3.456 \(\int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

Optimal. Leaf size=117 \[ \frac {(a+b)^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {(a+b)^3 \cot (c+d x)}{a^4 d}-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d} \]

[Out]

(a+b)^(7/2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/a^(9/2)/d+(a+b)^3*cot(d*x+c)/a^4/d-1/3*(a+b)^2*cot(d*x+c)^3
/a^3/d+1/5*(a+b)*cot(d*x+c)^5/a^2/d-1/7*cot(d*x+c)^7/a/d

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Rubi [A]  time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3195, 325, 205} \[ \frac {(a+b)^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b)^3 \cot (c+d x)}{a^4 d}-\frac {\cot ^7(c+d x)}{7 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]

[Out]

((a + b)^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*d) + ((a + b)^3*Cot[c + d*x])/(a^4*d) - ((
a + b)^2*Cot[c + d*x]^3)/(3*a^3*d) + ((a + b)*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a*d)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 3195

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{ff
= FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(p
+ 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^8 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=\frac {(a+b)^3 \cot (c+d x)}{a^4 d}-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {(a+b)^4 \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a^4 d}\\ &=\frac {(a+b)^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {(a+b)^3 \cot (c+d x)}{a^4 d}-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}\\ \end {align*}

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Mathematica [A]  time = 1.08, size = 135, normalized size = 1.15 \[ \frac {(a+b)^{7/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {\cot (c+d x) \left (-15 a^3 \csc ^6(c+d x)+176 a^3-a \left (122 a^2+112 a b+35 b^2\right ) \csc ^2(c+d x)+3 a^2 (22 a+7 b) \csc ^4(c+d x)+406 a^2 b+350 a b^2+105 b^3\right )}{105 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]

[Out]

((a + b)^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*d) + (Cot[c + d*x]*(176*a^3 + 406*a^2*b +
350*a*b^2 + 105*b^3 - a*(122*a^2 + 112*a*b + 35*b^2)*Csc[c + d*x]^2 + 3*a^2*(22*a + 7*b)*Csc[c + d*x]^4 - 15*a
^3*Csc[c + d*x]^6))/(105*a^4*d)

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fricas [B]  time = 0.49, size = 834, normalized size = 7.13 \[ \left [\frac {4 \, {\left (176 \, a^{3} + 406 \, a^{2} b + 350 \, a b^{2} + 105 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 28 \, {\left (58 \, a^{3} + 158 \, a^{2} b + 145 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (10 \, a^{3} + 29 \, a^{2} b + 28 \, a b^{2} + 9 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 105 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) - 420 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )}, \frac {2 \, {\left (176 \, a^{3} + 406 \, a^{2} b + 350 \, a b^{2} + 105 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 14 \, {\left (58 \, a^{3} + 158 \, a^{2} b + 145 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 70 \, {\left (10 \, a^{3} + 29 \, a^{2} b + 28 \, a b^{2} + 9 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 210 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/420*(4*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)*cos(d*x + c)^7 - 28*(58*a^3 + 158*a^2*b + 145*a*b^2 + 45
*b^3)*cos(d*x + c)^5 + 140*(10*a^3 + 29*a^2*b + 28*a*b^2 + 9*b^3)*cos(d*x + c)^3 + 105*((a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 +
3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 -
2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*cos(d*x + c))*sqrt(-(a
+ b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b +
b^2))*sin(d*x + c) - 420*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*
x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c)), 1/210*(2*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)
*cos(d*x + c)^7 - 14*(58*a^3 + 158*a^2*b + 145*a*b^2 + 45*b^3)*cos(d*x + c)^5 + 70*(10*a^3 + 29*a^2*b + 28*a*b
^2 + 9*b^3)*cos(d*x + c)^3 - 105*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt
((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))
*sin(d*x + c) - 210*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c
)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))]

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giac [B]  time = 0.33, size = 238, normalized size = 2.03 \[ \frac {\frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{\sqrt {a^{2} + a b} a^{4}} + \frac {105 \, a^{3} \tan \left (d x + c\right )^{6} + 315 \, a^{2} b \tan \left (d x + c\right )^{6} + 315 \, a b^{2} \tan \left (d x + c\right )^{6} + 105 \, b^{3} \tan \left (d x + c\right )^{6} - 35 \, a^{3} \tan \left (d x + c\right )^{4} - 70 \, a^{2} b \tan \left (d x + c\right )^{4} - 35 \, a b^{2} \tan \left (d x + c\right )^{4} + 21 \, a^{3} \tan \left (d x + c\right )^{2} + 21 \, a^{2} b \tan \left (d x + c\right )^{2} - 15 \, a^{3}}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="giac")

