Integrand size = 31, antiderivative size = 187 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {\left (2 a^2+b^2\right ) (A b-a B) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {2 a^3 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}-\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d} \]
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Time = 0.75 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4118, 4177, 4167, 4083, 3855, 3916, 2738, 214} \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=-\frac {2 a^3 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d \sqrt {a-b} \sqrt {a+b}}+\frac {\left (2 a^2+b^2\right ) (A b-a B) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {\left (-3 a^2 B+3 a A b-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac {B \tan (c+d x) \sec ^2(c+d x)}{3 b d} \]
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4118
Rule 4167
Rule 4177
Rubi steps \begin{align*} \text {integral}& = \frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a B+2 b B \sec (c+d x)+3 (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b} \\ & = \frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec (c+d x) \left (3 a (A b-a B)+b (3 A b+a B) \sec (c+d x)-2 \left (3 a A b-3 a^2 B-2 b^2 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^2} \\ & = -\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec (c+d x) \left (3 a b (A b-a B)+3 \left (2 a^2+b^2\right ) (A b-a B) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^3} \\ & = -\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^4}+\frac {\left (\left (2 a^2+b^2\right ) (A b-a B)\right ) \int \sec (c+d x) \, dx}{2 b^4} \\ & = \frac {\left (2 a^2+b^2\right ) (A b-a B) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^5} \\ & = \frac {\left (2 a^2+b^2\right ) (A b-a B) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d}-\frac {\left (2 a^3 (A b-a B)\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^5 d} \\ & = \frac {\left (2 a^2+b^2\right ) (A b-a B) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {2 a^3 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}-\frac {\left (3 a A b-3 a^2 B-2 b^2 B\right ) \tan (c+d x)}{3 b^3 d}+\frac {(A b-a B) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {B \sec ^2(c+d x) \tan (c+d x)}{3 b d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(422\) vs. \(2(187)=374\).
Time = 2.34 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.26 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {\frac {24 a^3 (A b-a B) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+6 \left (2 a^2+b^2\right ) (-A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 \left (2 a^2+b^2\right ) (-A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 (3 A b+(-3 a+b) B)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^3 B \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 b \left (-3 a A b+3 a^2 B+2 b^2 B\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 b^3 B \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {b^2 (3 A b+(-3 a+b) B)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 b \left (-3 a A b+3 a^2 B+2 b^2 B\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}}{12 b^4 d} \]
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Time = 1.17 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.79
method | result | size |
derivativedivides | \(\frac {-\frac {B}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {A b -B a -B b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}-B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {-2 A a b -A \,b^{2}+2 B \,a^{2}+B a b +2 b^{2} B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \left (A b -B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {B}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-A b +B a +B b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,a^{2} b -A \,b^{3}+2 B \,a^{3}+B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {-2 A a b -A \,b^{2}+2 B \,a^{2}+B a b +2 b^{2} B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(335\) |
default | \(\frac {-\frac {B}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {A b -B a -B b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}-B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {-2 A a b -A \,b^{2}+2 B \,a^{2}+B a b +2 b^{2} B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \left (A b -B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {B}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-A b +B a +B b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,a^{2} b -A \,b^{3}+2 B \,a^{3}+B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {-2 A a b -A \,b^{2}+2 B \,a^{2}+B a b +2 b^{2} B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(335\) |
risch | \(\frac {i \left (-3 A \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 B a b \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 A \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 B a b \,{\mathrm e}^{i \left (d x +c \right )}-6 A a b +6 B \,a^{2}+4 b^{2} B \right )}{3 d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{2}}{b^{3} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 b d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3}}{b^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{2}}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 b d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3}}{b^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a}{2 b^{2} d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, d \,b^{4}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, d \,b^{4}}\) | \(663\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (170) = 340\).
Time = 0.74 (sec) , antiderivative size = 743, normalized size of antiderivative = 3.97 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\left [-\frac {6 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )^{3} \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 ) + 3 \, {\left (2 \, B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, B a^{2} b^{3} - 2 \, B b^{5} + 2 \, {\left (3 \, B a^{4} b - 3 \, A a^{3} b^{2} - B a^{2} b^{3} + 3 \, A a b^{4} - 2 \, B b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}, \frac {12 \, {\left (B a^{4} - A a^{3} b\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 )^{3} - 3 \, {\left (2 \, B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, B a^{2} b^{3} - 2 \, B b^{5} + 2 \, {\left (3 \, B a^{4} b - 3 \, A a^{3} b^{2} - B a^{2} b^{3} + 3 \, A a b^{4} - 2 \, B b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}\right ] \]
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\[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (170) = 340\).
Time = 0.35 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.20 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (2 \, B a^{3} - 2 \, A a^{2} b + B a b^{2} - A b^{3}\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 \, B a^{3} - 2 \, A a^{2} b + B a b^{2} - A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {12 \, {\left (B a^{4} - A a^{3} b\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^{4}} + \frac {2 \, {\left (6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\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 = 19.28 (sec) , antiderivative size = 4667, normalized size of antiderivative = 24.96 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
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