3.18 \(\int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=140 \[ -\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac {\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x \]

[Out]

-a*x-1/9*cot(d*x+c)^9*(a+a*sec(d*x+c))/d+1/63*cot(d*x+c)^7*(9*a+8*a*sec(d*x+c))/d-1/105*cot(d*x+c)^5*(21*a+16*
a*sec(d*x+c))/d+1/315*cot(d*x+c)^3*(105*a+64*a*sec(d*x+c))/d-1/315*cot(d*x+c)*(315*a+128*a*sec(d*x+c))/d

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Rubi [A]  time = 0.14, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac {\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x]),x]

[Out]

-(a*x) - (Cot[c + d*x]^9*(a + a*Sec[c + d*x]))/(9*d) + (Cot[c + d*x]^7*(9*a + 8*a*Sec[c + d*x]))/(63*d) - (Cot
[c + d*x]^5*(21*a + 16*a*Sec[c + d*x]))/(105*d) + (Cot[c + d*x]^3*(105*a + 64*a*Sec[c + d*x]))/(315*d) - (Cot[
c + d*x]*(315*a + 128*a*Sec[c + d*x]))/(315*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {1}{9} \int \cot ^8(c+d x) (-9 a-8 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}+\frac {1}{63} \int \cot ^6(c+d x) (63 a+48 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac {\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac {1}{315} \int \cot ^4(c+d x) (-315 a-192 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac {\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac {\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}+\frac {1}{945} \int \cot ^2(c+d x) (945 a+384 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac {\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac {\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}-\frac {\cot (c+d x) (315 a+128 a \sec (c+d x))}{315 d}+\frac {1}{945} \int -945 a \, dx\\ &=-a x-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac {\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac {\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}-\frac {\cot (c+d x) (315 a+128 a \sec (c+d x))}{315 d}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 111, normalized size = 0.79 \[ -\frac {a \cot ^9(c+d x) \, _2F_1\left (-\frac {9}{2},1;-\frac {7}{2};-\tan ^2(c+d x)\right )}{9 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {4 a \csc ^7(c+d x)}{7 d}-\frac {6 a \csc ^5(c+d x)}{5 d}+\frac {4 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x]),x]

[Out]

-((a*Csc[c + d*x])/d) + (4*a*Csc[c + d*x]^3)/(3*d) - (6*a*Csc[c + d*x]^5)/(5*d) + (4*a*Csc[c + d*x]^7)/(7*d) -
(a*Csc[c + d*x]^9)/(9*d) - (a*Cot[c + d*x]^9*Hypergeometric2F1[-9/2, 1, -7/2, -Tan[c + d*x]^2])/(9*d)

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fricas [B]  time = 0.89, size = 279, normalized size = 1.99 \[ -\frac {563 \, a \cos \left (d x + c\right )^{8} - 248 \, a \cos \left (d x + c\right )^{7} - 1498 \, a \cos \left (d x + c\right )^{6} + 658 \, a \cos \left (d x + c\right )^{5} + 1610 \, a \cos \left (d x + c\right )^{4} - 602 \, a \cos \left (d x + c\right )^{3} - 763 \, a \cos \left (d x + c\right )^{2} + 187 \, a \cos \left (d x + c\right ) + 315 \, {\left (a d x \cos \left (d x + c\right )^{7} - a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{5} + 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{3} - 3 \, a d x \cos \left (d x + c\right )^{2} - a d x \cos \left (d x + c\right ) + a d x\right )} \sin \left (d x + c\right ) + 128 \, a}{315 \, {\left (d \cos \left (d x + c\right )^{7} - d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} + 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/315*(563*a*cos(d*x + c)^8 - 248*a*cos(d*x + c)^7 - 1498*a*cos(d*x + c)^6 + 658*a*cos(d*x + c)^5 + 1610*a*co
s(d*x + c)^4 - 602*a*cos(d*x + c)^3 - 763*a*cos(d*x + c)^2 + 187*a*cos(d*x + c) + 315*(a*d*x*cos(d*x + c)^7 -
a*d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^5 + 3*a*d*x*cos(d*x + c)^4 + 3*a*d*x*cos(d*x + c)^3 - 3*a*d*x*cos(
d*x + c)^2 - a*d*x*cos(d*x + c) + a*d*x)*sin(d*x + c) + 128*a)/((d*cos(d*x + c)^7 - d*cos(d*x + c)^6 - 3*d*cos
(d*x + c)^5 + 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c))

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giac [A]  time = 0.51, size = 140, normalized size = 1.00 \[ -\frac {45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4830 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80640 \, {\left (d x + c\right )} a - 40950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {80640 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13650 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2898 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 450 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/80640*(45*a*tan(1/2*d*x + 1/2*c)^7 - 630*a*tan(1/2*d*x + 1/2*c)^5 + 4830*a*tan(1/2*d*x + 1/2*c)^3 + 80640*(
d*x + c)*a - 40950*a*tan(1/2*d*x + 1/2*c) + (80640*a*tan(1/2*d*x + 1/2*c)^8 - 13650*a*tan(1/2*d*x + 1/2*c)^6 +
2898*a*tan(1/2*d*x + 1/2*c)^4 - 450*a*tan(1/2*d*x + 1/2*c)^2 + 35*a)/tan(1/2*d*x + 1/2*c)^9)/d

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maple [A]  time = 0.97, size = 205, normalized size = 1.46 \[ \frac {a \left (-\frac {\left (\cot ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{10}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^10*(a+a*sec(d*x+c)),x)

[Out]

1/d*(a*(-1/9*cot(d*x+c)^9+1/7*cot(d*x+c)^7-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+a*(-1/9/sin(d*x
+c)^9*cos(d*x+c)^10+1/63/sin(d*x+c)^7*cos(d*x+c)^10-1/105/sin(d*x+c)^5*cos(d*x+c)^10+1/63/sin(d*x+c)^3*cos(d*x
+c)^10-1/9/sin(d*x+c)*cos(d*x+c)^10-1/9*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x
+c)^2)*sin(d*x+c)))

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maxima [A]  time = 0.82, size = 119, normalized size = 0.85 \[ -\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a + \frac {{\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a}{\sin \left (d x + c\right )^{9}}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^10*(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/315*((315*d*x + 315*c + (315*tan(d*x + c)^8 - 105*tan(d*x + c)^6 + 63*tan(d*x + c)^4 - 45*tan(d*x + c)^2 +
35)/tan(d*x + c)^9)*a + (315*sin(d*x + c)^8 - 420*sin(d*x + c)^6 + 378*sin(d*x + c)^4 - 180*sin(d*x + c)^2 + 3
5)*a/sin(d*x + c)^9)/d

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mupad [B]  time = 3.18, size = 252, normalized size = 1.80 \[ -\frac {a\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+45\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4830\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-40950\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+80640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-13650\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2898\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+80640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{80640\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^10*(a + a/cos(c + d*x)),x)

[Out]

-(a*(35*cos(c/2 + (d*x)/2)^16 + 45*sin(c/2 + (d*x)/2)^16 - 630*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 48
30*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 - 40950*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 80640*cos(c
/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8 - 13650*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 2898*cos(c/2 + (d*x)
/2)^12*sin(c/2 + (d*x)/2)^4 - 450*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 80640*cos(c/2 + (d*x)/2)^7*sin(
c/2 + (d*x)/2)^9*(c + d*x)))/(80640*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^9)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**10*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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