Integrand size = 25, antiderivative size = 194 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {9 (3 c-2 d) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) (c+d)^3 \sqrt {c^2-d^2} f}+\frac {3 (c-d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))^3}-\frac {3 (c+6 d) \cos (e+f x)}{2 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {3 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{2 (c-d) d (c+d)^3 f (c+d \sin (e+f x))} \]
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Time = 0.22 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2841, 2833, 12, 2739, 632, 210} \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {a^2 (3 c-2 d) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f (c-d) (c+d)^3 \sqrt {c^2-d^2}}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 d f (c-d) (c+d)^3 (c+d \sin (e+f x))}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d f (c+d)^2 (c+d \sin (e+f x))^2}+\frac {a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2841
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a \int \frac {-6 a d-a (c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d (c+d)} \\ & = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}+\frac {a \int \frac {10 a (c-d) d+a (c-d) (c+6 d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 (c-d) d (c+d)^2} \\ & = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}-\frac {a \int -\frac {3 a (3 c-2 d) (c-d) d}{c+d \sin (e+f x)} \, dx}{6 (c-d)^2 d (c+d)^3} \\ & = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a^2 (3 c-2 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (c-d) (c+d)^3} \\ & = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a^2 (3 c-2 d)\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d) (c+d)^3 f} \\ & = \frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}-\frac {\left (2 a^2 (3 c-2 d)\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d) (c+d)^3 f} \\ & = \frac {a^2 (3 c-2 d) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) (c+d)^3 \sqrt {c^2-d^2} f}+\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))} \\ \end{align*}
Time = 1.60 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.98 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {3 \cos (e+f x) \left (-\frac {d (1+\sin (e+f x))^2}{(c+d \sin (e+f x))^3}-\frac {(3 c-2 d) \left (\frac {6 \text {arctanh}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{\sqrt {-c-d} \sqrt {c-d}}-\frac {\sqrt {\cos ^2(e+f x)} (4 c+d+(c+4 d) \sin (e+f x))}{(c+d \sin (e+f x))^2}\right )}{2 (c+d)^2 \sqrt {\cos ^2(e+f x)}}\right )}{(-c+d) (c+d) f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(522\) vs. \(2(196)=392\).
Time = 1.84 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.70
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (\frac {\frac {\left (c^{4}-6 c^{3} d +4 d^{3} c +2 d^{4}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (4 c^{5}-3 c^{4} d +18 c^{3} d^{2}-8 c^{2} d^{3}-12 c \,d^{4}-4 d^{5}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {d \left (36 c^{5}-21 c^{4} d +6 c^{3} d^{2}-20 c^{2} d^{3}-12 c \,d^{4}-4 d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (4 c^{5}-2 c^{4} d +12 c^{3} d^{2}-11 c^{2} d^{3}-6 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (c^{4}+18 c^{3} d -14 c^{2} d^{2}-8 d^{3} c -2 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {12 c^{3}-7 c^{2} d -6 c \,d^{2}-2 d^{3}}{6 \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (3 c -2 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(523\) |
default | \(\frac {2 a^{2} \left (\frac {\frac {\left (c^{4}-6 c^{3} d +4 d^{3} c +2 d^{4}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (4 c^{5}-3 c^{4} d +18 c^{3} d^{2}-8 c^{2} d^{3}-12 c \,d^{4}-4 d^{5}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {d \left (36 c^{5}-21 c^{4} d +6 c^{3} d^{2}-20 c^{2} d^{3}-12 c \,d^{4}-4 d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (4 c^{5}-2 c^{4} d +12 c^{3} d^{2}-11 c^{2} d^{3}-6 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {\left (c^{4}+18 c^{3} d -14 c^{2} d^{2}-8 d^{3} c -2 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}-\frac {12 c^{3}-7 c^{2} d -6 c \,d^{2}-2 d^{3}}{6 \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (3 c -2 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}+2 c^{3} d -2 d^{3} c -d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(523\) |
risch | \(\frac {i a^{2} \left (-6 i c \,d^{4}-6 i c^{4} d \,{\mathrm e}^{4 i \left (f x +e \right )}+45 i c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+10 i d^{5}-i c^{2} d^{3}+9 c \,d^{4} {\mathrm e}^{5 i \left (f x +e \right )}-6 d^{5} {\mathrm e}^{5 i \left (f x +e \right )}+6 i c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}-12 i c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-72 i c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+60 i c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+4 c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+24 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}-34 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+36 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-60 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-18 i c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 i d^{5} {\mathrm e}^{4 i \left (f x +e \right )}-24 i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}-6 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}-36 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}+51 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}+6 d^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{3} \left (c +d \right )^{3} \left (c -d \right ) f \,d^{2}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right ) f}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right ) f}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right ) f}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right ) f}\) | \(728\) |
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Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (196) = 392\).
Time = 0.34 (sec) , antiderivative size = 1366, normalized size of antiderivative = 7.04 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (196) = 392\).
Time = 0.40 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.87 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {3 \, {\left (3 \, a^{2} c - 2 \, a^{2} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{4} + 2 \, c^{3} d - 2 \, c d^{3} - d^{4}\right )} \sqrt {c^{2} - d^{2}}} + \frac {3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 9 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 54 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 24 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 36 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 72 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 42 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 72 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 66 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 42 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} c^{6} + 7 \, a^{2} c^{5} d + 6 \, a^{2} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3}}{{\left (c^{7} + 2 \, c^{6} d - 2 \, c^{4} d^{3} - c^{3} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{3}}}{3 \, f} \]
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Time = 10.84 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.79 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=-\frac {\frac {-12\,a^2\,c^3+7\,a^2\,c^2\,d+6\,a^2\,c\,d^2+2\,a^2\,d^3}{3\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (c^4-6\,c^3\,d+4\,c\,d^3+2\,d^4\right )}{c\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-4\,c^5+2\,c^4\,d-12\,c^3\,d^2+11\,c^2\,d^3+6\,c\,d^4+2\,d^5\right )}{c^2\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-4\,c^5+3\,c^4\,d-18\,c^3\,d^2+8\,c^2\,d^3+12\,c\,d^4+4\,d^5\right )}{c^2\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (-c^4-18\,c^3\,d+14\,c^2\,d^2+8\,c\,d^3+2\,d^4\right )}{c\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {2\,a^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (-12\,c^3+7\,c^2\,d+6\,c\,d^2+2\,d^3\right )}{3\,c^3\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}}{f\,\left (c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^2\,d+8\,d^3\right )+c^3+6\,c^2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,c^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}-\frac {a^2\,\mathrm {atan}\left (\frac {\left (\frac {a^2\,\left (3\,c-2\,d\right )\,\left (-2\,c^4\,d-4\,c^3\,d^2+4\,c\,d^4+2\,d^5\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,c-2\,d\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}}\right )\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}{3\,a^2\,c-2\,a^2\,d}\right )\,\left (3\,c-2\,d\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}} \]
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