Integrand size = 21, antiderivative size = 157 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {(2 a+3 b) \log (1-\sec (c+d x))}{4 (a+b)^2 d}-\frac {(2 a-3 b) \log (1+\sec (c+d x))}{4 (a-b)^2 d}-\frac {b^4 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sec (c+d x))}+\frac {1}{4 (a-b) d (1+\sec (c+d x))} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3970, 908} \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b^4 \log (a+b \sec (c+d x))}{a d \left (a^2-b^2\right )^2}+\frac {1}{4 d (a+b) (1-\sec (c+d x))}+\frac {1}{4 d (a-b) (\sec (c+d x)+1)}-\frac {(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac {(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac {\log (\cos (c+d x))}{a d} \]
[In]
[Out]
Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = \frac {b^4 \text {Subst}\left (\int \frac {1}{x (a+x) \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) (b-x)^2}+\frac {2 a+3 b}{4 b^4 (a+b)^2 (b-x)}+\frac {1}{a b^4 x}-\frac {1}{a (a-b)^2 (a+b)^2 (a+x)}-\frac {1}{4 (a-b) b^3 (b+x)^2}+\frac {-2 a+3 b}{4 (a-b)^2 b^4 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d} \\ & = -\frac {\log (\cos (c+d x))}{a d}-\frac {(2 a+3 b) \log (1-\sec (c+d x))}{4 (a+b)^2 d}-\frac {(2 a-3 b) \log (1+\sec (c+d x))}{4 (a-b)^2 d}-\frac {b^4 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sec (c+d x))}+\frac {1}{4 (a-b) d (1+\sec (c+d x))} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {b^4 \left (-\frac {4 \log (\cos (c+d x))}{a b^4}-\frac {(2 a+3 b) \log (1-\sec (c+d x))}{b^4 (a+b)^2}-\frac {(2 a-3 b) \log (1+\sec (c+d x))}{(a-b)^2 b^4}-\frac {4 \log (a+b \sec (c+d x))}{a (a-b)^2 (a+b)^2}-\frac {1}{b^4 (a+b) (-1+\sec (c+d x))}+\frac {1}{(a-b) b^4 (1+\sec (c+d x))}\right )}{4 d} \]
[In]
[Out]
Time = 0.94 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {-\frac {b^{4} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a}-\frac {1}{\left (4 a -4 b \right ) \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{2}}+\frac {1}{\left (4 a +4 b \right ) \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-2 a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}}{d}\) | \(126\) |
default | \(\frac {-\frac {b^{4} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a}-\frac {1}{\left (4 a -4 b \right ) \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{2}}+\frac {1}{\left (4 a +4 b \right ) \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-2 a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}}{d}\) | \(126\) |
risch | \(-\frac {3 i b c}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i b^{4} x}{a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {i a c}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {i a x}{a^{2}+2 a b +b^{2}}+\frac {3 i b c}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 i b x}{2 \left (a^{2}-2 a b +b^{2}\right )}-\frac {i x}{a}+\frac {i a x}{a^{2}-2 a b +b^{2}}+\frac {3 i b x}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 i b^{4} c}{d a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {i a c}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\) | \(465\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {2 \, a^{4} - 2 \, a^{2} b^{2} - 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 4 \, {\left (b^{4} \cos \left (d x + c\right )^{2} - b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + {\left (2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3} - {\left (2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3} - {\left (2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
[In]
[Out]
\[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {4 \, b^{4} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {{\left (2 \, a - 3 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (2 \, a + 3 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (b \cos \left (d x + c\right ) - a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (147) = 294\).
Time = 0.35 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.57 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {2 \, {\left (2 \, a + 3 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {4 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | a + b - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a + b + \frac {4 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 2 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left | a \right |} \right |}}{{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 2 \, {\left | a \right |} \right |}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left | a \right |}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a - b\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}}{8 \, d} \]
[In]
[Out]
Time = 14.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.11 \[ \int \frac {\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d\,\left (4\,a-4\,b\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a+3\,b\right )}{d\,\left (2\,a^2+4\,a\,b+2\,b^2\right )}-\frac {a-b}{2\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a+b\right )\,\left (4\,a-4\,b\right )}-\frac {b^4\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d\,{\left (a^2-b^2\right )}^2} \]
[In]
[Out]