Optimal. Leaf size=84 \[ -\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) (-5 a-4 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}+\frac {1}{15} \int \cot ^2(c+d x) (15 a+8 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}+\frac {1}{15} \int -15 a \, dx\\ &=-a x-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 79, normalized size = 0.94 \[ -\frac {a \cot ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(c+d x)\right )}{5 d}-\frac {b \csc ^5(c+d x)}{5 d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 130, normalized size = 1.55 \[ -\frac {23 \, a \cos \left (d x + c\right )^{5} + 15 \, b \cos \left (d x + c\right )^{4} - 35 \, a \cos \left (d x + c\right )^{3} - 20 \, b \cos \left (d x + c\right )^{2} + 15 \, a \cos \left (d x + c\right ) + 15 \, {\left (a d x \cos \left (d x + c\right )^{4} - 2 \, a d x \cos \left (d x + c\right )^{2} + a d x\right )} \sin \left (d x + c\right ) + 8 \, b}{15 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.37, size = 170, normalized size = 2.02 \[ \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, {\left (d x + c\right )} a + 330 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 150 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {330 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 150 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 25 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.82, size = 129, normalized size = 1.54 \[ \frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.69, size = 79, normalized size = 0.94 \[ -\frac {{\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a + \frac {{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} b}{\sin \left (d x + c\right )^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.39, size = 132, normalized size = 1.57 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a}{160}-\frac {b}{160}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\left (22\,a+10\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {7\,a}{3}-\frac {5\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{5}+\frac {b}{5}\right )}{32\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {7\,a}{96}-\frac {5\,b}{96}\right )}{d}-a\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {11\,a}{16}-\frac {5\,b}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{6}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________