Optimal. Leaf size=202 \[ -\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2} d}+\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]
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Rubi [A]
time = 0.53, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2802, 3135,
3134, 3080, 3855, 2739, 632, 210} \begin {gather*} \frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 d \left (a^2-b^2\right )^{3/2}}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2802
Rule 3080
Rule 3134
Rule 3135
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \frac {\csc ^2(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx\\ &=\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (5 a^4-11 a^2 b^2+6 b^4-a b \left (a^2-b^2\right ) \sin (c+d x)-\left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-6 b \left (a^2-b^2\right )^2-a \left (2 a^4-5 a^2 b^2+3 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {(3 b) \int \csc (c+d x) \, dx}{a^4}-\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )}\\ &=\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right ) d}\\ &=\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (2 \left (2 a^4-9 a^2 b^2+6 b^4\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} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right ) d}\\ &=-\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2} d}+\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 3.38, size = 195, normalized size = 0.97 \begin {gather*} \frac {-\frac {2 \left (2 a^4-9 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-a \cot \left (\frac {1}{2} (c+d x)\right )+6 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {a^2 b \cos (c+d x)}{(a+b \sin (c+d x))^2}+\frac {a b \left (-3 a^2+4 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))}+a \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 299, normalized size = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {2 \left (\frac {\frac {a \,b^{2} \left (5 a^{2}-6 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (4 a^{4}+3 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (11 a^{2}-14 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (4 a^{2}-5 b^{2}\right )}{2 a^{2}-2 b^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-9 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(299\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {2 \left (\frac {\frac {a \,b^{2} \left (5 a^{2}-6 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (4 a^{4}+3 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (11 a^{2}-14 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (4 a^{2}-5 b^{2}\right )}{2 a^{2}-2 b^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-9 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(299\) |
risch | \(\frac {i \left (-2 i a^{3} b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i a \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+20 i b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-24 i b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}-6 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+21 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{2} b^{2}-6 b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) d \,a^{3}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}\) | \(800\) |
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. 655 vs.
\(2 (191) = 382\).
time = 0.68, size = 1394, normalized size = 6.90 \begin {gather*} \left [-\frac {2 \, {\left (5 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (8 \, a^{6} b - 17 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (4 \, a^{5} b - 18 \, a^{3} b^{3} + 12 \, a b^{5} - 2 \, {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{6} - 7 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 6 \, b^{6} - {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (2 \, a^{7} + a^{5} b^{2} - 9 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right ) + 6 \, {\left (2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6 \, {\left (2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (2 \, {\left (a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d + {\left ({\left (a^{8} b^{2} - 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{10} - a^{8} b^{2} - a^{6} b^{4} + a^{4} b^{6}\right )} d\right )} \sin \left (d x + c\right )\right )}}, -\frac {{\left (5 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (8 \, a^{6} b - 17 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (4 \, a^{5} b - 18 \, a^{3} b^{3} + 12 \, a b^{5} - 2 \, {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{6} - 7 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 6 \, b^{6} - {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (2 \, a^{7} + a^{5} b^{2} - 9 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right ) + 3 \, {\left (2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (2 \, {\left (a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d + {\left ({\left (a^{8} b^{2} - 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{10} - a^{8} b^{2} - a^{6} b^{4} + a^{4} b^{6}\right )} d\right )} \sin \left (d x + c\right )\right )}}\right ] \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 {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 13.76, size = 339, normalized size = 1.68 \begin {gather*} -\frac {\frac {2 \, {\left (2 \, a^{4} - 9 \, a^{2} b^{2} + 6 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 5 \, a^{2} b^{3}\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} + \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {6 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.85, size = 1762, normalized size = 8.72 \begin {gather*} \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {a^2-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a\,b^3-6\,a^3\,b\right )}{a^2-b^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^4+9\,a^2\,b^2-12\,b^4\right )}{a^2-b^2}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^4+12\,a^2\,b^2-16\,b^4\right )}{a^2-b^2}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a^4\,b+a^2\,b^3-10\,b^5\right )}{a\,\left (a^2-b^2\right )}}{d\,\left (2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^5+8\,a^3\,b^2\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )\,\left (\frac {2\,a^8-15\,a^6\,b^2+12\,a^4\,b^4}{a^8-a^6\,b^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,a^8\,b-46\,a^6\,b^3+60\,a^4\,b^5-24\,a^2\,b^7\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}+\frac {\left (\frac {2\,a^{10}\,b-2\,a^8\,b^3}{a^8-a^6\,b^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-20\,a^{10}\,b^2+22\,a^8\,b^4-8\,a^6\,b^6\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}\right )\,1{}\mathrm {i}}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}+\frac {\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )\,\left (\frac {2\,a^8-15\,a^6\,b^2+12\,a^4\,b^4}{a^8-a^6\,b^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,a^8\,b-46\,a^6\,b^3+60\,a^4\,b^5-24\,a^2\,b^7\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}-\frac {\left (\frac {2\,a^{10}\,b-2\,a^8\,b^3}{a^8-a^6\,b^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-20\,a^{10}\,b^2+22\,a^8\,b^4-8\,a^6\,b^6\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}\right )\,1{}\mathrm {i}}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}}{\frac {2\,\left (6\,a^4\,b-27\,a^2\,b^3+18\,b^5\right )}{a^8-a^6\,b^2}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^6-24\,a^4\,b^2+39\,a^2\,b^4-18\,b^6\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}-\frac {\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )\,\left (\frac {2\,a^8-15\,a^6\,b^2+12\,a^4\,b^4}{a^8-a^6\,b^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,a^8\,b-46\,a^6\,b^3+60\,a^4\,b^5-24\,a^2\,b^7\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}+\frac {\left (\frac {2\,a^{10}\,b-2\,a^8\,b^3}{a^8-a^6\,b^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-20\,a^{10}\,b^2+22\,a^8\,b^4-8\,a^6\,b^6\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}+\frac {\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )\,\left (\frac {2\,a^8-15\,a^6\,b^2+12\,a^4\,b^4}{a^8-a^6\,b^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,a^8\,b-46\,a^6\,b^3+60\,a^4\,b^5-24\,a^2\,b^7\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}-\frac {\left (\frac {2\,a^{10}\,b-2\,a^8\,b^3}{a^8-a^6\,b^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-20\,a^{10}\,b^2+22\,a^8\,b^4-8\,a^6\,b^6\right )}{a^9-2\,a^7\,b^2+a^5\,b^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}\right )}{a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6}}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (a^4-\frac {9\,a^2\,b^2}{2}+3\,b^4\right )\,2{}\mathrm {i}}{d\,\left (a^{10}-3\,a^8\,b^2+3\,a^6\,b^4-a^4\,b^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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