Integrand size = 38, antiderivative size = 180 \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (3 a b B-a^2 C-2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} d}+\frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Time = 0.45 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {4157, 4094, 4088, 12, 3916, 2738, 214} \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (a^2 (-C)+3 a b B-2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{5/2} (a+b)^{5/2}}+\frac {a (b B-a C) \tan (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \tan (c+d x)}{2 b d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))} \]
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Rule 12
Rule 214
Rule 2738
Rule 3916
Rule 4088
Rule 4094
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^2(c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx \\ & = \frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) \left (-2 b (b B-a C)+\left (a b B+a^2 C-2 b^2 C\right ) \sec (c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {b \left (3 a b B-a^2 C-2 b^2 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b \left (a^2-b^2\right )^2} \\ & = \frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (3 a b B-a^2 C-2 b^2 C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = \frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (3 a b B-a^2 C-2 b^2 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b \left (a^2-b^2\right )^2} \\ & = \frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (3 a b B-a^2 C-2 b^2 C\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 \left (a^2-b^2\right )^2 d} \\ & = -\frac {\left (3 a b B-a^2 C-2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} d}+\frac {a (b B-a C) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \tan (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.87 \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {-\frac {2 \left (-3 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {(-b B+a C) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}+\frac {\left (2 a^2 B+b^2 B-3 a b C\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}}{2 d} \]
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Time = 0.39 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (2 B \,a^{2}+B a b +2 B \,b^{2}-C \,a^{2}-4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2 B \,a^{2}-B a b +2 B \,b^{2}+C \,a^{2}-4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (3 B a b -C \,a^{2}-2 C \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(238\) |
default | \(\frac {\frac {-\frac {\left (2 B \,a^{2}+B a b +2 B \,b^{2}-C \,a^{2}-4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2 B \,a^{2}-B a b +2 B \,b^{2}+C \,a^{2}-4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (3 B a b -C \,a^{2}-2 C \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(238\) |
risch | \(\frac {i \left (3 B \,a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}-C \,a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-2 C \,a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+2 B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+5 B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 b^{4} B \,{\mathrm e}^{2 i \left (d x +c \right )}-3 C \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 C a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+5 B \,a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+4 B a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+C \,a^{4} {\mathrm e}^{i \left (d x +c \right )}-10 C \,a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+2 B \,a^{4}+B \,a^{2} b^{2}-3 a^{3} b C \right )}{a \left (-a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {3 \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 ) B a b}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{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}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{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 \,b^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {3 a b \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 ) B}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {a^{2} \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}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{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 \,b^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(765\) |
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (165) = 330\).
Time = 0.33 (sec) , antiderivative size = 750, normalized size of antiderivative = 4.17 \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\left [\frac {{\left (C a^{2} b^{2} - 3 \, B a b^{3} + 2 \, C b^{4} + {\left (C a^{4} - 3 \, B a^{3} b + 2 \, C a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{3} b - 3 \, B a^{2} b^{2} + 2 \, C a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \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 ) + 2 \, {\left (C a^{5} + B a^{4} b - 5 \, C a^{3} b^{2} + B a^{2} b^{3} + 4 \, C a b^{4} - 2 \, B b^{5} + {\left (2 \, B a^{5} - 3 \, C a^{4} b - B a^{3} b^{2} + 3 \, C a^{2} b^{3} - B a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d\right )}}, \frac {{\left (C a^{2} b^{2} - 3 \, B a b^{3} + 2 \, C b^{4} + {\left (C a^{4} - 3 \, B a^{3} b + 2 \, C a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{3} b - 3 \, B a^{2} b^{2} + 2 \, C a b^{3}\right )} \cos \left (d x + c\right )\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 ) + {\left (C a^{5} + B a^{4} b - 5 \, C a^{3} b^{2} + B a^{2} b^{3} + 4 \, C a b^{4} - 2 \, B b^{5} + {\left (2 \, B a^{5} - 3 \, C a^{4} b - B a^{3} b^{2} + 3 \, C a^{2} b^{3} - B a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d\right )}}\right ] \]
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\[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (165) = 330\).
Time = 0.36 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.22 \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {{\left (C a^{2} - 3 \, B a b + 2 \, C b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \]
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Time = 20.65 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.39 \[ \int \frac {\sec (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^2-2\,a\,b+b^2\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{5/2}}\right )\,\left (C\,a^2-3\,B\,a\,b+2\,C\,b^2\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B\,a^2+2\,B\,b^2-C\,a^2+B\,a\,b-4\,C\,a\,b\right )}{{\left (a+b\right )}^2\,\left (a-b\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B\,a^2+2\,B\,b^2+C\,a^2-B\,a\,b-4\,C\,a\,b\right )}{\left (a+b\right )\,\left (a^2-2\,a\,b+b^2\right )}}{d\,\left (2\,a\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2-2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2-2\,a\,b+b^2\right )+a^2+b^2\right )} \]
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