Optimal. Leaf size=298 \[ -\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^3 (c-d)^4 (c+d) \sqrt {c^2-d^2} f}-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.49, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2845, 3057,
2833, 12, 2739, 632, 210} \begin {gather*} -\frac {2 d^3 (4 c+3 d) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a^3 f (c-d)^4 (c+d) \sqrt {c^2-d^2}}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))}-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2845
Rule 3057
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {\int \frac {-2 a (c-3 d)-3 a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx}{5 a^2 (c-d)}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}+\frac {\int \frac {a^2 \left (2 c^2-8 c d+27 d^2\right )+2 a^2 (2 c-9 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\int \frac {-2 a^3 (c-36 d) d^2-a^3 d \left (2 c^2-12 c d+45 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}+\frac {\int -\frac {15 a^3 d^3 (4 c+3 d)}{c+d \sin (e+f x)} \, dx}{15 a^6 (c-d)^4 (c+d)}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\left (d^3 (4 c+3 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{a^3 (c-d)^4 (c+d)}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}-\frac {\left (2 d^3 (4 c+3 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 )}{a^3 (c-d)^4 (c+d) f}\\ &=-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}+\frac {\left (4 d^3 (4 c+3 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 )}{a^3 (c-d)^4 (c+d) f}\\ &=-\frac {2 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^3 (c-d)^4 (c+d) \sqrt {c^2-d^2} f}-\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \cos (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))}-\frac {\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))}-\frac {(2 c-9 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac {\left (2 c^2-12 c d+45 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 1.64, size = 361, normalized size = 1.21 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c-6 d) (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 (c-6 d) (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2 \left (2 c^2-14 c d+57 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-\frac {30 d^3 (4 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d) \sqrt {c^2-d^2}}-\frac {15 d^4 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d) (c+d \sin (e+f x))}\right )}{15 a^3 (c-d)^4 f (1+\sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 273, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (8 c -12 d \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8 d -4 c}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (c^{2}-4 c d +6 d^{2}\right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{5 \left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 d^{3} \left (\frac {\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}+\frac {d}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (4 c +3 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{f \,a^{3}}\) | \(273\) |
default | \(\frac {-\frac {2 \left (8 c -12 d \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8 d -4 c}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (c^{2}-4 c d +6 d^{2}\right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{5 \left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (c -d \right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 d^{3} \left (\frac {\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}+\frac {d}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (4 c +3 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{f \,a^{3}}\) | \(273\) |
risch | \(-\frac {2 \left (-2 c^{3} d +12 c^{2} d^{2}-43 d^{3} c +60 i c^{2} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+345 i c \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-480 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+567 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+20 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+299 i c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+26 i c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+170 i c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}-1220 i c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-14 i c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}-410 i c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-40 i c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+225 i d^{4} {\mathrm e}^{5 i \left (f x +e \right )}+40 c^{3} {\mathrm e}^{4 i \left (f x +e \right )} d +60 c \,d^{3} {\mathrm e}^{6 i \left (f x +e \right )}-600 i d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+315 i d^{4} {\mathrm e}^{i \left (f x +e \right )}-270 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-865 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-98 d \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+238 d^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+848 d^{3} c \,{\mathrm e}^{2 i \left (f x +e \right )}+45 d^{4} {\mathrm e}^{6 i \left (f x +e \right )}+4 i c^{4} {\mathrm e}^{i \left (f x +e \right )}-72 d^{4}\right )}{15 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} \left (c +d \right ) \left (d \,{\mathrm e}^{2 i \left (f x +e \right )}-d +2 i c \,{\mathrm e}^{i \left (f x +e \right )}\right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {4 d^{3} \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}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {3 d^{4} \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 )}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}+\frac {4 d^{3} \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}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}+\frac {3 d^{4} \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 )}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}\) | \(804\) |
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. 1601 vs.
\(2 (297) = 594\).
time = 0.48, size = 3292, normalized size = 11.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 517, normalized size = 1.73 \begin {gather*} -\frac {2 \, {\left (\frac {15 \, {\left (4 \, c d^{3} + 3 \, d^{4}\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 (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 )} \sqrt {c^{2} - d^{2}}} + \frac {15 \, {\left (d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c d^{4}\right )}}{{\left (a^{3} c^{6} - 3 \, a^{3} c^{5} d + 2 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} - 3 \, a^{3} c^{2} d^{4} + a^{3} c d^{5}\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 )}} + \frac {15 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 90 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 150 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 300 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 190 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 420 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 110 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 270 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{2} - 34 \, c d + 72 \, d^{2}}{{\left (a^{3} c^{4} - 4 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} - 4 \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}\right )}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.30, size = 987, normalized size = 3.31 \begin {gather*} -\frac {\frac {2\,\left (7\,c^4-27\,c^3\,d+38\,c^2\,d^2+72\,c\,d^3+15\,d^4\right )}{15\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (5\,c^4-18\,c^3\,d+19\,c^2\,d^2+84\,c\,d^3+15\,d^4\right )}{3\,c\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^5-76\,c^4\,d+106\,c^3\,d^2+346\,c^2\,d^3+219\,c\,d^4+15\,d^5\right )}{15\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (c^5-3\,c^4\,d+2\,c^3\,d^2+6\,c^2\,d^3+d^5\right )}{c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,c^5-6\,c^4\,d+4\,c^3\,d^2+24\,c^2\,d^3+13\,c\,d^4+5\,d^5\right )}{c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (11\,c^5-27\,c^4\,d+4\,c^3\,d^2+162\,c^2\,d^3+135\,c\,d^4+30\,d^5\right )}{3\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (47\,c^5-137\,c^4\,d+88\,c^3\,d^2+812\,c^2\,d^3+690\,c\,d^4+75\,d^5\right )}{15\,c\,\left (c+d\right )\,\left (c-d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}}{f\,\left (a^3\,c+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,a^3\,c+2\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (5\,a^3\,c+2\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,a^3\,c+10\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (11\,a^3\,c+10\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (15\,a^3\,c+20\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (15\,a^3\,c+20\,a^3\,d\right )+a^3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}-\frac {2\,d^3\,\mathrm {atan}\left (\frac {\frac {d^3\,\left (4\,c+3\,d\right )\,\left (2\,a^3\,c^5\,d-6\,a^3\,c^4\,d^2+4\,a^3\,c^3\,d^3+4\,a^3\,c^2\,d^4-6\,a^3\,c\,d^5+2\,a^3\,d^6\right )}{a^3\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}}+\frac {2\,c\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c+3\,d\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 )}{a^3\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}}}{6\,d^4+8\,c\,d^3}\right )\,\left (4\,c+3\,d\right )}{a^3\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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