Optimal. Leaf size=197 \[ -\frac {\log (\cos (c+d x))}{a^2 d}-\frac {(a+2 b) \log (1-\sec (c+d x))}{2 (a+b)^3 d}-\frac {(a-2 b) \log (1+\sec (c+d x))}{2 (a-b)^3 d}-\frac {b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sec (c+d x))}+\frac {1}{4 (a-b)^2 d (1+\sec (c+d x))}+\frac {b^4}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.17, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3970, 908}
\begin {gather*} \frac {b^4}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac {\log (\cos (c+d x))}{a^2 d}+\frac {1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac {1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac {(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac {(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 3970
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\frac {b^4 \text {Subst}\left (\int \frac {1}{x (a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac {b^4 \text {Subst}\left (\int \left (\frac {1}{4 b^3 (a+b)^2 (b-x)^2}+\frac {a+2 b}{2 b^4 (a+b)^3 (b-x)}+\frac {1}{a^2 b^4 x}-\frac {1}{a (a-b)^2 (a+b)^2 (a+x)^2}+\frac {-5 a^2+b^2}{a^2 (a-b)^3 (a+b)^3 (a+x)}-\frac {1}{4 (a-b)^2 b^3 (b+x)^2}+\frac {-a+2 b}{2 (a-b)^3 b^4 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac {\log (\cos (c+d x))}{a^2 d}-\frac {(a+2 b) \log (1-\sec (c+d x))}{2 (a+b)^3 d}-\frac {(a-2 b) \log (1+\sec (c+d x))}{2 (a-b)^3 d}-\frac {b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sec (c+d x))}+\frac {1}{4 (a-b)^2 d (1+\sec (c+d x))}+\frac {b^4}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.30, size = 351, normalized size = 1.78 \begin {gather*} \frac {(b+a \cos (c+d x)) \left (-\frac {8 b^5}{a^2 (a-b)^2 (a+b)^2}-\frac {16 i \left (a^4-3 a^2 b^2-2 b^4\right ) (c+d x) (b+a \cos (c+d x))}{(a-b)^3 (a+b)^3}+\frac {8 i (a-2 b) \text {ArcTan}(\tan (c+d x)) (b+a \cos (c+d x))}{(a-b)^3}+\frac {8 i (a+2 b) \text {ArcTan}(\tan (c+d x)) (b+a \cos (c+d x))}{(a+b)^3}-\frac {(b+a \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}+\frac {4 (a-2 b) (b+a \cos (c+d x)) \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(-a+b)^3}+\frac {8 b^4 \left (-5 a^2+b^2\right ) (b+a \cos (c+d x)) \log (b+a \cos (c+d x))}{a^2 \left (a^2-b^2\right )^3}-\frac {4 (a+2 b) (b+a \cos (c+d x)) \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^3}-\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}\right ) \sec ^2(c+d x)}{8 d (a+b \sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 164, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {-\frac {b^{5}}{a^{2} \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}-b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{2}}+\frac {1}{4 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a +2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}}{d}\) | \(164\) |
default | \(\frac {-\frac {b^{5}}{a^{2} \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}-b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{2}}+\frac {1}{4 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a +2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}}{d}\) | \(164\) |
risch | \(\frac {i a x}{a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}}+\frac {10 i b^{4} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {i x}{a^{2}}-\frac {2 i b x}{a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}}+\frac {2 i b x}{a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}}+\frac {i a c}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}+\frac {2 i b c}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}+\frac {i a c}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}-\frac {2 i b c}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}-\frac {2 i b^{6} c}{a^{2} d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {i a x}{a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}}+\frac {10 i b^{4} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 i b^{6} x}{a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {-2 a^{4} b \,{\mathrm e}^{5 i \left (d x +c \right )}-2 b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+2 a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-2 a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+4 a^{2} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+4 b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+2 a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-2 a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 a^{4} b \,{\mathrm e}^{i \left (d x +c \right )}-2 b^{5} {\mathrm e}^{i \left (d x +c \right )}}{\left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right ) \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}-\frac {5 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(886\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 303, normalized size = 1.54 \begin {gather*} -\frac {\frac {2 \, {\left (5 \, a^{2} b^{4} - b^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} + \frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {a^{4} b + a^{2} b^{3} + 2 \, b^{5} - 2 \, {\left (a^{4} b + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} - {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )}}{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. 693 vs.
\(2 (187) = 374\).
time = 3.62, size = 693, normalized size = 3.52 \begin {gather*} \frac {a^{6} b + a^{2} b^{5} - 2 \, b^{7} - 2 \, {\left (a^{6} b - a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, a^{2} b^{5} - b^{7} - {\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + {\left (a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right ) - {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d\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 ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d 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. 656 vs.
\(2 (187) = 374\).
time = 0.54, size = 656, normalized size = 3.33 \begin {gather*} -\frac {\frac {4 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {8 \, {\left (5 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac {a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3} + \frac {3 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {20 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (\frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {8 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.48, size = 313, normalized size = 1.59 \begin {gather*} \frac {\frac {a^2-2\,a\,b+b^2}{2\,\left (a+b\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^5-4\,a^4\,b+6\,a^3\,b^2-4\,a^2\,b^3+a\,b^4-16\,b^5\right )}{2\,a\,{\left (a+b\right )}^2\,\left (a-b\right )}}{d\,\left (\left (4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-4\,a^3+4\,a^2\,b+4\,a\,b^2-4\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d\,{\left (a-b\right )}^2}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a+2\,b\right )}{d\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}-\frac {b^4\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (5\,a^2-b^2\right )}{a^2\,d\,{\left (a^2-b^2\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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