Optimal. Leaf size=254 \[ \frac {a^3 x}{c^3}-\frac {(b c-a d) \left (2 a b c d \left (4 c^2-d^2\right )-b^2 c^2 \left (c^2+2 d^2\right )-a^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 (c-d)^{5/2} (c+d)^{5/2} f}+\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {(b c-a d)^2 \left (5 a c^2-3 b c d-2 a d^2\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.79, antiderivative size = 254, 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 = {4026, 2871,
3100, 2814, 2738, 214} \begin {gather*} \frac {a^3 x}{c^3}-\frac {(b c-a d) \left (-\left (a^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right )+2 a b c d \left (4 c^2-d^2\right )-b^2 c^2 \left (c^2+2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac {(b c-a d)^2 \left (5 a c^2-2 a d^2-3 b c d\right ) \sin (e+f x)}{2 c^2 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)}+\frac {(b c-a d)^2 \sin (e+f x) (a \cos (e+f x)+b)}{2 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 2738
Rule 2814
Rule 2871
Rule 3100
Rule 4026
Rubi steps
\begin {align*} \int \frac {(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^3} \, dx &=\int \frac {(b+a \cos (e+f x))^3}{(d+c \cos (e+f x))^3} \, dx\\ &=\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {\int \frac {5 a b^2 c^2-4 a^2 b c d-2 b^3 c d+a^3 d^2+\left (b^3 c^2-2 a^3 c d-4 a b^2 c d+a^2 b \left (6 c^2-d^2\right )\right ) \cos (e+f x)+2 a^3 \left (c^2-d^2\right ) \cos ^2(e+f x)}{(d+c \cos (e+f x))^2} \, dx}{2 c \left (c^2-d^2\right )}\\ &=\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {(b c-a d)^2 \left (5 a c^2-3 b c d-2 a d^2\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}+\frac {\int \frac {-c \left (9 a b^2 c^2 d-3 a^2 b c \left (2 c^2+d^2\right )-b^3 c \left (c^2+2 d^2\right )+a^3 \left (4 c^2 d-d^3\right )\right )+2 a^3 \left (c^2-d^2\right )^2 \cos (e+f x)}{d+c \cos (e+f x)} \, dx}{2 c^2 \left (c^2-d^2\right )^2}\\ &=\frac {a^3 x}{c^3}+\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {(b c-a d)^2 \left (5 a c^2-3 b c d-2 a d^2\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}-\frac {\left (9 a b^2 c^4 d-3 a^2 b c^3 \left (2 c^2+d^2\right )-b^3 c^3 \left (c^2+2 d^2\right )+a^3 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \int \frac {1}{d+c \cos (e+f x)} \, dx}{2 c^3 \left (c^2-d^2\right )^2}\\ &=\frac {a^3 x}{c^3}+\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {(b c-a d)^2 \left (5 a c^2-3 b c d-2 a d^2\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}-\frac {\left (9 a b^2 c^4 d-3 a^2 b c^3 \left (2 c^2+d^2\right )-b^3 c^3 \left (c^2+2 d^2\right )+a^3 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \text {Subst}\left (\int \frac {1}{c+d+(-c+d) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{c^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {a^3 x}{c^3}+\frac {(b c-a d) \left (6 a^2 c^4+b^2 c^4-8 a b c^3 d-5 a^2 c^2 d^2+2 b^2 c^2 d^2+2 a b c d^3+2 a^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 (c-d)^{5/2} (c+d)^{5/2} f}+\frac {(b c-a d)^2 (b+a \cos (e+f x)) \sin (e+f x)}{2 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}+\frac {(b c-a d)^2 \left (5 a c^2-3 b c d-2 a d^2\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(517\) vs. \(2(254)=508\).
