Integrand size = 21, antiderivative size = 176 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {3 \left (2 a^2+b^2\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{2 b^4 d \sqrt {\sec ^2(c+d x)}}+\frac {3 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))} \]
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Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3593, 747, 829, 858, 221, 739, 212} \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {3 \left (2 a^2+b^2\right ) \sec (c+d x) \text {arcsinh}(\tan (c+d x))}{2 b^4 d \sqrt {\sec ^2(c+d x)}}+\frac {3 a \sqrt {a^2+b^2} \sec (c+d x) \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))} \]
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Rule 212
Rule 221
Rule 739
Rule 747
Rule 829
Rule 858
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) \text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{3/2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(3 \sec (c+d x)) \text {Subst}\left (\int \frac {x \sqrt {1+\frac {x^2}{b^2}}}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(3 \sec (c+d x)) \text {Subst}\left (\int \frac {-\frac {a}{b^2}+\frac {\left (2 a^2+b^2\right ) x}{b^4}}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))}-\frac {\left (3 a \left (a^2+b^2\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b^5 d \sqrt {\sec ^2(c+d x)}}+\frac {\left (3 \left (2 a^2+b^2\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b^5 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {3 \left (2 a^2+b^2\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{2 b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))}+\frac {\left (3 a \left (a^2+b^2\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{b^5 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {3 \left (2 a^2+b^2\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{2 b^4 d \sqrt {\sec ^2(c+d x)}}+\frac {3 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac {\sec ^3(c+d x)}{b d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.58 (sec) , antiderivative size = 709, normalized size of antiderivative = 4.03 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {(a-i b) (a+i b) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{b^3 d (a+b \tan (c+d x))^2}-\frac {2 a \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d (a+b \tan (c+d x))^2}-\frac {6 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a^2+b^2} \left (-b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \cos \left (\frac {1}{2} (c+d x)\right )+b^2 \cos \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^4 d (a+b \tan (c+d x))^2}-\frac {3 \left (2 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}+\frac {3 \left (2 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}-\frac {2 a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2}-\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}+\frac {2 a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2} \]
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Time = 32.08 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a -b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-6 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {1}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {4 a -b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (6 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}+\frac {\frac {2 \left (\frac {\left (a^{2}+b^{2}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+b \left (a^{2}+b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-6 a \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}}{d}\) | \(259\) |
default | \(\frac {\frac {1}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a -b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-6 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {1}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {4 a -b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (6 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}+\frac {\frac {2 \left (\frac {\left (a^{2}+b^{2}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+b \left (a^{2}+b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-6 a \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}}{d}\) | \(259\) |
risch | \(-\frac {-3 i a b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+3 i a b \,{\mathrm e}^{i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) d \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d \,b^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d \,b^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d \,b^{2}}+\frac {3 \sqrt {a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{d \,b^{4}}-\frac {3 \sqrt {a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{d \,b^{4}}\) | \(353\) |
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Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (163) = 326\).
Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.02 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {6 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, b^{3} + 6 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (a^{2} \cos \left (d x + c\right )^{3} + a b \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 3 \, {\left ({\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{4 \, {\left (a b^{4} d \cos \left (d x + c\right )^{3} + b^{5} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (163) = 326\).
Time = 0.33 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.68 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (6 \, a^{3} + 2 \, a b^{2} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (9 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, {\left (2 \, a^{3} + a b^{2}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, {\left (3 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} b^{3} + \frac {2 \, a b^{4} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{2} b^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a b^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} b^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2 \, a b^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{2} b^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {6 \, \sqrt {a^{2} + b^{2}} a \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{b^{4}} - \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{4}} + \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{4}}}{2 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.59 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} + \frac {6 \, {\left (a^{3} + a b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} + \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} b^{3}} + \frac {4 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} + a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} a b^{3}}}{2 \, d} \]
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Time = 5.94 (sec) , antiderivative size = 585, normalized size of antiderivative = 3.32 \[ \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\mathrm {atanh}\left (\frac {648\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{216\,a\,b^2+648\,a^3+\frac {432\,a^5}{b^2}}+\frac {432\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{432\,a^5+648\,a^3\,b^2+216\,a\,b^4}+\frac {216\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{216\,a+\frac {648\,a^3}{b^2}+\frac {432\,a^5}{b^4}}\right )\,\left (6\,a^2+3\,b^2\right )}{b^4\,d}-\frac {\frac {2\,\left (3\,a^2+b^2\right )}{b^3}+\frac {6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{b^3}-\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2+b^2\right )}{b^3}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (9\,a^2+2\,b^2\right )}{a\,b^2}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^2+b^2\right )}{a\,b^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^2+2\,b^2\right )}{a\,b^2}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {6\,a\,\mathrm {atanh}\left (\frac {432\,a^3\,\sqrt {a^2+b^2}}{432\,a^3\,b+\frac {432\,a^5}{b}+864\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+864\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {864\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}}{432\,a^3+\frac {432\,a^5}{b^2}+864\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {864\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b}}+\frac {432\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}}{432\,a^5+864\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b+432\,a^3\,b^2+864\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3}\right )\,\sqrt {a^2+b^2}}{b^4\,d} \]
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