Integrand size = 31, antiderivative size = 276 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\frac {a^2 \left (12 c^2-16 c d+7 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{4 (c-d)^{5/2} (c+d)^{9/2} f}-\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sec (e+f x))} \]
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Time = 0.70 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 100, 156, 12, 95, 211} \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=-\frac {a^3 \left (12 c^2-16 c d+7 d^2\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{4 f (c-d)^{5/2} (c+d)^{9/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sec (e+f x))}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d f (c+d)^2 (c+d \sec (e+f x))^3}-\frac {a^2 (c-d) \tan (e+f x)}{4 d f (c+d) (c+d \sec (e+f x))^4} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 211
Rule 4072
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x} (c+d x)^5} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {(a \tan (e+f x)) \text {Subst}\left (\int \frac {-8 a^3 d-a^3 (c+7 d) x}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)^4} \, dx,x,\sec (e+f x)\right )}{4 d (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-21 a^5 (c-d) d-2 a^5 (c-d) (c+8 d) x}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{12 a d (c+d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-2 a^7 (19 c-16 d) (c-d) d+a^7 (c-d) \left (21 d^2-2 c (c+8 d)\right ) x}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{24 a^3 d (c+d) \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int -\frac {3 a^9 (c-d) d \left (12 c^2-16 c d+7 d^2\right )}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{24 a^5 d (c+d) \left (c^2-d^2\right )^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sec (e+f x))}-\frac {\left (a^4 (c-d) \left (12 c^2-16 c d+7 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{8 (c+d) \left (c^2-d^2\right )^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sec (e+f x))}-\frac {\left (a^4 (c-d) \left (12 c^2-16 c d+7 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {a-a \sec (e+f x)}}\right )}{4 (c+d) \left (c^2-d^2\right )^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {a^3 \left (12 c^2-16 c d+7 d^2\right ) \arctan \left (\frac {\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right ) \tan (e+f x)}{4 (c-d)^{5/2} (c+d)^{9/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {a^2 (c-d) \tan (e+f x)}{4 d (c+d) f (c+d \sec (e+f x))^4}+\frac {a^2 (c+8 d) \tan (e+f x)}{12 d (c+d)^2 f (c+d \sec (e+f x))^3}+\frac {a^2 \left (2 c^2+16 c d-21 d^2\right ) \tan (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sec (e+f x))^2}+\frac {a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \tan (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sec (e+f x))} \\ \end{align*}
Time = 8.01 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.17 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\frac {a^2 \left (-\frac {24 \left (12 c^2-16 c d+7 d^2\right ) \text {arctanh}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+\frac {\left (24 c^5+192 c^4 d-446 c^3 d^2+128 c^2 d^3-148 c d^4+160 d^5+\left (144 c^5-172 c^4 d+208 c^3 d^2-785 c^2 d^3+368 c d^4+102 d^5\right ) \cos (e+f x)+2 \left (12 c^5+96 c^4 d-227 c^3 d^2+32 c^2 d^3+44 c d^4+16 d^5\right ) \cos (2 (e+f x))+48 c^5 \cos (3 (e+f x))-68 c^4 d \cos (3 (e+f x))-16 c^3 d^2 \cos (3 (e+f x))+5 c^2 d^3 \cos (3 (e+f x))+16 c d^4 \cos (3 (e+f x))+6 d^5 \cos (3 (e+f x))\right ) \sin (e+f x)}{(d+c \cos (e+f x))^4}\right )}{96 (c-d)^2 (c+d)^4 f} \]
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Time = 1.44 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {8 a^{2} \left (-\frac {\frac {\left (12 c^{2}-16 c d +7 d^{2}\right ) \left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{32 c^{4}+128 c^{3} d +192 c^{2} d^{2}+128 c \,d^{3}+32 d^{4}}-\frac {11 \left (12 c^{2}-16 c d +7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{96 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {\left (156 c^{2}-272 c d +83 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{96 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}-\frac {\left (20 c^{2}-48 c d +25 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{32 \left (c +d \right ) \left (c^{2}-2 c d +d^{2}\right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{4}}+\frac {\left (12 c^{2}-16 c d +7 d^{2}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{32 \left (c^{6}+2 c^{5} d -c^{4} d^{2}-4 c^{3} d^{3}-c^{2} d^{4}+2 c \,d^{5}+d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{f}\) | \(352\) |
default | \(\frac {8 a^{2} \left (-\frac {\frac {\left (12 c^{2}-16 c d +7 d^{2}\right ) \left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{32 c^{4}+128 c^{3} d +192 c^{2} d^{2}+128 c \,d^{3}+32 d^{4}}-\frac {11 \left (12 c^{2}-16 c d +7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{96 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {\left (156 c^{2}-272 c d +83 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{96 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}-\frac {\left (20 c^{2}-48 c d +25 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{32 \left (c +d \right ) \left (c^{2}-2 c d +d^{2}\right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{4}}+\frac {\left (12 c^{2}-16 c d +7 d^{2}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{32 \left (c^{6}+2 c^{5} d -c^{4} d^{2}-4 c^{3} d^{3}-c^{2} d^{4}+2 c \,d^{5}+d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{f}\) | \(352\) |
risch | \(\text {Expression too large to display}\) | \(1557\) |
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Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (257) = 514\).
