Optimal. Leaf size=306 \[ -\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{3/2} f}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]
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Rubi [A]
time = 0.66, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2871, 3112,
3102, 2814, 2739, 632, 210} \begin {gather*} \frac {2 \left (3 a^2 d+a b c-4 b^2 d\right ) (b c-a d)^3 \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \left (a^2-b^2\right )^{3/2}}+\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b^2 f \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {d^2 x \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{2 b^4}+\frac {d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b^3 f \left (a^2-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2871
Rule 3102
Rule 3112
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {(c+d \sin (e+f x)) \left (4 b^2 c^2 d+2 a^2 d^3-a b c \left (c^2+5 d^2\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+d \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {8 b^3 c^3 d+8 a^2 b c d^3-3 a^3 d^4-a b^2 \left (2 c^4+12 c^2 d^2-d^4\right )+b d \left (b^2 d \left (12 c^2+d^2\right )-4 a b c \left (c^2+2 d^2\right )-a^2 \left (2 c^2 d-d^3\right )\right ) \sin (e+f x)-2 d (2 b c-a d) \left (b^2 c^2-2 a b c d+3 a^2 d^2-2 b^2 d^2\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {b \left (8 b^3 c^3 d+8 a^2 b c d^3-3 a^3 d^4-a b^2 \left (2 c^4+12 c^2 d^2-d^4\right )\right )+\left (a^2-b^2\right ) d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=-\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left ((b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^4 \left (a^2-b^2\right )}\\ &=-\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^4 \left (a^2-b^2\right ) f}\\ &=-\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (4 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^4 \left (a^2-b^2\right ) f}\\ &=-\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{3/2} f}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 2.10, size = 199, normalized size = 0.65 \begin {gather*} -\frac {-2 d^2 \left (-16 a b c d+6 a^2 d^2+b^2 \left (12 c^2+d^2\right )\right ) (e+f x)+\frac {8 (-b c+a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+8 b d^3 (2 b c-a d) \cos (e+f x)-\frac {4 b (b c-a d)^4 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+b^2 d^4 \sin (2 (e+f x))}{4 b^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 456, normalized size = 1.49
method | result | size |
derivativedivides | \(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} b^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\) | \(456\) |
default | \(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} b^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\) | \(456\) |
risch | \(\text {Expression too large to display}\) | \(1666\) |
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. 692 vs.
\(2 (304) = 608\).
time = 0.44, size = 1472, normalized size = 4.81 \begin {gather*} \left [\frac {{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} d^{4} \cos \left (f x + e\right )^{3} + {\left (12 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} c^{2} d^{2} - 16 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c d^{3} + {\left (6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right )} d^{4}\right )} f x + {\left (a^{2} b^{4} c^{4} - 4 \, a b^{5} c^{3} d - 6 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3}\right )} c d^{3} - {\left (3 \, a^{6} - 4 \, a^{4} b^{2}\right )} d^{4} + {\left (a b^{5} c^{4} - 4 \, b^{6} c^{3} d - 6 \, {\left (a^{3} b^{3} - 2 \, a b^{5}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} c d^{3} - {\left (3 \, a^{5} b - 4 \, a^{3} b^{3}\right )} d^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + {\left (2 \, {\left (a^{2} b^{5} - b^{7}\right )} c^{4} - 8 \, {\left (a^{3} b^{4} - a b^{6}\right )} c^{3} d + 12 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} c^{2} d^{2} - 8 \, {\left (2 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + a b^{6}\right )} c d^{3} + {\left (6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right )} d^{4}\right )} \cos \left (f x + e\right ) + {\left ({\left (12 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} c^{2} d^{2} - 16 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} c d^{3} + {\left (6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right )} d^{4}\right )} f x - {\left (8 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} c d^{3} - 3 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} f\right )}}, \frac {{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} d^{4} \cos \left (f x + e\right )^{3} + {\left (12 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} c^{2} d^{2} - 16 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c d^{3} + {\left (6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right )} d^{4}\right )} f x - 2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a b^{5} c^{3} d - 6 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3}\right )} c d^{3} - {\left (3 \, a^{6} - 4 \, a^{4} b^{2}\right )} d^{4} + {\left (a b^{5} c^{4} - 4 \, b^{6} c^{3} d - 6 \, {\left (a^{3} b^{3} - 2 \, a b^{5}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} c d^{3} - {\left (3 \, a^{5} b - 4 \, a^{3} b^{3}\right )} d^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + {\left (2 \, {\left (a^{2} b^{5} - b^{7}\right )} c^{4} - 8 \, {\left (a^{3} b^{4} - a b^{6}\right )} c^{3} d + 12 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} c^{2} d^{2} - 8 \, {\left (2 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + a b^{6}\right )} c d^{3} + {\left (6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right )} d^{4}\right )} \cos \left (f x + e\right ) + {\left ({\left (12 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} c^{2} d^{2} - 16 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} c d^{3} + {\left (6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right )} d^{4}\right )} f x - {\left (8 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} c d^{3} - 3 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} f\right )}}\right ] \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 = 0.50, size = 517, normalized size = 1.69 \begin {gather*} \frac {\frac {4 \, {\left (a b^{4} c^{4} - 4 \, b^{5} c^{3} d - 6 \, a^{3} b^{2} c^{2} d^{2} + 12 \, a b^{4} c^{2} d^{2} + 8 \, a^{4} b c d^{3} - 12 \, a^{2} b^{3} c d^{3} - 3 \, a^{5} d^{4} + 4 \, a^{3} b^{2} d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (b^{5} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a b^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} b^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a^{3} b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{4} b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + 6 \, a^{3} b^{2} c^{2} d^{2} - 4 \, a^{4} b c d^{3} + a^{5} d^{4}\right )}}{{\left (a^{3} b^{3} - a b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}} + \frac {{\left (12 \, b^{2} c^{2} d^{2} - 16 \, a b c d^{3} + 6 \, a^{2} d^{4} + b^{2} d^{4}\right )} {\left (f x + e\right )}}{b^{4}} + \frac {2 \, {\left (b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, b c d^{3} + 4 \, a d^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} b^{3}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 20.45, size = 2500, normalized size = 8.17 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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