[Out]

1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((
a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*a^4) + (105*a^3*tan(d*x + c)^6 + 315*a^2*b
*tan(d*x + c)^6 + 315*a*b^2*tan(d*x + c)^6 + 105*b^3*tan(d*x + c)^6 - 35*a^3*tan(d*x + c)^4 - 70*a^2*b*tan(d*x
+ c)^4 - 35*a*b^2*tan(d*x + c)^4 + 21*a^3*tan(d*x + c)^2 + 21*a^2*b*tan(d*x + c)^2 - 15*a^3)/(a^4*tan(d*x + c
)^7))/d

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maple [B]  time = 0.68, size = 342, normalized size = 2.92 \[ \frac {\arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{d \sqrt {a \left (a +b \right )}}+\frac {4 \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b}{d a \sqrt {a \left (a +b \right )}}+\frac {6 \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b^{2}}{d \,a^{2} \sqrt {a \left (a +b \right )}}+\frac {4 \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b^{3}}{d \,a^{3} \sqrt {a \left (a +b \right )}}+\frac {\arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b^{4}}{d \,a^{4} \sqrt {a \left (a +b \right )}}+\frac {1}{d a \tan \left (d x +c \right )}+\frac {3 b}{d \,a^{2} \tan \left (d x +c \right )}+\frac {3 b^{2}}{d \,a^{3} \tan \left (d x +c \right )}+\frac {b^{3}}{d \,a^{4} \tan \left (d x +c \right )}+\frac {1}{5 d a \tan \left (d x +c \right )^{5}}+\frac {b}{5 d \,a^{2} \tan \left (d x +c \right )^{5}}-\frac {1}{7 d a \tan \left (d x +c \right )^{7}}-\frac {1}{3 d a \tan \left (d x +c \right )^{3}}-\frac {2 b}{3 d \,a^{2} \tan \left (d x +c \right )^{3}}-\frac {b^{2}}{3 d \,a^{3} \tan \left (d x +c \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x)

[Out]

1/d/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))+4/d/a/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*
(a+b))^(1/2))*b+6/d/a^2/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b^2+4/d/a^3/(a*(a+b))^(1/2)*a
rctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b^3+1/d/a^4/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b
^4+1/d/a/tan(d*x+c)+3/d/a^2/tan(d*x+c)*b+3/d/a^3/tan(d*x+c)*b^2+1/d/a^4/tan(d*x+c)*b^3+1/5/d/a/tan(d*x+c)^5+1/
5/d/a^2/tan(d*x+c)^5*b-1/7/d/a/tan(d*x+c)^7-1/3/d/a/tan(d*x+c)^3-2/3/d/a^2/tan(d*x+c)^3*b-1/3/d/a^3/tan(d*x+c)
^3*b^2

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maxima [A]  time = 0.48, size = 154, normalized size = 1.32 \[ \frac {\frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{4}} + \frac {105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{6} - 35 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a^{3} + 21 \, {\left (a^{3} + a^{2} b\right )} \tan \left (d x + c\right )^{2}}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/(sqrt((a +
b)*a)*a^4) + (105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*tan(d*x + c)^6 - 35*(a^3 + 2*a^2*b + a*b^2)*tan(d*x + c)^4
- 15*a^3 + 21*(a^3 + a^2*b)*tan(d*x + c)^2)/(a^4*tan(d*x + c)^7))/d

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mupad [B]  time = 18.91, size = 100, normalized size = 0.85 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,{\left (a+b\right )}^{7/2}}{a^{9/2}\,d}-\frac {\frac {1}{7\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{5\,a^2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (a+b\right )}^2}{3\,a^3}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+b\right )}^3}{a^4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^8/(a + b*sin(c + d*x)^2),x)

[Out]

(atan((tan(c + d*x)*(a + b)^(1/2))/a^(1/2))*(a + b)^(7/2))/(a^(9/2)*d) - (1/(7*a) - (tan(c + d*x)^2*(a + b))/(
5*a^2) + (tan(c + d*x)^4*(a + b)^2)/(3*a^3) - (tan(c + d*x)^6*(a + b)^3)/a^4)/(d*tan(c + d*x)^7)

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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)**8/(a+b*sin(d*x+c)**2),x)

[Out]

Timed out

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