Optimal. Leaf size=181 \[ -\frac {2 d^3 \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{7/2} \sqrt {c+d} f}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right )} \]
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Rubi [A]
time = 0.23, antiderivative size = 235, normalized size of antiderivative = 1.30, number of steps
used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 106, 157,
12, 95, 211} \begin {gather*} \frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 f (c-d)^3 \left (a^3 \sec (e+f x)+a^3\right )}+\frac {2 d^3 \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 f (c-d)^{7/2} \sqrt {c+d} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(2 c-7 d) \tan (e+f x)}{15 a f (c-d)^2 (a \sec (e+f x)+a)^2}+\frac {\tan (e+f x)}{5 f (c-d) (a \sec (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 211
Rule 4072
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^2 (2 c-5 d)-2 a^2 d x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{5 a (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^4 \left (2 c^2-7 c d+15 d^2\right )+a^4 (2 c-7 d) d x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^4 (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {15 a^6 d^3}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^7 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {\left (d^3 \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 )}{a (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {\left (2 d^3 \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 )}{a (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3}+\frac {(2 c-7 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2}+\frac {2 d^3 \tan ^{-1}\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)}{a^2 (c-d)^{7/2} \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 c^2-9 c d+22 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.06, size = 345, normalized size = 1.91 \begin {gather*} \frac {\cos \left (\frac {1}{2} (e+f x)\right ) \left (\frac {480 d^3 \text {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 ) \cos ^5\left (\frac {1}{2} (e+f x)\right ) (i \cos (e)+\sin (e))}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\sec \left (\frac {e}{2}\right ) \left (5 \left (8 c^2-27 c d+37 d^2\right ) \sin \left (\frac {f x}{2}\right )-15 \left (2 c^2-7 c d+9 d^2\right ) \sin \left (e+\frac {f x}{2}\right )+20 c^2 \sin \left (e+\frac {3 f x}{2}\right )-75 c d \sin \left (e+\frac {3 f x}{2}\right )+115 d^2 \sin \left (e+\frac {3 f x}{2}\right )-15 c^2 \sin \left (2 e+\frac {3 f x}{2}\right )+45 c d \sin \left (2 e+\frac {3 f x}{2}\right )-45 d^2 \sin \left (2 e+\frac {3 f x}{2}\right )+7 c^2 \sin \left (2 e+\frac {5 f x}{2}\right )-24 c d \sin \left (2 e+\frac {5 f x}{2}\right )+32 d^2 \sin \left (2 e+\frac {5 f x}{2}\right )\right )\right )}{30 a^3 (c-d)^3 f (1+\cos (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 203, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {\frac {\frac {c^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c d \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {d^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+2 c d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4 d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 c d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+7 d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c -d \right )^{3}}-\frac {8 d^{3} \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3} \sqrt {\left (c +d \right ) \left (c -d \right )}}}{4 f \,a^{3}}\) | \(203\) |
default | \(\frac {\frac {\frac {c^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c d \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {d^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+2 c d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4 d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 c d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+7 d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c -d \right )^{3}}-\frac {8 d^{3} \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3} \sqrt {\left (c +d \right ) \left (c -d \right )}}}{4 f \,a^{3}}\) | \(203\) |
risch | \(\frac {2 i \left (15 c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-45 c d \,{\mathrm e}^{4 i \left (f x +e \right )}+45 d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+30 c^{2} {\mathrm e}^{3 i \left (f x +e \right )}-105 c d \,{\mathrm e}^{3 i \left (f x +e \right )}+135 d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+40 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-135 c d \,{\mathrm e}^{2 i \left (f x +e \right )}+185 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+20 c^{2} {\mathrm e}^{i \left (f x +e \right )}-75 d \,{\mathrm e}^{i \left (f x +e \right )} c +115 d^{2} {\mathrm e}^{i \left (f x +e \right )}+7 c^{2}-24 c d +32 d^{2}\right )}{15 f \,a^{3} \left (c -d \right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c -d \right )^{3} f \,a^{3}}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c -d \right )^{3} f \,a^{3}}\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 483 vs.
\(2 (173) = 346\).
time = 3.02, size = 1027, normalized size = 5.67 \begin {gather*} \left [-\frac {15 \, {\left (d^{3} \cos \left (f x + e\right )^{3} + 3 \, d^{3} \cos \left (f x + e\right )^{2} + 3 \, d^{3} \cos \left (f x + e\right ) + 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 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4} + {\left (7 \, c^{4} - 24 \, c^{3} d + 25 \, c^{2} d^{2} + 24 \, c d^{3} - 32 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, c^{4} - 9 \, c^{3} d + 15 \, c^{2} d^{2} + 9 \, c d^{3} - 17 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\left ({\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right )^{2} + 3 \, {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right ) + {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f\right )}}, -\frac {15 \, {\left (d^{3} \cos \left (f x + e\right )^{3} + 3 \, d^{3} \cos \left (f x + e\right )^{2} + 3 \, d^{3} \cos \left (f x + e\right ) + 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 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4} + {\left (7 \, c^{4} - 24 \, c^{3} d + 25 \, c^{2} d^{2} + 24 \, c d^{3} - 32 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, c^{4} - 9 \, c^{3} d + 15 \, c^{2} d^{2} + 9 \, c d^{3} - 17 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left ({\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right )^{2} + 3 \, {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f \cos \left (f x + e\right ) + {\left (a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right )} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec {\left (e + f x \right )}}{c \sec ^{3}{\left (e + f x \right )} + 3 c \sec ^{2}{\left (e + f x \right )} + 3 c \sec {\left (e + f x \right )} + c + d \sec ^{4}{\left (e + f x \right )} + 3 d \sec ^{3}{\left (e + f x \right )} + 3 d \sec ^{2}{\left (e + f x \right )} + d \sec {\left (e + f x \right )}}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 471 vs.
\(2 (166) = 332\).
time = 0.53, size = 471, normalized size = 2.60 \begin {gather*} -\frac {\frac {120 \, {\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 )} d^{3}}{{\left (a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {3 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{12} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 18 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 50 \, a^{12} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 90 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 70 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 20 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 90 \, a^{12} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 240 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 270 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 105 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{5} - 5 \, a^{15} c^{4} d + 10 \, a^{15} c^{3} d^{2} - 10 \, a^{15} c^{2} d^{3} + 5 \, a^{15} c d^{4} - a^{15} d^{5}}}{60 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.95, size = 228, normalized size = 1.26 \begin {gather*} \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3}{4\,a^3\,\left (c-d\right )}-\frac {\left (c+d\right )\,\left (\frac {3}{4\,a^3\,\left (c-d\right )}-\frac {c+d}{4\,a^3\,{\left (c-d\right )}^2}\right )}{c-d}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {1}{4\,a^3\,\left (c-d\right )}-\frac {c+d}{12\,a^3\,{\left (c-d\right )}^2}\right )}{f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,a^3\,f\,\left (c-d\right )}-\frac {2\,d^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-2\,d\right )\,\left (a^3\,c^3-3\,a^3\,c^2\,d+3\,a^3\,c\,d^2-a^3\,d^3\right )}{2\,a^3\,\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )}{a^3\,f\,\sqrt {c+d}\,{\left (c-d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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