sqrt {a^{2}-b^{2}}\, d \,b^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{b^{2} d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{b^{2} d}\) | \(360\) |
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (86) = 172\).
Time = 0.63 (sec) , antiderivative size = 424, normalized size of antiderivative = 4.46 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=\left [\frac {{\left (C a^{2} + A b^{2}\right )} \sqrt {a^{2} - b^{2}} \cos \left (d x + c\right ) \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (C a^{3} - C a b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{3} - C a b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} d \cos \left (d x + c\right )}, \frac {2 \, {\left (C a^{2} + A b^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right ) - {\left (C a^{3} - C a b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{3} - C a b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} d \cos \left (d x + c\right )}\right ] \]
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\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.35 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.72 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=-\frac {\frac {C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{2}} - \frac {C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{2}} + \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} b} + \frac {2 \, {\left (C a^{2} + A b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} b^{2}}}{d} \]
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Time = 18.81 (sec) , antiderivative size = 934, normalized size of antiderivative = 9.83 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx=-\frac {2\,C\,a\,\mathrm {atanh}\left (\frac {64\,C^3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,C^3\,a^3+128\,A\,C^2\,a^3-\frac {64\,C^3\,a^4}{b}+64\,A^2\,C\,a\,b^2-64\,A^2\,C\,a^2\,b-\frac {128\,A\,C^2\,a^4}{b}}+\frac {64\,C^3\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,C\,a^2\,b^2-64\,A^2\,C\,a\,b^3+128\,A\,C^2\,a^4-128\,A\,C^2\,a^3\,b+64\,C^3\,a^4-64\,C^3\,a^3\,b}+\frac {128\,A\,C^2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,C\,a^2\,b^2-64\,A^2\,C\,a\,b^3+128\,A\,C^2\,a^4-128\,A\,C^2\,a^3\,b+64\,C^3\,a^4-64\,C^3\,a^3\,b}+\frac {64\,A^2\,C\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,C\,a^2-\frac {64\,C^3\,a^3}{b}+\frac {64\,C^3\,a^4}{b^2}-64\,A^2\,C\,a\,b-\frac {128\,A\,C^2\,a^3}{b}+\frac {128\,A\,C^2\,a^4}{b^2}}+\frac {128\,A\,C^2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,C^3\,a^3+128\,A\,C^2\,a^3-\frac {64\,C^3\,a^4}{b}+64\,A^2\,C\,a\,b^2-64\,A^2\,C\,a^2\,b-\frac {128\,A\,C^2\,a^4}{b}}-\frac {64\,A^2\,C\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,C\,a^2-\frac {64\,C^3\,a^3}{b}+\frac {64\,C^3\,a^4}{b^2}-64\,A^2\,C\,a\,b-\frac {128\,A\,C^2\,a^3}{b}+\frac {128\,A\,C^2\,a^4}{b^2}}\right )}{b^2\,d}-\frac {\ln \left (b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sqrt {a^2-b^2}\right )\,\left (A\,b^2\,\sqrt {a^2-b^2}+C\,a^2\,\sqrt {a^2-b^2}\right )}{b^2\,d\,\left (a^2-b^2\right )}-\frac {\ln \left (\frac {\sqrt {\left (a+b\right )\,\left (a-b\right )}\,\left (C\,a^2+A\,b^2\right )\,\left (\frac {32\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-b\right )\,\left (A^2\,b^4+2\,A\,C\,a^2\,b^2+2\,C^2\,a^4-2\,C^2\,a^3\,b+C^2\,a^2\,b^2\right )}{b^2}+\frac {32\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\,\left (C\,a^2+A\,b^2\right )\,\left (a-b\right )\,\left (A\,a^2\,b^2-A\,b^4+C\,a\,b^3-C\,a^3\,b+2\,C\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2-b^2}+2\,A\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2-b^2}\right )}{\left (b^4-a^2\,b^2\right )\,\left (a+b\right )}\right )}{b^4-a^2\,b^2}+\frac {32\,C\,a\,\left (a-b\right )\,\left (A^2\,b^3+A\,C\,a^2\,b+A\,C\,a\,b^2+C^2\,a^3\right )}{b^3}\right )\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\,\left (C\,a^2+A\,b^2\right )}{d\,\left (b^4-a^2\,b^2\right )}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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