3.55 \(\int \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=213 \[ -\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {3 a^3 \csc (c+d x)}{d}+a^3 x \]

[Out]

a^3*x+a^3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d+1/5*a^3*cot(d*x+c)^5/d-1/7*a^3*cot(d*x+c)^7/d+1/9*a^3*cot(d*x+c)
^9/d-4/11*a^3*cot(d*x+c)^11/d+3*a^3*csc(d*x+c)/d-16/3*a^3*csc(d*x+c)^3/d+34/5*a^3*csc(d*x+c)^5/d-36/7*a^3*csc(
d*x+c)^7/d+19/9*a^3*csc(d*x+c)^9/d-4/11*a^3*csc(d*x+c)^11/d

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Rubi [A]  time = 0.22, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {3 a^3 \csc (c+d x)}{d}+a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/
(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) + (3*a^3*Csc[c + d*x])/d - (16*a^3*Csc[c +
d*x]^3)/(3*d) + (34*a^3*Csc[c + d*x]^5)/(5*d) - (36*a^3*Csc[c + d*x]^7)/(7*d) + (19*a^3*Csc[c + d*x]^9)/(9*d)
- (4*a^3*Csc[c + d*x]^11)/(11*d)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3886

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^{12}(c+d x)+3 a^3 \cot ^{11}(c+d x) \csc (c+d x)+3 a^3 \cot ^{10}(c+d x) \csc ^2(c+d x)+a^3 \cot ^9(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^{12}(c+d x) \, dx+a^3 \int \cot ^9(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^{11}(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^{10}(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot ^{11}(c+d x)}{11 d}-a^3 \int \cot ^{10}(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^{10} \, dx,x,-\cot (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+a^3 \int \cot ^8(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}-a^3 \int \cot ^6(c+d x) \, dx\\ &=\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}+a^3 \int \cot ^4(c+d x) \, dx\\ &=-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}-a^3 \int \cot ^2(c+d x) \, dx\\ &=\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}+a^3 \int 1 \, dx\\ &=a^3 x+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {16 a^3 \csc ^3(c+d x)}{3 d}+\frac {34 a^3 \csc ^5(c+d x)}{5 d}-\frac {36 a^3 \csc ^7(c+d x)}{7 d}+\frac {19 a^3 \csc ^9(c+d x)}{9 d}-\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}\\ \end {align*}

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Mathematica [A]  time = 6.05, size = 268, normalized size = 1.26 \[ -\frac {a^3 \tan \left (\frac {c}{2}\right ) (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (20 \cot ^2\left (\frac {c}{2}\right ) (-4528480 \cos (c+d x)+2388316 \cos (2 (c+d x))-750112 \cos (3 (c+d x))+112229 \cos (4 (c+d x))+2786111) \csc ^{10}\left (\frac {1}{2} (c+d x)\right )+7392 \csc \left (\frac {c}{2}\right ) \left (-3060 \sin \left (c+\frac {d x}{2}\right )+2860 \sin \left (c+\frac {3 d x}{2}\right )-855 \sin \left (2 c+\frac {3 d x}{2}\right )+743 \sin \left (2 c+\frac {5 d x}{2}\right )+4370 \sin \left (\frac {d x}{2}\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )-5 \cot \left (\frac {c}{2}\right ) \left (\csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) (54812150 \cos (c+d x)-32118776 \cos (2 (c+d x))[phone] \cos (3 (c+d x))-3023754 \cos (4 (c+d x))+347267 \cos (5 (c+d x))-32611198) \csc ^{11}\left (\frac {1}{2} (c+d x)\right )[phone] d x\right )\right )}{3633315840 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]

[Out]

-1/3633315840*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(20*(2786111 - 4528480*Cos[c + d*x] + 2388316*Cos[2
*(c + d*x)] - 750112*Cos[3*(c + d*x)] + 112229*Cos[4*(c + d*x)])*Cot[c/2]^2*Csc[(c + d*x)/2]^10 - 5*Cot[c/2]*(
90832896*d*x + (-32611198 + 54812150*Cos[c + d*x] - 32118776*Cos[2*(c + d*x)] + 12626567*Cos[3*(c + d*x)] - 30
23754*Cos[4*(c + d*x)] + 347267*Cos[5*(c + d*x)])*Csc[c/2]*Csc[(c + d*x)/2]^11*Sin[(d*x)/2]) + 7392*Csc[c/2]*S
ec[(c + d*x)/2]^5*(4370*Sin[(d*x)/2] - 3060*Sin[c + (d*x)/2] + 2860*Sin[c + (3*d*x)/2] - 855*Sin[2*c + (3*d*x)
/2] + 743*Sin[2*c + (5*d*x)/2]))*Tan[c/2])/d

