Integrand size = 31, antiderivative size = 178 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{\sqrt {c-d} (c+d)^{7/2} f}+\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}-\frac {5 a^3 (c-d) \tan (e+f x)}{6 d (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {5 a^3 (c+4 d) \tan (e+f x)}{6 d (c+d)^3 f (c+d \sec (e+f x))} \]
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Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4072, 96, 95, 211} \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=-\frac {5 a^4 \tan (e+f x) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{f \sqrt {c-d} (c+d)^{7/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {5 a^3 \tan (e+f x)}{2 f (c+d)^3 (c+d \sec (e+f x))}+\frac {5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{6 f (c+d)^2 (c+d \sec (e+f x))^2}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f (c+d) (c+d \sec (e+f x))^3} \]
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Rule 95
Rule 96
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)^{5/2}}{\sqrt {a-a x} (c+d x)^4} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}-\frac {\left (5 a^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{6 (c+d)^2 f (c+d \sec (e+f x))^2}-\frac {\left (5 a^4 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+a x}}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{6 (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {5 a^3 \tan (e+f x)}{2 (c+d)^3 f (c+d \sec (e+f x))}-\frac {\left (5 a^5 \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 )}{2 (c+d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{6 (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {5 a^3 \tan (e+f x)}{2 (c+d)^3 f (c+d \sec (e+f x))}-\frac {\left (5 a^5 \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 )}{(c+d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {5 a^4 \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)}{\sqrt {c-d} (c+d)^{7/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{3 (c+d) f (c+d \sec (e+f x))^3}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{6 (c+d)^2 f (c+d \sec (e+f x))^2}+\frac {5 a^3 \tan (e+f x)}{2 (c+d)^3 f (c+d \sec (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.88 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.24 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=\frac {a^3 (d+c \cos (e+f x)) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) (1+\sec (e+f x))^3 \left (-\frac {120 i \arctan \left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (d+c \cos (e+f x))^3 (\cos (e)-i \sin (e))}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {c \sec (e) \left (6 \left (8 c^4+6 c^3 d+30 c^2 d^2+9 c d^3+2 d^4\right ) \sin (f x)-3 \left (6 c^4-3 c^3 d+30 c^2 d^2+18 c d^3+4 d^4\right ) \sin (2 e+f x)+c \left (3 \left (3 c^3+38 c^2 d+12 c d^2+2 d^3\right ) \sin (e+2 f x)+3 \left (3 c^3-6 c^2 d-6 c d^2-2 d^3\right ) \sin (3 e+2 f x)+c \left (22 c^2+9 c d+2 d^2\right ) \sin (2 e+3 f x)\right )\right )-2 d \left (66 c^4+27 c^3 d+50 c^2 d^2+18 c d^3+4 d^4\right ) \tan (e)}{c^3}\right )}{192 (c+d)^3 f (c+d \sec (e+f x))^4} \]
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Time = 1.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {16 a^{3} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{6 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{3}}-\frac {5 \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{2}}-\frac {3 \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}+\frac {\operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c +d \right )}\right )}{6 \left (c +d \right )}\right )}{f}\) | \(227\) |
default | \(\frac {16 a^{3} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{6 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{3}}-\frac {5 \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{2}}-\frac {3 \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}+\frac {\operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c +d \right )}\right )}{6 \left (c +d \right )}\right )}{f}\) | \(227\) |
risch | \(\frac {i a^{3} \left (22 c^{5}+132 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}+54 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+100 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+36 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+36 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+180 c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+54 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+12 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+114 c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}+36 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}+6 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}+18 c^{3} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+6 c^{2} d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-9 c^{4} d \,{\mathrm e}^{4 i \left (f x +e \right )}+18 c^{4} d \,{\mathrm e}^{5 i \left (f x +e \right )}+90 c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+54 c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+12 c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+2 c^{3} d^{2}+48 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+9 c^{4} d +8 d^{5} {\mathrm e}^{3 i \left (f x +e \right )}-9 c^{5} {\mathrm e}^{5 i \left (f x +e \right )}+18 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}+9 c^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{3} \left (c +d \right )^{3} f \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )^{3}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{2 \sqrt {c^{2}-d^{2}}\, \left (c +d \right )^{3} f}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{2 \sqrt {c^{2}-d^{2}}\, \left (c +d \right )^{3} f}\) | \(580\) |
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Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (163) = 326\).
Time = 0.34 (sec) , antiderivative size = 1012, normalized size of antiderivative = 5.69 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=\left [\frac {15 \, {\left (a^{3} c^{3} \cos \left (f x + e\right )^{3} + 3 \, a^{3} c^{2} d \cos \left (f x + e\right )^{2} + 3 \, a^{3} c d^{2} \cos \left (f x + e\right ) + a^{3} d^{3}\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4} + {\left (22 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d - 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 2 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left ({\left (c^{8} + 3 \, c^{7} d + 2 \, c^{6} d^{2} - 2 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - c^{3} d^{5}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (c^{7} d + 3 \, c^{6} d^{2} + 2 \, c^{5} d^{3} - 2 \, c^{4} d^{4} - 3 \, c^{3} d^{5} - c^{2} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 3 \, {\left (c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f\right )}}, \frac {15 \, {\left (a^{3} c^{3} \cos \left (f x + e\right )^{3} + 3 \, a^{3} c^{2} d \cos \left (f x + e\right )^{2} + 3 \, a^{3} c d^{2} \cos \left (f x + e\right ) + a^{3} d^{3}\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left (2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4} + {\left (22 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d - 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 2 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left ({\left (c^{8} + 3 \, c^{7} d + 2 \, c^{6} d^{2} - 2 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - c^{3} d^{5}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (c^{7} d + 3 \, c^{6} d^{2} + 2 \, c^{5} d^{3} - 2 \, c^{4} d^{4} - 3 \, c^{3} d^{5} - c^{2} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 3 \, {\left (c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f\right )}}\right ] \]
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\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{c^{4} + 4 c^{3} d \sec {\left (e + f x \right )} + 6 c^{2} d^{2} \sec ^{2}{\left (e + f x \right )} + 4 c d^{3} \sec ^{3}{\left (e + f x \right )} + d^{4} \sec ^{4}{\left (e + f x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.39 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.72 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=-\frac {\frac {15 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )} a^{3}}{{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {15 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 30 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 33 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 66 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 33 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{3}}}{3 \, f} \]
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Time = 17.09 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.48 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx=\frac {\frac {5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (a^3\,c^2-2\,a^3\,c\,d+a^3\,d^2\right )}{{\left (c+d\right )}^3}+\frac {11\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{c+d}-\frac {40\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (a^3\,c-a^3\,d\right )}{3\,{\left (c+d\right )}^2}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-3\,c^3-3\,c^2\,d+3\,c\,d^2+3\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-3\,c^3+3\,c^2\,d+3\,c\,d^2-3\,d^3\right )+3\,c\,d^2+3\,c^2\,d+c^3+d^3-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )\right )}+\frac {5\,a^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {c-d}}{\sqrt {c+d}}\right )}{f\,{\left (c+d\right )}^{7/2}\,\sqrt {c-d}} \]
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