Optimal. Leaf size=379 \[ \frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} b^3 (a+b)^{3/2} f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f} \]
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Rubi [A]
time = 0.48, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {4073, 3031,
2743, 12, 2738, 214, 3855, 3852, 8, 3853} \begin {gather*} \frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \tan (e+f x)}{b^4 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {d^2 \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^5 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^4 (5 b c-2 a d) \tan (e+f x) \sec (e+f x)}{2 b^3 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^3 f (a-b)^{3/2} (a+b)^{3/2}}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {d^5 \tan (e+f x)}{b^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 214
Rule 2738
Rule 2743
Rule 3031
Rule 3852
Rule 3853
Rule 3855
Rule 4073
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx &=\int \frac {(d+c \cos (e+f x))^5 \sec ^4(e+f x)}{(b+a \cos (e+f x))^2} \, dx\\ &=\int \left (\frac {(-b c+a d)^5}{a b^4 (b+a \cos (e+f x))^2}+\frac {(-b c+a d)^4 (b c+4 a d)}{a b^5 (b+a \cos (e+f x))}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \sec (e+f x)}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \sec ^2(e+f x)}{b^4}+\frac {d^4 (5 b c-2 a d) \sec ^3(e+f x)}{b^3}+\frac {d^5 \sec ^4(e+f x)}{b^2}\right ) \, dx\\ &=\frac {d^5 \int \sec ^4(e+f x) \, dx}{b^2}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec ^3(e+f x) \, dx}{b^3}-\frac {(b c-a d)^5 \int \frac {1}{(b+a \cos (e+f x))^2} \, dx}{a b^4}+\frac {\left ((b c-a d)^4 (b c+4 a d)\right ) \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^5}+\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \int \sec ^2(e+f x) \, dx}{b^4}+\frac {\left (d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )\right ) \int \sec (e+f x) \, dx}{b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec (e+f x) \, dx}{2 b^3}+\frac {(b c-a d)^5 \int \frac {b}{b+a \cos (e+f x)} \, dx}{a b^4 \left (a^2-b^2\right )}-\frac {d^5 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{b^2 f}+\frac {\left (2 (b c-a d)^4 (b c+4 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a b^5 f}-\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b^4 f}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {(b c-a d)^5 \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^3 \left (a^2-b^2\right )}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {\left (2 (b c-a d)^5\right ) \text {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a b^3 \left (a^2-b^2\right ) f}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} b^3 (a+b)^{3/2} f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1137\) vs. \(2(379)=758\).
time = 6.53, size = 1137, normalized size = 3.00 \begin {gather*} -\frac {2 (b c-a d)^4 \left (-a b c-4 a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 (c+d \sec (e+f x))^5}{b^5 \sqrt {a^2-b^2} \left (-a^2+b^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {\left (-20 b^3 c^3 d^2+40 a b^2 c^2 d^3-30 a^2 b c d^4-5 b^3 c d^4+8 a^3 d^5+2 a b^2 d^5\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {\left (20 b^3 c^3 d^2-40 a b^2 c^2 d^3+30 a^2 b c d^4+5 b^3 c d^4-8 a^3 d^5-2 a b^2 d^5\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {(b+a \cos (e+f x)) (c+d \sec (e+f x))^5 \left (-60 a^2 b^3 c^2 d^3 \sin (e+f x)+60 b^5 c^2 d^3 \sin (e+f x)+45 a^3 b^2 c d^4 \sin (e+f x)-45 a b^4 c d^4 \sin (e+f x)-12 a^4 b d^5 \sin (e+f x)+12 b^5 d^5 \sin (e+f x)+6 b^5 c^5 \sin (2 (e+f x))-30 a b^4 c^4 d \sin (2 (e+f x))+60 a^2 b^3 c^3 d^2 \sin (2 (e+f x))-120 a^3 b^2 c^2 d^3 \sin (2 (e+f x))+60 a b^4 c^2 d^3 \sin (2 (e+f x))+90 a^4 b c d^4 \sin (2 (e+f x))-90 a^2 b^3 c d^4 \sin (2 (e+f x))+30 b^5 c d^4 \sin (2 (e+f x))-24 a^5 d^5 \sin (2 (e+f x))+22 a^3 b^2 d^5 \sin (2 (e+f x))-4 a b^4 d^5 \sin (2 (e+f x))-60 a^2 b^3 c^2 d^3 \sin (3 (e+f x))+60 b^5 c^2 d^3 \sin (3 (e+f x))+45 a^3 b^2 c d^4 \sin (3 (e+f x))-45 a b^4 c d^4 \sin (3 (e+f x))-12 a^4 b d^5 \sin (3 (e+f x))+8 a^2 b^3 d^5 \sin (3 (e+f x))+4 b^5 d^5 \sin (3 (e+f x))+3 b^5 c^5 \sin (4 (e+f x))-15 a b^4 c^4 d \sin (4 (e+f x))+30 a^2 b^3 c^3 d^2 \sin (4 (e+f x))-60 a^3 b^2 c^2 d^3 \sin (4 (e+f x))+30 a b^4 c^2 d^3 \sin (4 (e+f x))+45 a^4 b c d^4 \sin (4 (e+f x))-30 a^2 b^3 c d^4 \sin (4 (e+f x))-12 a^5 d^5 \sin (4 (e+f x))+7 a^3 b^2 d^5 \sin (4 (e+f x))+2 a b^4 d^5 \sin (4 (e+f x))\right )}{24 b^4 \left (-a^2+b^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.46, size = 689, normalized size = 1.82
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a -b \right )}-\frac {\left (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\) | \(689\) |
default | \(\frac {-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a -b \right )}-\frac {\left (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\) | \(689\) |
risch | \(\text {Expression too large to display}\) | \(3699\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{5} \sec {\left (e + f x \right )}}{\left (a + b \sec {\left (e + f x \right )}\right )^{2}}\, dx \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. 857 vs.
\(2 (355) = 710\).
time = 0.62, size = 857, normalized size = 2.26 \begin {gather*} -\frac {\frac {12 \, {\left (a b^{5} c^{5} - 5 \, b^{6} c^{4} d - 10 \, a^{3} b^{3} c^{3} d^{2} + 20 \, a b^{5} c^{3} d^{2} + 20 \, a^{4} b^{2} c^{2} d^{3} - 30 \, a^{2} b^{4} c^{2} d^{3} - 15 \, a^{5} b c d^{4} + 20 \, a^{3} b^{3} c d^{4} + 4 \, a^{6} d^{5} - 5 \, a^{4} b^{2} d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {12 \, {\left (b^{5} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a b^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 10 \, a^{2} b^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 10 \, a^{3} b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, a^{4} b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{5} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )}} - \frac {3 \, {\left (20 \, b^{3} c^{3} d^{2} - 40 \, a b^{2} c^{2} d^{3} + 30 \, a^{2} b c d^{4} + 5 \, b^{3} c d^{4} - 8 \, a^{3} d^{5} - 2 \, a b^{2} d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{5}} + \frac {3 \, {\left (20 \, b^{3} c^{3} d^{2} - 40 \, a b^{2} c^{2} d^{3} + 30 \, a^{2} b c d^{4} + 5 \, b^{3} c d^{4} - 8 \, a^{3} d^{5} - 2 \, a b^{2} d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{5}} + \frac {2 \, {\left (60 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 60 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, b^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 18 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 36 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 60 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, b^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} b^{4}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.95, size = 2500, normalized size = 6.60 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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