Optimal. Leaf size=211 \[ -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {3 b \log (\sin (c+d x))}{a^4 d}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.43, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3650, 3730,
3732, 3611, 3556} \begin {gather*} -\frac {3 b \log (\sin (c+d x))}{a^4 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3611
Rule 3650
Rule 3730
Rule 3732
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (3 b+a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{a}\\ &=-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (6 b \left (a^2+b^2\right )+2 a^3 \tan (c+d x)+2 b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (6 b \left (a^2+b^2\right )^2+2 a^3 \left (a^2-b^2\right ) \tan (c+d x)+2 b \left (a^4+6 a^2 b^2+3 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {(3 b) \int \cot (c+d x) \, dx}{a^4}+\frac {\left (b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^3}\\ &=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {3 b \log (\sin (c+d x))}{a^4 d}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.93, size = 178, normalized size = 0.84 \begin {gather*} -\frac {\frac {2 \cot (c+d x)}{a^3}+\frac {b^5}{a^4 \left (a^2+b^2\right ) (b+a \cot (c+d x))^2}-\frac {2 b^4 \left (5 a^2+3 b^2\right )}{a^4 \left (a^2+b^2\right )^2 (b+a \cot (c+d x))}+\frac {\log (i-\cot (c+d x))}{(i a+b)^3}+\frac {\log (i+\cot (c+d x))}{(-i a+b)^3}-\frac {2 b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (b+a \cot (c+d x))}{a^4 \left (a^2+b^2\right )^3}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 201, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 b^{2} a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(201\) |
default | \(\frac {-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 b^{2} a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(201\) |
norman | \(\frac {\frac {b \left (3 a^{4} b +11 a^{2} b^{3}+6 b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {1}{d a}+\frac {b^{2} \left (4 a^{4} b +17 a^{2} b^{3}+9 b^{5}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(422\) |
risch | \(\frac {x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 b^{2} a}+\frac {6 i b x}{a^{4}}+\frac {6 i b c}{a^{4} d}-\frac {20 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {20 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} x}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} c}{a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} x}{a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} c}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i \left (a^{7}-a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-10 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{6} b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 i b^{7} {\mathrm e}^{2 i \left (d x +c \right )}+3 i b^{7}+i a^{6} b +3 i a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+3 a \,b^{6}+3 a^{5} b^{2}-5 i a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-15 i a^{2} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-6 i a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{6} b \,{\mathrm e}^{4 i \left (d x +c \right )}+8 i a^{2} b^{5}+3 i a^{4} b^{3}+3 i b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+8 a^{3} b^{4}+6 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+3 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (-i b +a \right )^{3} a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {9 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(862\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 349, normalized size = 1.65 \begin {gather*} -\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, a^{4} b^{3} + 9 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, a^{6} + 4 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + 2 \, {\left (a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, a^{5} b + 17 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} + \frac {6 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \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. 585 vs.
\(2 (209) = 418\).
time = 1.03, size = 585, normalized size = 2.77 \begin {gather*} -\frac {2 \, a^{9} + 6 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 2 \, a^{3} b^{6} - {\left (9 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - 2 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} - 2 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 3 \, a b^{8} + 2 \, {\left (a^{8} b - 3 \, a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (10 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 3 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (10 \, a^{5} b^{4} + 9 \, a^{3} b^{6} + 3 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (10 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + 3 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (4 \, a^{8} b + 12 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 2 \, {\left (a^{9} - 3 \, a^{7} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 357, normalized size = 1.69 \begin {gather*} -\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, a^{4} b^{4} + 9 \, a^{2} b^{6} + 3 \, b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} + \frac {30 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} + 27 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 9 \, b^{9} \tan \left (d x + c\right )^{2} + 68 \, a^{5} b^{4} \tan \left (d x + c\right ) + 66 \, a^{3} b^{6} \tan \left (d x + c\right ) + 22 \, a b^{8} \tan \left (d x + c\right ) + 39 \, a^{6} b^{3} + 41 \, a^{4} b^{5} + 14 \, a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac {6 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {2 \, {\left (3 \, b \tan \left (d x + c\right ) - a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.80, size = 293, normalized size = 1.39 \begin {gather*} \frac {b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^4+9\,a^2\,b^2+3\,b^4\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^4\,b^2+6\,a^2\,b^4+3\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^4\,b+17\,a^2\,b^3+9\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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