Optimal. Leaf size=211 \[ \frac {2 d^2 (3 c+2 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^2 (c-d)^{7/2} (c+d)^{3/2} f}+\frac {d \left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sec (e+f x))}+\frac {(c-6 d) \tan (e+f x)}{3 a^2 (c-d)^2 f (1+\sec (e+f x)) (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{3 (c-d) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))} \]
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Rubi [A]
time = 0.25, antiderivative size = 260, normalized size of antiderivative = 1.23, 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, 105, 157,
12, 95, 211} \begin {gather*} \frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 f (c-d)^3 (c+d) \left (a^2 \sec (e+f x)+a^2\right )}-\frac {2 d^2 (3 c+2 d) \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 f (c-d)^{7/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}+\frac {(c+4 d) \tan (e+f x)}{3 f (c-d)^2 (c+d) (a \sec (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 211
Rule 4072
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^2 (c+2 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 )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^4 (c-6 d) (c+d)-a^4 d (c+4 d) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {3 a^6 d^2 (3 c+2 d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^6 (c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (d^2 (3 c+2 d) \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 )}{(c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (2 d^2 (3 c+2 d) \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)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {2 d^2 (3 c+2 d) \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 (c-d)^{7/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.93, size = 376, normalized size = 1.78 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec ^4(e+f x) \left (\frac {12 d^2 (3 c+2 d) \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 ^3\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) (i \cos (e)+\sin (e))}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+(c-d) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )-4 (c-4 d) \cos ^2\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\frac {6 d^3 \cos ^3\left (\frac {1}{2} (e+f x)\right ) (-d \sin (e)+c \sin (f x))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )}+(c-d) \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \tan \left (\frac {e}{2}\right )\right )}{3 a^2 (-c+d)^3 f (1+\sec (e+f x))^2 (c+d \sec (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 203, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (3 c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\) | \(203\) |
default | \(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (3 c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\) | \(203\) |
risch | \(\frac {2 i \left (-3 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}+9 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+21 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-5 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}+41 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+39 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{i \left (f x +e \right )}+8 c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+25 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{i \left (f x +e \right )}-2 c^{4}+6 c^{3} d +8 c^{2} d^{2}+3 c \,d^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right ) \left (-c^{2}+d^{2}\right ) c f \,a^{2} \left (-c +d \right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {3 d^{2} \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 ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}+\frac {2 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 ) \left (c -d \right )^{3} f \,a^{2}}-\frac {3 d^{2} \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 ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}-\frac {2 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 ) \left (c -d \right )^{3} f \,a^{2}}\) | \(729\) |
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. 603 vs.
\(2 (205) = 410\).
time = 2.84, size = 1268, normalized size = 6.01 \begin {gather*} \left [-\frac {3 \, {\left (3 \, c d^{3} + 2 \, d^{4} + {\left (3 \, c^{2} d^{2} + 2 \, c d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{2} d^{2} + 7 \, c d^{3} + 2 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right )} \cos \left (f x + e\right )\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 (c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5} + {\left (2 \, c^{5} - 6 \, c^{4} d - 10 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 8 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left ({\left (a^{2} c^{7} - 2 \, a^{2} c^{6} d - a^{2} c^{5} d^{2} + 4 \, a^{2} c^{4} d^{3} - a^{2} c^{3} d^{4} - 2 \, a^{2} c^{2} d^{5} + a^{2} c d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (2 \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - 4 \, a^{2} c^{5} d^{2} + 7 \, a^{2} c^{4} d^{3} + 2 \, a^{2} c^{3} d^{4} - 5 \, a^{2} c^{2} d^{5} + a^{2} d^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right )} f\right )}}, \frac {3 \, {\left (3 \, c d^{3} + 2 \, d^{4} + {\left (3 \, c^{2} d^{2} + 2 \, c d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{2} d^{2} + 7 \, c d^{3} + 2 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right )} \cos \left (f x + e\right )\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 (c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5} + {\left (2 \, c^{5} - 6 \, c^{4} d - 10 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 8 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left ({\left (a^{2} c^{7} - 2 \, a^{2} c^{6} d - a^{2} c^{5} d^{2} + 4 \, a^{2} c^{4} d^{3} - a^{2} c^{3} d^{4} - 2 \, a^{2} c^{2} d^{5} + a^{2} c d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (2 \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - 4 \, a^{2} c^{5} d^{2} + 7 \, a^{2} c^{4} d^{3} + 2 \, a^{2} c^{3} d^{4} - 5 \, a^{2} c^{2} d^{5} + a^{2} d^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\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^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{3}{\left (e + f x \right )} + 4 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{4}{\left (e + f x \right )} + 2 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{2}} \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. 474 vs.
\(2 (196) = 392\).
time = 0.53, size = 474, normalized size = 2.25 \begin {gather*} \frac {\frac {12 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\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 )}} + \frac {12 \, {\left (3 \, c d^{2} + 2 \, d^{3}\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 (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 48 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{6} - 6 \, a^{6} c^{5} d + 15 \, a^{6} c^{4} d^{2} - 20 \, a^{6} c^{3} d^{3} + 15 \, a^{6} c^{2} d^{4} - 6 \, a^{6} c d^{5} + a^{6} d^{6}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.18, size = 314, normalized size = 1.49 \begin {gather*} \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3}{2\,a^2\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{a^2\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6\,a^2\,f\,{\left (c-d\right )}^2}+\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^2\,d^4-a^2\,c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^2\,c^4-4\,a^2\,c^3\,d+6\,a^2\,c^2\,d^2-4\,a^2\,c\,d^3+a^2\,d^4\right )-2\,a^2\,c\,d^3+2\,a^2\,c^3\,d\right )}-\frac {d^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d+6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^2-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^3+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^4}{\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )\,\left (3\,c+2\,d\right )\,2{}\mathrm {i}}{a^2\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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