Time = 0.40 (sec) , antiderivative size = 1908, normalized size of antiderivative = 6.91 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{5} + 5 c^{4} d \sec {\left (e + f x \right )} + 10 c^{3} d^{2} \sec ^{2}{\left (e + f x \right )} + 10 c^{2} d^{3} \sec ^{3}{\left (e + f x \right )} + 5 c d^{4} \sec ^{4}{\left (e + f x \right )} + d^{5} \sec ^{5}{\left (e + f x \right )}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{c^{5} + 5 c^{4} d \sec {\left (e + f x \right )} + 10 c^{3} d^{2} \sec ^{2}{\left (e + f x \right )} + 10 c^{2} d^{3} \sec ^{3}{\left (e + f x \right )} + 5 c d^{4} \sec ^{4}{\left (e + f x \right )} + d^{5} \sec ^{5}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{c^{5} + 5 c^{4} d \sec {\left (e + f x \right )} + 10 c^{3} d^{2} \sec ^{2}{\left (e + f x \right )} + 10 c^{2} d^{3} \sec ^{3}{\left (e + f x \right )} + 5 c d^{4} \sec ^{4}{\left (e + f x \right )} + d^{5} \sec ^{5}{\left (e + f x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (257) = 514\).
Time = 0.51 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.57 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\frac {\frac {3 \, {\left (12 \, a^{2} c^{2} - 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{6} + 2 \, c^{5} d - c^{4} d^{2} - 4 \, c^{3} d^{3} - c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {36 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 156 \, a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 273 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 243 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 111 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 21 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 132 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 308 \, a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 121 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 231 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 253 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 77 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 156 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 116 \, a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 345 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 199 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 189 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 83 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 60 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 177 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 147 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 81 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 75 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{6} + 2 \, c^{5} d - c^{4} d^{2} - 4 \, c^{3} d^{3} - c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{4}}}{12 \, f} \]
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Time = 17.02 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.59 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx=\frac {\frac {11\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (12\,a^2\,c^2-16\,a^2\,c\,d+7\,a^2\,d^2\right )}{12\,{\left (c+d\right )}^3}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (12\,a^2\,c^3-28\,a^2\,c^2\,d+23\,a^2\,c\,d^2-7\,a^2\,d^3\right )}{4\,{\left (c+d\right )}^4}-\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (156\,c^2-272\,c\,d+83\,d^2\right )}{12\,{\left (c+d\right )}^2\,\left (c-d\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2-48\,c\,d+25\,d^2\right )}{4\,\left (c+d\right )\,\left (c^2-2\,c\,d+d^2\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,c^4-12\,c^2\,d^2+6\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-4\,c^4-8\,c^3\,d+8\,c\,d^3+4\,d^4\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,c^4-8\,c^3\,d+8\,c\,d^3-4\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (c^4-4\,c^3\,d+6\,c^2\,d^2-4\,c\,d^3+d^4\right )+4\,c\,d^3+4\,c^3\,d+c^4+d^4+6\,c^2\,d^2\right )}+\frac {a^2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-2\,d\right )\,\left (c^2-2\,c\,d+d^2\right )}{2\,\sqrt {c+d}\,{\left (c-d\right )}^{5/2}}\right )\,\left (12\,c^2-16\,c\,d+7\,d^2\right )}{4\,f\,{\left (c+d\right )}^{9/2}\,{\left (c-d\right )}^{5/2}} \]
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