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) - (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)

________________________________________________________________________________________

Rubi [A]  time = 0.144726, antiderivative size = 140, normalized size of antiderivative = 1., 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 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]

Rule 8

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

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*}

Mathematica [C]  time = 0.0636214, size = 111, normalized size = 0.79 \[ -\frac{a \cot ^9(c+d x) \text{Hypergeometric2F1}\left (-\frac{9}{2},1,-\frac{7}{2},-\tan ^2(c+d x)\right )}{9 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{4 a \csc ^7(c+d x)}{7 d}-\frac{6 a \csc ^5(c+d x)}{5 d}+\frac{4 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.136, size = 205, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}-\cot \left ( dx+c \right ) -dx-c \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{9\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{9} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) \right ) } \end{align*}

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)))

________________________________________________________________________________________

Maxima [A]  time = 1.69648, size = 161, normalized size = 1.15 \begin{align*} -\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} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 0.907772, size = 745, normalized size = 5.32 \begin{align*} -\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 )} \end{align*}

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))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.51179, size = 189, normalized size = 1.35 \begin{align*} -\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} \end{align*}

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