time = 2.21, size = 517, normalized size = 2.04 \begin {gather*} \frac {-\frac {4 \left (-9 a b^2 c^4 d+3 a^2 b c^3 \left (2 c^2+d^2\right )+b^3 c^3 \left (c^2+2 d^2\right )+a^3 \left (-6 c^4 d+5 c^2 d^3-2 d^5\right )\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{5/2}}+\frac {2 a^3 c^6 e-6 a^3 c^2 d^4 e+4 a^3 d^6 e+2 a^3 c^6 f x-6 a^3 c^2 d^4 f x+4 a^3 d^6 f x+8 a^3 c d \left (c^2-d^2\right )^2 (e+f x) \cos (e+f x)+2 a^3 \left (c^3-c d^2\right )^2 (e+f x) \cos (2 (e+f x))+2 b^3 c^6 \sin (e+f x)+6 a b^2 c^5 d \sin (e+f x)-18 a^2 b c^4 d^2 \sin (e+f x)-8 b^3 c^4 d^2 \sin (e+f x)+10 a^3 c^3 d^3 \sin (e+f x)+12 a b^2 c^3 d^3 \sin (e+f x)-4 a^3 c d^5 \sin (e+f x)+6 a b^2 c^6 \sin (2 (e+f x))-12 a^2 b c^5 d \sin (2 (e+f x))-3 b^3 c^5 d \sin (2 (e+f x))+6 a^3 c^4 d^2 \sin (2 (e+f x))+3 a b^2 c^4 d^2 \sin (2 (e+f x))+3 a^2 b c^3 d^3 \sin (2 (e+f x))-3 a^3 c^2 d^4 \sin (2 (e+f x))}{\left (c^2-d^2\right )^2 (d+c \cos (e+f x))^2}}{4 c^3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.51, size = 458, normalized size = 1.80
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{3} c^{2} d^{2}+a^{3} c \,d^{3}-2 a^{3} d^{4}-12 a^{2} b \,c^{3} d -3 a^{2} b \,c^{2} d^{2}+6 a \,b^{2} c^{4}+3 a \,b^{2} c^{3} d +6 a \,b^{2} c^{2} d^{2}-b^{3} c^{4}-4 b^{3} c^{3} d \right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}+\frac {c \left (6 a^{3} c^{2} d^{2}-a^{3} c \,d^{3}-2 a^{3} d^{4}-12 a^{2} b \,c^{3} d +3 a^{2} b \,c^{2} d^{2}+6 a \,b^{2} c^{4}-3 a \,b^{2} c^{3} d +6 a \,b^{2} c^{2} d^{2}+b^{3} c^{4}-4 b^{3} c^{3} d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c -d \right )^{2}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (6 a^{3} c^{4} d -5 a^{3} c^{2} d^{3}+2 a^{3} d^{5}-6 a^{2} b \,c^{5}-3 a^{2} b \,c^{3} d^{2}+9 a \,b^{2} c^{4} d -b^{3} c^{5}-2 b^{3} c^{3} d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{3}}+\frac {2 a^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{3}}}{f}\) | \(458\) |
default | \(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{3} c^{2} d^{2}+a^{3} c \,d^{3}-2 a^{3} d^{4}-12 a^{2} b \,c^{3} d -3 a^{2} b \,c^{2} d^{2}+6 a \,b^{2} c^{4}+3 a \,b^{2} c^{3} d +6 a \,b^{2} c^{2} d^{2}-b^{3} c^{4}-4 b^{3} c^{3} d \right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}+\frac {c \left (6 a^{3} c^{2} d^{2}-a^{3} c \,d^{3}-2 a^{3} d^{4}-12 a^{2} b \,c^{3} d +3 a^{2} b \,c^{2} d^{2}+6 a \,b^{2} c^{4}-3 a \,b^{2} c^{3} d +6 a \,b^{2} c^{2} d^{2}+b^{3} c^{4}-4 b^{3} c^{3} d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c -d \right )^{2}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (6 a^{3} c^{4} d -5 a^{3} c^{2} d^{3}+2 a^{3} d^{5}-6 a^{2} b \,c^{5}-3 a^{2} b \,c^{3} d^{2}+9 a \,b^{2} c^{4} d -b^{3} c^{5}-2 b^{3} c^{3} d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{3}}+\frac {2 a^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{3}}}{f}\) | \(458\) |
risch | \(\text {Expression too large to display}\) | \(2042\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 795 vs.
\(2 (244) = 488\).
