3.455 \(\int \frac{\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

Optimal. Leaf size=96 \[ -\frac{(a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d}+\frac{(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b)^2 \cot (c+d x)}{a^3 d}-\frac{\cot ^5(c+d x)}{5 a d} \]

[Out]

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

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Rubi [A]  time = 0.0961273, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3195, 325, 205} \[ -\frac{(a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d}+\frac{(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b)^2 \cot (c+d x)}{a^3 d}-\frac{\cot ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]

[Out]

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

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]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.897472, size = 101, normalized size = 1.05 \[ \frac{-\sqrt{a} \cot (c+d x) \left (3 a^2 \csc ^4(c+d x)+23 a^2-a (11 a+5 b) \csc ^2(c+d x)+35 a b+15 b^2\right )-15 (a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{15 a^{7/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]

[Out]

(-15*(a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] - Sqrt[a]*Cot[c + d*x]*(23*a^2 + 35*a*b + 15*b^2
- a*(11*a + 5*b)*Csc[c + d*x]^2 + 3*a^2*Csc[c + d*x]^4))/(15*a^(7/2)*d)

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Maple [B]  time = 0.126, size = 239, normalized size = 2.5 \begin{align*} -{\frac{1}{d}\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}}-3\,{\frac{b}{da\sqrt{a \left ( a+b \right ) }}\arctan \left ({\frac{ \left ( a+b \right ) \tan \left ( dx+c \right ) }{\sqrt{a \left ( a+b \right ) }}} \right ) }-3\,{\frac{{b}^{2}}{{a}^{2}d\sqrt{a \left ( a+b \right ) }}\arctan \left ({\frac{ \left ( a+b \right ) \tan \left ( dx+c \right ) }{\sqrt{a \left ( a+b \right ) }}} \right ) }-{\frac{{b}^{3}}{d{a}^{3}}\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}}-{\frac{1}{da\tan \left ( dx+c \right ) }}-2\,{\frac{b}{{a}^{2}d\tan \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\tan \left ( dx+c \right ) }}-{\frac{1}{5\,da \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{3\,da \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{b}{3\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+sin(d*x+c)^2*b),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90358, size = 1413, normalized size = 14.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/60*(4*(23*a^2 + 35*a*b + 15*b^2)*cos(d*x + c)^5 - 20*(7*a^2 + 13*a*b + 6*b^2)*cos(d*x + c)^3 - 15*((a^2 +
2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^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) + 60*(a^2 + 2*a*b + b^2)*cos(d*x + c))/((a^3*d*cos(d*
x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c)), -1/30*(2*(23*a^2 + 35*a*b + 15*b^2)*cos(d*x + c)^5 -
10*(7*a^2 + 13*a*b + 6*b^2)*cos(d*x + c)^3 - 15*((a^2 + 2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*c
os(d*x + c)^2 + a^2 + 2*a*b + b^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) + 30*(a^2 + 2*a*b + b^2)*cos(d*x + c))/((a^3*d*cos(d*x +
c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))]

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

[Out]

Timed out

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Giac [B]  time = 1.1902, size = 231, normalized size = 2.41 \begin{align*} -\frac{\frac{15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\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^{3}} + \frac{15 \, a^{2} \tan \left (d x + c\right )^{4} + 30 \, a b \tan \left (d x + c\right )^{4} + 15 \, b^{2} \tan \left (d x + c\right )^{4} - 5 \, a^{2} \tan \left (d x + c\right )^{2} - 5 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{5}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm="giac")

[Out]

-1/15*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(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^3) + (15*a^2*tan(d*x + c)^4 + 30*a*b*tan(d*x + c)^4 +
15*b^2*tan(d*x + c)^4 - 5*a^2*tan(d*x + c)^2 - 5*a*b*tan(d*x + c)^2 + 3*a^2)/(a^3*tan(d*x + c)^5))/d