Optimal. Leaf size=125 \[ -\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.27, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2954,
2952, 2687, 276, 2686, 14} \begin {gather*} -\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 276
Rule 2686
Rule 2687
Rule 2952
Rule 2954
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc ^6(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\int (-a+a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^8(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^9(c+d x)+a^2 \cot ^2(c+d x) \csc ^{10}(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^2}\\ &=\frac {\text {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=\frac {\text {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int \left (x^2+4 x^4+6 x^6+4 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.92, size = 233, normalized size = 1.86 \begin {gather*} -\frac {\csc (c) \csc ^7(c+d x) \sec ^2(c+d x) (630784 \sin (c)-1103872 \sin (d x)-218834 \sin (c+d x)-79576 \sin (2 (c+d x))+119364 \sin (3 (c+d x))+79576 \sin (4 (c+d x))-28420 \sin (5 (c+d x))-34104 \sin (6 (c+d x))-1421 \sin (7 (c+d x))+5684 \sin (8 (c+d x))+1421 \sin (9 (c+d x))+1419264 \sin (2 c+d x)+114688 \sin (c+2 d x)-172032 \sin (2 c+3 d x)-114688 \sin (3 c+4 d x)+40960 \sin (4 c+5 d x)+49152 \sin (5 c+6 d x)+2048 \sin (6 c+7 d x)-8192 \sin (7 c+8 d x)-2048 \sin (8 c+9 d x))}{22708224 a^2 d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 112, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{512 d \,a^{2}}\) | \(112\) |
default | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{512 d \,a^{2}}\) | \(112\) |
risch | \(\frac {32 i \left (693 \,{\mathrm e}^{10 i \left (d x +c \right )}+308 \,{\mathrm e}^{9 i \left (d x +c \right )}+539 \,{\mathrm e}^{8 i \left (d x +c \right )}-56 \,{\mathrm e}^{7 i \left (d x +c \right )}+84 \,{\mathrm e}^{6 i \left (d x +c \right )}+56 \,{\mathrm e}^{5 i \left (d x +c \right )}-20 \,{\mathrm e}^{4 i \left (d x +c \right )}-24 \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{693 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7}}\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 174, normalized size = 1.39 \begin {gather*} -\frac {\frac {\frac {9702 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3234 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {792 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2}} + \frac {33 \, {\left (\frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {56 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{354816 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.60, size = 204, normalized size = 1.63 \begin {gather*} \frac {16 \, \cos \left (d x + c\right )^{9} + 32 \, \cos \left (d x + c\right )^{8} - 40 \, \cos \left (d x + c\right )^{7} - 112 \, \cos \left (d x + c\right )^{6} + 14 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{3} - 70 \, \cos \left (d x + c\right )^{2} + 56 \, \cos \left (d x + c\right ) + 28}{693 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} + 2 \, a^{2} d \cos \left (d x + c\right )^{7} - 2 \, a^{2} d \cos \left (d x + c\right )^{6} - 6 \, a^{2} d \cos \left (d x + c\right )^{5} + 6 \, a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.49, size = 134, normalized size = 1.07 \begin {gather*} -\frac {\frac {33 \, {\left (56 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {63 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 385 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 792 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3234 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9702 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{22}}}{354816 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.69, size = 201, normalized size = 1.61 \begin {gather*} -\frac {99\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1848\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9702\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3234\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-385\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-63\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{354816\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________