Optimal. Leaf size=278 \[ \frac {\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac {b^6}{a d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^4}+\frac {\log (\cos (c+d x))}{a^2 d}-\frac {5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac {5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac {1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac {1}{16 d (a-b)^2 (\sec (c+d x)+1)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 278, 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 = {3885, 894} \[ \frac {b^6}{a d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^4}+\frac {\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac {\log (\cos (c+d x))}{a^2 d}-\frac {5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac {5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac {1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac {1}{16 d (a-b)^2 (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 894
Rule 3885
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=-\frac {b^6 \operatorname {Subst}\left (\int \frac {1}{x (a+x)^2 \left (b^2-x^2\right )^3} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac {b^6 \operatorname {Subst}\left (\int \left (\frac {1}{8 b^4 (a+b)^2 (b-x)^3}+\frac {5 a+9 b}{16 b^5 (a+b)^3 (b-x)^2}+\frac {4 a^2+13 a b+12 b^2}{8 b^6 (a+b)^4 (b-x)}+\frac {1}{a^2 b^6 x}+\frac {1}{a (a-b)^3 (a+b)^3 (a+x)^2}+\frac {7 a^2-b^2}{a^2 (a-b)^4 (a+b)^4 (a+x)}-\frac {1}{8 (a-b)^2 b^4 (b+x)^3}+\frac {-5 a+9 b}{16 (a-b)^3 b^5 (b+x)^2}+\frac {-4 a^2+13 a b-12 b^2}{8 (a-b)^4 b^6 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 (a+b)^4 d}+\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (1+\sec (c+d x))}{8 (a-b)^4 d}-\frac {b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^4 d}-\frac {1}{16 (a+b)^2 d (1-\sec (c+d x))^2}-\frac {5 a+9 b}{16 (a+b)^3 d (1-\sec (c+d x))}-\frac {1}{16 (a-b)^2 d (1+\sec (c+d x))^2}-\frac {5 a-9 b}{16 (a-b)^3 d (1+\sec (c+d x))}+\frac {b^6}{a \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.14, size = 473, normalized size = 1.70 \[ \frac {\sec ^2(c+d x) (a \cos (c+d x)+b) \left (\frac {64 b^7}{a^2 (b-a)^3 (a+b)^3}+\frac {8 \left (4 a^2-13 a b+12 b^2\right ) \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a-b)^4}-\frac {16 i \left (4 a^2-13 a b+12 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a-b)^4}-\frac {16 i \left (4 a^2+13 a b+12 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a+b)^4}+\frac {8 \left (4 a^2+13 a b+12 b^2\right ) \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a+b)^4}+\frac {64 \left (b^8-7 a^2 b^6\right ) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a^2 \left (a^2-b^2\right )^4}+\frac {128 i \left (a^6-4 a^4 b^2+6 a^2 b^4+3 b^6\right ) (c+d x) (a \cos (c+d x)+b)}{(a-b)^4 (a+b)^4}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac {2 (7 a+11 b) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^3}-\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}+\frac {2 (7 a-11 b) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^3}\right )}{64 d (a+b \sec (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.10, size = 1378, normalized size = 4.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.19, size = 795, normalized size = 2.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.74, size = 367, normalized size = 1.32 \[ -\frac {b^{7}}{d \,a^{2} \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}-\frac {7 b^{6} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{8} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4} a^{2}}-\frac {1}{16 d \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {7 a}{16 d \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}-\frac {11 b}{16 d \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right ) a^{2}}{2 d \left (a +b \right )^{4}}+\frac {13 \ln \left (-1+\cos \left (d x +c \right )\right ) a b}{8 d \left (a +b \right )^{4}}+\frac {3 \ln \left (-1+\cos \left (d x +c \right )\right ) b^{2}}{2 d \left (a +b \right )^{4}}-\frac {1}{16 d \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {7 a}{16 d \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}-\frac {11 b}{16 d \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (1+\cos \left (d x +c \right )\right ) a^{2}}{2 d \left (a -b \right )^{4}}-\frac {13 \ln \left (1+\cos \left (d x +c \right )\right ) a b}{8 d \left (a -b \right )^{4}}+\frac {3 \ln \left (1+\cos \left (d x +c \right )\right ) b^{2}}{2 d \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.38, size = 558, normalized size = 2.01 \[ -\frac {\frac {8 \, {\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}} - \frac {{\left (4 \, a^{2} - 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (4 \, a^{2} + 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (3 \, a^{6} b - 6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, b^{7} + {\left (5 \, a^{6} b - 13 \, a^{4} b^{3} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - {\left (4 \, a^{7} - 11 \, a^{5} b^{2} + 7 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (7 \, a^{6} b - 17 \, a^{4} b^{3} - 6 \, a^{2} b^{5} - 8 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7} + {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.97, size = 471, normalized size = 1.69 \[ \frac {\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{4\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-13\,a^4+20\,a^3\,b+18\,a^2\,b^2-44\,a\,b^3+19\,b^4\right )}{4\,{\left (a+b\right )}^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^7-10\,a^6\,b+5\,a^5\,b^2+20\,a^4\,b^3-35\,a^3\,b^4+22\,a^2\,b^5-5\,a\,b^6+32\,b^7\right )}{a\,{\left (a+b\right )}^3\,\left (a-b\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (16\,a^4-64\,a^3\,b+96\,a^2\,b^2-64\,a\,b^3+16\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a^4-32\,a^3\,b+32\,a\,b^3-16\,b^4\right )\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d\,{\left (a-b\right )}^2}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {16\,a^2+32\,a\,b-48\,b^2}{512\,{\left (a-b\right )}^4}-\frac {7}{32\,{\left (a-b\right )}^2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a^2+13\,a\,b+12\,b^2\right )}{d\,\left (4\,a^4+16\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3+4\,b^4\right )}-\frac {b^6\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (7\,a^2-b^2\right )}{a^2\,d\,{\left (a^2-b^2\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________