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fricas [A]  time = 0.52, size = 314, normalized size = 1.47 \[ \frac {7453 \, a^{3} \cos \left (d x + c\right )^{8} - 11964 \, a^{3} \cos \left (d x + c\right )^{7} - 11866 \, a^{3} \cos \left (d x + c\right )^{6} + 30542 \, a^{3} \cos \left (d x + c\right )^{5} + 90 \, a^{3} \cos \left (d x + c\right )^{4} - 26438 \, a^{3} \cos \left (d x + c\right )^{3} + 8539 \, a^{3} \cos \left (d x + c\right )^{2} + 7671 \, a^{3} \cos \left (d x + c\right ) - 3712 \, a^{3} + 3465 \, {\left (a^{3} d x \cos \left (d x + c\right )^{7} - 3 \, a^{3} d x \cos \left (d x + c\right )^{6} + a^{3} d x \cos \left (d x + c\right )^{5} + 5 \, a^{3} d x \cos \left (d x + c\right )^{4} - 5 \, a^{3} d x \cos \left (d x + c\right )^{3} - a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5} + 5 \, d \cos \left (d x + c\right )^{4} - 5 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3465*(7453*a^3*cos(d*x + c)^8 - 11964*a^3*cos(d*x + c)^7 - 11866*a^3*cos(d*x + c)^6 + 30542*a^3*cos(d*x + c)
^5 + 90*a^3*cos(d*x + c)^4 - 26438*a^3*cos(d*x + c)^3 + 8539*a^3*cos(d*x + c)^2 + 7671*a^3*cos(d*x + c) - 3712
*a^3 + 3465*(a^3*d*x*cos(d*x + c)^7 - 3*a^3*d*x*cos(d*x + c)^6 + a^3*d*x*cos(d*x + c)^5 + 5*a^3*d*x*cos(d*x +
c)^4 - 5*a^3*d*x*cos(d*x + c)^3 - a^3*d*x*cos(d*x + c)^2 + 3*a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d
*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + d*cos(d*x + c)^5 + 5*d*cos(d*x + c)^4 - 5*d*cos(d*x + c)^3 - d*cos(d*x
+ c)^2 + 3*d*cos(d*x + c) - d)*sin(d*x + c))

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giac [A]  time = 0.65, size = 161, normalized size = 0.76 \[ -\frac {693 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11550 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 887040 \, {\left (d x + c\right )} a^{3} + 159390 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {5 \, {\left (264726 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 59136 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18018 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4554 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 770 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{887040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/887040*(693*a^3*tan(1/2*d*x + 1/2*c)^5 - 11550*a^3*tan(1/2*d*x + 1/2*c)^3 - 887040*(d*x + c)*a^3 + 159390*a
^3*tan(1/2*d*x + 1/2*c) - 5*(264726*a^3*tan(1/2*d*x + 1/2*c)^10 - 59136*a^3*tan(1/2*d*x + 1/2*c)^8 + 18018*a^3
*tan(1/2*d*x + 1/2*c)^6 - 4554*a^3*tan(1/2*d*x + 1/2*c)^4 + 770*a^3*tan(1/2*d*x + 1/2*c)^2 - 63*a^3)/tan(1/2*d
*x + 1/2*c)^11)/d

