Optimal. Leaf size=255 \[ \frac {b^3 x}{d^3}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{5/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.46, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2871, 3100,
2814, 2739, 632, 210} \begin {gather*} -\frac {\left (-a^3 d^3 \left (2 c^2+d^2\right )+9 a^2 b c d^4-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{5/2}}+\frac {(b c-a d)^2 \left (3 a c d+2 b c^2-5 b d^2\right ) \cos (e+f x)}{2 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}+\frac {b^3 x}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2871
Rule 3100
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}-\frac {\int \frac {b^3 c^2-2 a^3 c d-4 a b^2 c d+5 a^2 b d^2-\left (4 a^2 b c d+2 b^3 c d-a^3 d^2+a b^2 \left (c^2-6 d^2\right )\right ) \sin (e+f x)-2 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d \left (c^2-d^2\right )}\\ &=\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\int \frac {-d \left (9 a^2 b c d^2-a^3 d \left (2 c^2+d^2\right )-3 a b^2 d \left (c^2+2 d^2\right )-b^3 \left (c^3-4 c d^2\right )\right )+2 b^3 \left (c^2-d^2\right )^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2 \left (c^2-d^2\right )^2}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^3 \left (c^2-d^2\right )^2}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\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 )}{d^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\left (2 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right )\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 )}{d^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {b^3 x}{d^3}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{5/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(521\) vs. \(2(255)=510\).
time = 2.50, size = 521, normalized size = 2.04 \begin {gather*} \frac {-\frac {4 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{5/2}}+\frac {4 b^3 c^6 e-6 b^3 c^4 d^2 e+2 b^3 d^6 e+4 b^3 c^6 f x-6 b^3 c^4 d^2 f x+2 b^3 d^6 f x-2 d (b c-a d)^2 \left (-2 b c^3-4 a c^2 d+5 b c d^2+a d^3\right ) \cos (e+f x)-2 b^3 \left (-c^2 d+d^3\right )^2 (e+f x) \cos (2 (e+f x))+8 b^3 c^5 d e \sin (e+f x)-16 b^3 c^3 d^3 e \sin (e+f x)+8 b^3 c d^5 e \sin (e+f x)+8 b^3 c^5 d f x \sin (e+f x)-16 b^3 c^3 d^3 f x \sin (e+f x)+8 b^3 c d^5 f x \sin (e+f x)+3 b^3 c^4 d^2 \sin (2 (e+f x))-3 a b^2 c^3 d^3 \sin (2 (e+f x))-3 a^2 b c^2 d^4 \sin (2 (e+f x))-6 b^3 c^2 d^4 \sin (2 (e+f x))+3 a^3 c d^5 \sin (2 (e+f x))+12 a b^2 c d^5 \sin (2 (e+f x))-6 a^2 b d^6 \sin (2 (e+f x))}{\left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}}{4 d^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs.
\(2(246)=492\).
time = 0.76, size = 670, normalized size = 2.63
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (5 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-9 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d +6 a \,b^{2} c^{2} d^{3}+b^{3} c^{5}-4 b^{3} c^{3} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{4} d^{3}+7 a^{3} c^{2} d^{5}-2 a^{3} d^{7}-6 a^{2} b \,c^{5} d^{2}-15 a^{2} b \,c^{3} d^{4}-6 a^{2} b c \,d^{6}+9 a \,b^{2} c^{4} d^{3}+18 a \,b^{2} c^{2} d^{5}+2 b^{3} c^{7}-b^{3} c^{5} d^{2}-10 b^{3} c^{3} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-15 a^{2} b \,c^{3} d^{2}-12 a^{2} b c \,d^{4}-3 a \,b^{2} c^{4} d +30 a \,b^{2} c^{2} d^{3}+7 b^{3} c^{5}-16 b^{3} c^{3} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{2} d^{3}-a^{3} d^{5}-6 a^{2} b \,c^{3} d^{2}-3 a^{2} b c \,d^{4}+9 a \,b^{2} c^{2} d^{3}+2 b^{3} c^{5}-5 b^{3} c^{3} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{3} c^{2} d^{3}+a^{3} d^{5}-9 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}+6 a \,b^{2} d^{5}-2 b^{3} c^{5}+5 b^{3} c^{3} d^{2}-6 d^{4} b^{3} c \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {2 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}}{f}\) | \(670\) |
default | \(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (5 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-9 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d +6 a \,b^{2} c^{2} d^{3}+b^{3} c^{5}-4 b^{3} c^{3} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{4} d^{3}+7 a^{3} c^{2} d^{5}-2 a^{3} d^{7}-6 a^{2} b \,c^{5} d^{2}-15 a^{2} b \,c^{3} d^{4}-6 a^{2} b c \,d^{6}+9 a \,b^{2} c^{4} d^{3}+18 a \,b^{2} c^{2} d^{5}+2 b^{3} c^{7}-b^{3} c^{5} d^{2}-10 b^{3} c^{3} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-15 a^{2} b \,c^{3} d^{2}-12 a^{2} b c \,d^{4}-3 a \,b^{2} c^{4} d +30 a \,b^{2} c^{2} d^{3}+7 b^{3} c^{5}-16 b^{3} c^{3} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{2} d^{3}-a^{3} d^{5}-6 a^{2} b \,c^{3} d^{2}-3 a^{2} b c \,d^{4}+9 a \,b^{2} c^{2} d^{3}+2 b^{3} c^{5}-5 b^{3} c^{3} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{3} c^{2} d^{3}+a^{3} d^{5}-9 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}+6 a \,b^{2} d^{5}-2 b^{3} c^{5}+5 b^{3} c^{3} d^{2}-6 d^{4} b^{3} c \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {2 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}}{f}\) | \(670\) |
risch | \(\text {Expression too large to display}\) | \(2015\) |
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. 822 vs.
