3.6.46 \(\int \cot (c+d x) (a+b \sin ^2(c+d x))^p \, dx\) [546]

Optimal. Leaf size=54 \[ -\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b \sin ^2(c+d x)}{a}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a d (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3273, 67} \begin {gather*} -\frac {\left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^2(c+d x)}{a}+1\right )}{2 a d (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3273

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 54, normalized size = 1.00 \begin {gather*} -\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b \sin ^2(c+d x)}{a}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a d (1+p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \cot \left (d x +c \right ) \left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cot(d*x+c)*(a+sin(d*x+c)^2*b)^p,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(cot(d*x+c)*(a+b*sin(d*x+c)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c), x)

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Fricas [F]
time = 0.41, size = 25, normalized size = 0.46 \begin {gather*} {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \left (d x + c\right ), x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*sin(d*x+c)**2)**p,x)

[Out]

Integral((a + b*sin(c + d*x)**2)**p*cot(c + d*x), 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(cot(d*x+c)*(a+b*sin(d*x+c)^2)^p,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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