3.2.34 \(\int \cot ^5(e+f x) (a+a \sin (e+f x))^m \, dx\) [134]

Optimal. Leaf size=123 \[ \frac {(9-m) \csc ^3(e+f x) (a+a \sin (e+f x))^{3+m}}{12 a^3 f}-\frac {\csc ^4(e+f x) (a+a \sin (e+f x))^{3+m}}{4 a^3 f}-\frac {\left (12-9 m+m^2\right ) \, _2F_1(3,3+m;4+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{3+m}}{12 a^3 f (3+m)} \]

[Out]

1/12*(9-m)*csc(f*x+e)^3*(a+a*sin(f*x+e))^(3+m)/a^3/f-1/4*csc(f*x+e)^4*(a+a*sin(f*x+e))^(3+m)/a^3/f-1/12*(m^2-9
*m+12)*hypergeom([3, 3+m],[4+m],1+sin(f*x+e))*(a+a*sin(f*x+e))^(3+m)/a^3/f/(3+m)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2786, 91, 79, 67} \begin {gather*} -\frac {\left (m^2-9 m+12\right ) (a \sin (e+f x)+a)^{m+3} \, _2F_1(3,m+3;m+4;\sin (e+f x)+1)}{12 a^3 f (m+3)}-\frac {\csc ^4(e+f x) (a \sin (e+f x)+a)^{m+3}}{4 a^3 f}+\frac {(9-m) \csc ^3(e+f x) (a \sin (e+f x)+a)^{m+3}}{12 a^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^5*(a + a*Sin[e + f*x])^m,x]

[Out]

((9 - m)*Csc[e + f*x]^3*(a + a*Sin[e + f*x])^(3 + m))/(12*a^3*f) - (Csc[e + f*x]^4*(a + a*Sin[e + f*x])^(3 + m
))/(4*a^3*f) - ((12 - 9*m + m^2)*Hypergeometric2F1[3, 3 + m, 4 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(3
+ m))/(12*a^3*f*(3 + m))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 2786

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \cot ^5(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)^{2+m}}{x^5} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {\csc ^4(e+f x) (a+a \sin (e+f x))^{3+m}}{4 a^3 f}+\frac {\text {Subst}\left (\int \frac {(a+x)^{2+m} \left (-a^2 (9-m)+4 a x\right )}{x^4} \, dx,x,a \sin (e+f x)\right )}{4 a f}\\ &=\frac {(9-m) \csc ^3(e+f x) (a+a \sin (e+f x))^{3+m}}{12 a^3 f}-\frac {\csc ^4(e+f x) (a+a \sin (e+f x))^{3+m}}{4 a^3 f}+\frac {\left (12 a^2-a^2 (9-m) m\right ) \text {Subst}\left (\int \frac {(a+x)^{2+m}}{x^3} \, dx,x,a \sin (e+f x)\right )}{12 a^2 f}\\ &=\frac {(9-m) \csc ^3(e+f x) (a+a \sin (e+f x))^{3+m}}{12 a^3 f}-\frac {\csc ^4(e+f x) (a+a \sin (e+f x))^{3+m}}{4 a^3 f}-\frac {(12-(9-m) m) \, _2F_1(3,3+m;4+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{3+m}}{12 a^3 f (3+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 83, normalized size = 0.67 \begin {gather*} -\frac {\left ((3+m) \csc ^3(e+f x) (-9+m+3 \csc (e+f x))+\left (12-9 m+m^2\right ) \, _2F_1(3,3+m;4+m;1+\sin (e+f x))\right ) (1+\sin (e+f x))^3 (a (1+\sin (e+f x)))^m}{12 f (3+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^5*(a + a*Sin[e + f*x])^m,x]

[Out]

-1/12*(((3 + m)*Csc[e + f*x]^3*(-9 + m + 3*Csc[e + f*x]) + (12 - 9*m + m^2)*Hypergeometric2F1[3, 3 + m, 4 + m,
1 + Sin[e + f*x]])*(1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^m)/(f*(3 + m))

________________________________________________________________________________________

Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (\cot ^{5}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^m*cot(f*x + e)^5, x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^5, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{5}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**5*(a+a*sin(f*x+e))**m,x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*cot(e + f*x)**5, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^m*cot(f*x + e)^5, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^5\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^5*(a + a*sin(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^5*(a + a*sin(e + f*x))^m, x)

________________________________________________________________________________________