3.134 \(\int \cos ^4(a+b x) \cot ^2(a+b x) \, dx\)

Optimal. Leaf size=61 \[ -\frac{15 \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}+\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}-\frac{15 x}{8} \]

[Out]

(-15*x)/8 - (15*Cot[a + b*x])/(8*b) + (5*Cos[a + b*x]^2*Cot[a + b*x])/(8*b) + (Cos[a + b*x]^4*Cot[a + b*x])/(4
*b)

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Rubi [A]  time = 0.0455062, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2591, 288, 321, 203} \[ -\frac{15 \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}+\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}-\frac{15 x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4*Cot[a + b*x]^2,x]

[Out]

(-15*x)/8 - (15*Cot[a + b*x])/(8*b) + (5*Cos[a + b*x]^2*Cot[a + b*x])/(8*b) + (Cos[a + b*x]^4*Cot[a + b*x])/(4
*b)

Rule 2591

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

Rule 288

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

Rule 321

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cos ^4(a+b x) \cot ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (a+b x)\right )}{b}\\ &=\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}-\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (a+b x)\right )}{4 b}\\ &=\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}-\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (a+b x)\right )}{8 b}\\ &=-\frac{15 \cot (a+b x)}{8 b}+\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (a+b x)\right )}{8 b}\\ &=-\frac{15 x}{8}-\frac{15 \cot (a+b x)}{8 b}+\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.138363, size = 41, normalized size = 0.67 \[ -\frac{16 \sin (2 (a+b x))+\sin (4 (a+b x))+32 \cot (a+b x)+60 a+60 b x}{32 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4*Cot[a + b*x]^2,x]

[Out]

-(60*a + 60*b*x + 32*Cot[a + b*x] + 16*Sin[2*(a + b*x)] + Sin[4*(a + b*x)])/(32*b)

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Maple [A]  time = 0.011, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{\sin \left ( bx+a \right ) }}- \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) \sin \left ( bx+a \right ) -{\frac{15\,bx}{8}}-{\frac{15\,a}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^6/sin(b*x+a)^2,x)

[Out]

1/b*(-1/sin(b*x+a)*cos(b*x+a)^7-(cos(b*x+a)^5+5/4*cos(b*x+a)^3+15/8*cos(b*x+a))*sin(b*x+a)-15/8*b*x-15/8*a)

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Maxima [A]  time = 1.48495, size = 85, normalized size = 1.39 \begin{align*} -\frac{15 \, b x + 15 \, a + \frac{15 \, \tan \left (b x + a\right )^{4} + 25 \, \tan \left (b x + a\right )^{2} + 8}{\tan \left (b x + a\right )^{5} + 2 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(15*b*x + 15*a + (15*tan(b*x + a)^4 + 25*tan(b*x + a)^2 + 8)/(tan(b*x + a)^5 + 2*tan(b*x + a)^3 + tan(b*x
+ a)))/b

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Fricas [A]  time = 1.84227, size = 135, normalized size = 2.21 \begin{align*} \frac{2 \, \cos \left (b x + a\right )^{5} + 5 \, \cos \left (b x + a\right )^{3} - 15 \, b x \sin \left (b x + a\right ) - 15 \, \cos \left (b x + a\right )}{8 \, b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*cos(b*x + a)^5 + 5*cos(b*x + a)^3 - 15*b*x*sin(b*x + a) - 15*cos(b*x + a))/(b*sin(b*x + a))

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Sympy [A]  time = 4.80812, size = 119, normalized size = 1.95 \begin{align*} \begin{cases} - \frac{15 x \sin ^{4}{\left (a + b x \right )}}{8} - \frac{15 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} - \frac{15 x \cos ^{4}{\left (a + b x \right )}}{8} - \frac{15 \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{8 b} - \frac{25 \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac{\cos ^{5}{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{6}{\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**6/sin(b*x+a)**2,x)

[Out]

Piecewise((-15*x*sin(a + b*x)**4/8 - 15*x*sin(a + b*x)**2*cos(a + b*x)**2/4 - 15*x*cos(a + b*x)**4/8 - 15*sin(
a + b*x)**3*cos(a + b*x)/(8*b) - 25*sin(a + b*x)*cos(a + b*x)**3/(8*b) - cos(a + b*x)**5/(b*sin(a + b*x)), Ne(
b, 0)), (x*cos(a)**6/sin(a)**2, True))

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Giac [A]  time = 1.24547, size = 74, normalized size = 1.21 \begin{align*} -\frac{15 \, b x + 15 \, a + \frac{7 \, \tan \left (b x + a\right )^{3} + 9 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{2}} + \frac{8}{\tan \left (b x + a\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(15*b*x + 15*a + (7*tan(b*x + a)^3 + 9*tan(b*x + a))/(tan(b*x + a)^2 + 1)^2 + 8/tan(b*x + a))/b