3.10.33 \(\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx\) [933]

Optimal. Leaf size=42 \[ \frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[Out]

hypergeom([2, 1+m],[2+m],1+sin(d*x+c))*(a+a*sin(d*x+c))^(1+m)/a/d/(1+m)

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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 67} \begin {gather*} \frac {(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^m,x]

[Out]

(Hypergeometric2F1[2, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {a^2 (a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \text {Subst}\left (\int \frac {(a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 42, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1(2,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^m,x]

[Out]

(Hypergeometric2F1[2, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x)

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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(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^2, x)

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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(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

integral((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**2*(a+a*sin(d*x+c))**m,x)

[Out]

Integral((a*(sin(c + d*x) + 1))**m*cos(c + d*x)*csc(c + d*x)**2, x)

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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(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^2,x)

[Out]

int((cos(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^2, x)

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