time = 3.80, size = 1653, normalized size = 6.51 \begin {gather*} \left [\frac {4 \, {\left (a^{3} c^{8} - 3 \, a^{3} c^{6} d^{2} + 3 \, a^{3} c^{4} d^{4} - a^{3} c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 8 \, {\left (a^{3} c^{7} d - 3 \, a^{3} c^{5} d^{3} + 3 \, a^{3} c^{3} d^{5} - a^{3} c d^{7}\right )} f x \cos \left (f x + e\right ) + 4 \, {\left (a^{3} c^{6} d^{2} - 3 \, a^{3} c^{4} d^{4} + 3 \, a^{3} c^{2} d^{6} - a^{3} d^{8}\right )} f x - {\left (5 \, a^{3} c^{2} d^{5} - 2 \, a^{3} d^{7} + {\left (6 \, a^{2} b + b^{3}\right )} c^{5} d^{2} - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{3} d^{4} + {\left (5 \, a^{3} c^{4} d^{3} - 2 \, a^{3} c^{2} d^{5} + {\left (6 \, a^{2} b + b^{3}\right )} c^{7} - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{6} d + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, a^{3} c^{3} d^{4} - 2 \, a^{3} c d^{6} + {\left (6 \, a^{2} b + b^{3}\right )} c^{6} d - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{5} d^{2} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{4} d^{3}\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 (b^{3} c^{8} + 3 \, a b^{2} c^{7} d + 2 \, a^{3} c d^{7} - {\left (9 \, a^{2} b + 5 \, b^{3}\right )} c^{6} d^{2} + {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c^{5} d^{3} + {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{4} d^{4} - {\left (7 \, a^{3} + 6 \, a b^{2}\right )} c^{3} d^{5} + 3 \, {\left (2 \, a b^{2} c^{8} - a^{2} b c^{3} d^{5} + a^{3} c^{2} d^{6} - {\left (4 \, a^{2} b + b^{3}\right )} c^{7} d + {\left (2 \, a^{3} - a b^{2}\right )} c^{6} d^{2} + {\left (5 \, a^{2} b + b^{3}\right )} c^{5} d^{3} - {\left (3 \, a^{3} + a b^{2}\right )} c^{4} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}, \frac {2 \, {\left (a^{3} c^{8} - 3 \, a^{3} c^{6} d^{2} + 3 \, a^{3} c^{4} d^{4} - a^{3} c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 4 \, {\left (a^{3} c^{7} d - 3 \, a^{3} c^{5} d^{3} + 3 \, a^{3} c^{3} d^{5} - a^{3} c d^{7}\right )} f x \cos \left (f x + e\right ) + 2 \, {\left (a^{3} c^{6} d^{2} - 3 \, a^{3} c^{4} d^{4} + 3 \, a^{3} c^{2} d^{6} - a^{3} d^{8}\right )} f x + {\left (5 \, a^{3} c^{2} d^{5} - 2 \, a^{3} d^{7} + {\left (6 \, a^{2} b + b^{3}\right )} c^{5} d^{2} - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{3} d^{4} + {\left (5 \, a^{3} c^{4} d^{3} - 2 \, a^{3} c^{2} d^{5} + {\left (6 \, a^{2} b + b^{3}\right )} c^{7} - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{6} d + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, a^{3} c^{3} d^{4} - 2 \, a^{3} c d^{6} + {\left (6 \, a^{2} b + b^{3}\right )} c^{6} d - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{5} d^{2} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{4} d^{3}\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 (b^{3} c^{8} + 3 \, a b^{2} c^{7} d + 2 \, a^{3} c d^{7} - {\left (9 \, a^{2} b + 5 \, b^{3}\right )} c^{6} d^{2} + {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c^{5} d^{3} + {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{4} d^{4} - {\left (7 \, a^{3} + 6 \, a b^{2}\right )} c^{3} d^{5} + 3 \, {\left (2 \, a b^{2} c^{8} - a^{2} b c^{3} d^{5} + a^{3} c^{2} d^{6} - {\left (4 \, a^{2} b + b^{3}\right )} c^{7} d + {\left (2 \, a^{3} - a b^{2}\right )} c^{6} d^{2} + {\left (5 \, a^{2} b + b^{3}\right )} c^{5} d^{3} - {\left (3 \, a^{3} + a b^{2}\right )} c^{4} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{3}}{\left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 818 vs.
\(2 (238) = 476\).
time = 0.63, size = 818, normalized size = 3.22 \begin {gather*} \frac {\frac {{\left (f x + e\right )} a^{3}}{c^{3}} + \frac {{\left (6 \, a^{2} b c^{5} + b^{3} c^{5} - 6 \, a^{3} c^{4} d - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + 2 \, b^{3} c^{3} d^{2} + 5 \, a^{3} c^{2} d^{3} - 2 \, a^{3} d^{5}\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 (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {6 \, a b^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{2} b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a b^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, b^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{2} b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a b^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, b^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a b^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a^{2} b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, b^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, a^{2} b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, b^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a^{2} b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{6} - 2 \, c^{4} d^{2} + c^{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 )}^{2}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 14.50, size = 2500, normalized size = 9.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________