Optimal. Leaf size=274 \[ \frac {\left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.41, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3067, 3100,
2833, 12, 2738, 211} \begin {gather*} -\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\left (a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a \left (-4 a^3 B+a^2 A b+9 a b^2 B-6 A b^3\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {\left (2 a^5 B+a^4 A b-5 a^3 b^2 B-10 a^2 A b^3+18 a b^4 B-6 A b^5\right ) \sin (c+d x)}{6 b^2 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 2738
Rule 2833
Rule 3067
Rule 3100
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx &=-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {3 a b (A b-a B)+\left (a^2-3 b^2\right ) (A b-a B) \cos (c+d x)+3 b \left (a^2-b^2\right ) B \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\int \frac {-2 b^2 \left (2 a^2 A b+3 A b^3+a^3 B-6 a b^2 B\right )-b \left (a^3 A b-6 a A b^3+2 a^4 B-3 a^2 b^2 B+6 b^4 B\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {3 b^3 \left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right )}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )^3}\\ &=-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=\frac {\left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.35, size = 251, normalized size = 0.92 \begin {gather*} \frac {-\frac {24 \left (a^3 A+4 a A b^2-3 a^2 b B-2 b^3 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {2 \left (-25 a^4 A b-14 a^2 A b^3-6 A b^5+10 a^5 B+17 a^3 b^2 B+18 a b^4 B+6 a \left (a^4 A-9 a^2 A b^2-2 A b^4+a^3 b B+9 a b^3 B\right ) \cos (c+d x)+\left (a^4 A b-10 a^2 A b^3-6 A b^5+2 a^5 B-5 a^3 b^2 B+18 a b^4 B\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{(a+b \cos (c+d x))^3}}{24 \left (a^2-b^2\right )^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.71, size = 371, normalized size = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (A \,a^{3}+6 A \,a^{2} b +2 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -3 a^{2} b B -6 B a \,b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {4 \left (7 A \,a^{2} b +3 A \,b^{3}-a^{3} B -9 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3}-6 A \,a^{2} b +2 A a \,b^{2}-2 A \,b^{3}+2 a^{3} B -3 a^{2} b B +6 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (A \,a^{3}+4 A a \,b^{2}-3 a^{2} b B -2 b^{3} B \right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(371\) |
default | \(\frac {\frac {-\frac {\left (A \,a^{3}+6 A \,a^{2} b +2 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -3 a^{2} b B -6 B a \,b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {4 \left (7 A \,a^{2} b +3 A \,b^{3}-a^{3} B -9 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3}-6 A \,a^{2} b +2 A a \,b^{2}-2 A \,b^{3}+2 a^{3} B -3 a^{2} b B +6 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (A \,a^{3}+4 A a \,b^{2}-3 a^{2} b B -2 b^{3} B \right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(371\) |
risch | \(\text {Expression too large to display}\) | \(1386\) |
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. 575 vs.
\(2 (260) = 520\).
time = 0.47, size = 1220, normalized size = 4.45 \begin {gather*} \left [-\frac {3 \, {\left (A a^{6} - 3 \, B a^{5} b + 4 \, A a^{4} b^{2} - 2 \, B a^{3} b^{3} + {\left (A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 4 \, A a b^{5} - 2 \, B b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} - 2 \, B a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (A a^{5} b - 3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3} - 2 \, B a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (4 \, B a^{7} - 13 \, A a^{6} b + 7 \, B a^{5} b^{2} + 11 \, A a^{4} b^{3} - 11 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + {\left (2 \, B a^{7} + A a^{6} b - 7 \, B a^{5} b^{2} - 11 \, A a^{4} b^{3} + 23 \, B a^{3} b^{4} + 4 \, A a^{2} b^{5} - 18 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (A a^{7} + B a^{6} b - 10 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}}, \frac {3 \, {\left (A a^{6} - 3 \, B a^{5} b + 4 \, A a^{4} b^{2} - 2 \, B a^{3} b^{3} + {\left (A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 4 \, A a b^{5} - 2 \, B b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} - 2 \, B a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (A a^{5} b - 3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3} - 2 \, B a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) + {\left (4 \, B a^{7} - 13 \, A a^{6} b + 7 \, B a^{5} b^{2} + 11 \, A a^{4} b^{3} - 11 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + {\left (2 \, B a^{7} + A a^{6} b - 7 \, B a^{5} b^{2} - 11 \, A a^{4} b^{3} + 23 \, B a^{3} b^{4} + 4 \, A a^{2} b^{5} - 18 \, B a b^{6} + 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (A a^{7} + B a^{6} b - 10 \, A a^{5} b^{2} + 8 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 689 vs.
\(2 (260) = 520\).
time = 0.50, size = 689, normalized size = 2.51 \begin {gather*} -\frac {\frac {3 \, {\left (A a^{3} - 3 \, B a^{2} b + 4 \, A a b^{2} - 2 \, B b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, A a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, B a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 28 \, A a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.15, size = 440, normalized size = 1.61 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}}\right )\,\left (A\,a^3-3\,B\,a^2\,b+4\,A\,a\,b^2-2\,B\,b^3\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}-\frac {\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-B\,a^3+7\,A\,a^2\,b-9\,B\,a\,b^2+3\,A\,b^3\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A\,a^3+2\,A\,b^3-2\,B\,a^3+2\,A\,a\,b^2+6\,A\,a^2\,b-6\,B\,a\,b^2-3\,B\,a^2\,b\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^3-2\,A\,b^3+2\,B\,a^3+2\,A\,a\,b^2-6\,A\,a^2\,b+6\,B\,a\,b^2-3\,B\,a^2\,b\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{d\,\left (3\,a\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )+3\,a^2\,b+a^3+b^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________