\(2 (252) = 504\).
time = 0.48, size = 1732, normalized size = 6.79 \begin {gather*} \left [\frac {4 \, {\left (b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x \cos \left (f x + e\right )^{2} - 4 \, {\left (b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x - {\left (2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7} - {\left (2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} d^{4} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{2} d^{5} - {\left (a^{3} + 6 \, a b^{2}\right )} c d^{6}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 2 \, {\left (2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left (6 \, a^{2} b + 7 \, b^{3}\right )} c^{5} d^{3} + {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{4} d^{4} + {\left (3 \, a^{2} b + 5 \, b^{3}\right )} c^{3} d^{5} - {\left (5 \, a^{3} + 9 \, a b^{2}\right )} c^{2} d^{6}\right )} \cos \left (f x + e\right ) - 2 \, {\left (4 \, {\left (b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right )} f x + 3 \, {\left (b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left (a^{2} b + 3 \, b^{3}\right )} c^{4} d^{4} + {\left (a^{3} + 5 \, a b^{2}\right )} c^{3} d^{5} - {\left (a^{2} b - 2 \, b^{3}\right )} c^{2} d^{6} - {\left (a^{3} + 4 \, a b^{2}\right )} c d^{7}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right )} f \sin \left (f x + e\right ) - {\left (c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\right )} f\right )}}, \frac {2 \, {\left (b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x \cos \left (f x + e\right )^{2} - 2 \, {\left (b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x - {\left (2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7} - {\left (2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} d^{4} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{2} d^{5} - {\left (a^{3} + 6 \, a b^{2}\right )} c d^{6}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) - {\left (2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left (6 \, a^{2} b + 7 \, b^{3}\right )} c^{5} d^{3} + {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{4} d^{4} + {\left (3 \, a^{2} b + 5 \, b^{3}\right )} c^{3} d^{5} - {\left (5 \, a^{3} + 9 \, a b^{2}\right )} c^{2} d^{6}\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right )} f x + 3 \, {\left (b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left (a^{2} b + 3 \, b^{3}\right )} c^{4} d^{4} + {\left (a^{3} + 5 \, a b^{2}\right )} c^{3} d^{5} - {\left (a^{2} b - 2 \, b^{3}\right )} c^{2} d^{6} - {\left (a^{3} + 4 \, a b^{2}\right )} c d^{7}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right )} f \sin \left (f x + e\right ) - {\left (c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 889 vs.
\(2 (252) = 504\).
time = 0.54, size = 889, normalized size = 3.49 \begin {gather*} \frac {\frac {{\left (f x + e\right )} b^{3}}{d^{3}} - \frac {{\left (2 \, b^{3} c^{5} - 5 \, b^{3} c^{3} d^{2} - 2 \, a^{3} c^{2} d^{3} - 3 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + 6 \, b^{3} c d^{4} - a^{3} d^{5} - 6 \, a b^{2} d^{5}\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 (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} \sqrt {c^{2} - d^{2}}} + \frac {b^{3} c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a b^{2} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{2} b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a b^{2} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, b^{3} c^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{2} b c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a b^{2} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{2} b c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, b^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18 \, a b^{2} c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{2} b c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, b^{3} c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{2} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{2} b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, b^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, a^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, a b^{2} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, b^{3} c^{7} - 6 \, a^{2} b c^{5} d^{2} - 5 \, b^{3} c^{5} d^{2} + 4 \, a^{3} c^{4} d^{3} + 9 \, a b^{2} c^{4} d^{3} - 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}}{{\left (c^{6} d^{2} - 2 \, c^{4} d^{4} + c^{2} d^{6}\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 )}^{2}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 20.46, size = 2500, normalized size = 9.80 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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