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maple [B]  time = 1.27, size = 425, normalized size = 2.00 \[ \frac {a^{3} \left (-\frac {\left (\cot ^{11}\left (d x +c \right )\right )}{11}+\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 )+3 a^{3} \left (-\frac {\cos ^{12}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}+\frac {\cos ^{12}\left (d x +c \right )}{99 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{12}\left (d x +c \right )}{231 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{12}\left (d x +c \right )}{231 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{12}\left (d x +c \right )}{99 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{12}\left (d x +c \right )}{11 \sin \left (d x +c \right )}+\frac {\left (\frac {256}{63}+\cos ^{10}\left (d x +c \right )+\frac {10 \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {80 \left (\cos ^{6}\left (d x +c \right )\right )}{63}+\frac {32 \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {128 \left (\cos ^{2}\left (d x +c \right )\right )}{63}\right ) \sin \left (d x +c \right )}{11}\right )-\frac {3 a^{3} \left (\cos ^{11}\left (d x +c \right )\right )}{11 \sin \left (d x +c \right )^{11}}+a^{3} \left (-\frac {\cos ^{10}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {\cos ^{10}\left (d x +c \right )}{99 \sin \left (d x +c \right )^{9}}+\frac {\cos ^{10}\left (d x +c \right )}{693 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{10}\left (d x +c \right )}{1155 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{10}\left (d x +c \right )}{693 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{10}\left (d x +c \right )}{99 \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 )}{99}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/11*cot(d*x+c)^11+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)+3*a^3*(-1/11/sin(d*x+c)^11*cos(d*x+c)^12+1/99/sin(d*x+c)^9*cos(d*x+c)^12-1/231/sin(d*x+c)^7*cos(d*x+c)^1
2+1/231/sin(d*x+c)^5*cos(d*x+c)^12-1/99/sin(d*x+c)^3*cos(d*x+c)^12+1/11/sin(d*x+c)*cos(d*x+c)^12+1/11*(256/63+
cos(d*x+c)^10+10/9*cos(d*x+c)^8+80/63*cos(d*x+c)^6+32/21*cos(d*x+c)^4+128/63*cos(d*x+c)^2)*sin(d*x+c))-3/11*a^
3/sin(d*x+c)^11*cos(d*x+c)^11+a^3*(-1/11/sin(d*x+c)^11*cos(d*x+c)^10-1/99/sin(d*x+c)^9*cos(d*x+c)^10+1/693/sin
(d*x+c)^7*cos(d*x+c)^10-1/1155/sin(d*x+c)^5*cos(d*x+c)^10+1/693/sin(d*x+c)^3*cos(d*x+c)^10-1/99/sin(d*x+c)*cos
(d*x+c)^10-1/99*(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 = 0.44, size = 212, normalized size = 1.00 \[ \frac {{\left (3465 \, d x + 3465 \, c + \frac {3465 \, \tan \left (d x + c\right )^{10} - 1155 \, \tan \left (d x + c\right )^{8} + 693 \, \tan \left (d x + c\right )^{6} - 495 \, \tan \left (d x + c\right )^{4} + 385 \, \tan \left (d x + c\right )^{2} - 315}{\tan \left (d x + c\right )^{11}}\right )} a^{3} + \frac {15 \, {\left (693 \, \sin \left (d x + c\right )^{10} - 1155 \, \sin \left (d x + c\right )^{8} + 1386 \, \sin \left (d x + c\right )^{6} - 990 \, \sin \left (d x + c\right )^{4} + 385 \, \sin \left (d x + c\right )^{2} - 63\right )} a^{3}}{\sin \left (d x + c\right )^{11}} - \frac {{\left (1155 \, \sin \left (d x + c\right )^{8} - 2772 \, \sin \left (d x + c\right )^{6} + 2970 \, \sin \left (d x + c\right )^{4} - 1540 \, \sin \left (d x + c\right )^{2} + 315\right )} a^{3}}{\sin \left (d x + c\right )^{11}} - \frac {945 \, a^{3}}{\tan \left (d x + c\right )^{11}}}{3465 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/3465*((3465*d*x + 3465*c + (3465*tan(d*x + c)^10 - 1155*tan(d*x + c)^8 + 693*tan(d*x + c)^6 - 495*tan(d*x +
c)^4 + 385*tan(d*x + c)^2 - 315)/tan(d*x + c)^11)*a^3 + 15*(693*sin(d*x + c)^10 - 1155*sin(d*x + c)^8 + 1386*s
in(d*x + c)^6 - 990*sin(d*x + c)^4 + 385*sin(d*x + c)^2 - 63)*a^3/sin(d*x + c)^11 - (1155*sin(d*x + c)^8 - 277
2*sin(d*x + c)^6 + 2970*sin(d*x + c)^4 - 1540*sin(d*x + c)^2 + 315)*a^3/sin(d*x + c)^11 - 945*a^3/tan(d*x + c)
^11)/d

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mupad [B]  time = 3.03, size = 254, normalized size = 1.19 \[ -\frac {a^3\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+693\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-11550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+159390\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-1323630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-90090\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+22770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3850\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-887040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (c+d\,x\right )\right )}{887040\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^12*(a + a/cos(c + d*x))^3,x)

[Out]

-(a^3*(315*cos(c/2 + (d*x)/2)^16 + 693*sin(c/2 + (d*x)/2)^16 - 11550*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^1
4 + 159390*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 - 1323630*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 2
95680*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8 - 90090*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 22770*cos
(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 3850*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 - 887040*cos(c/2 + (
d*x)/2)^5*sin(c/2 + (d*x)/2)^11*(c + d*x)))/(887040*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^11)

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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)**12*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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