3.551 \(\int \cot ^4(c+d x) (a+b \sin ^2(c+d x))^p \, dx\)

Optimal. Leaf size=101 \[ -\frac {\sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{3 d} \]

[Out]

-1/3*AppellF1(-3/2,-3/2,-p,-1/2,sin(d*x+c)^2,-b*sin(d*x+c)^2/a)*csc(d*x+c)^3*sec(d*x+c)*(a+b*sin(d*x+c)^2)^p*(
cos(d*x+c)^2)^(1/2)/d/((1+b*sin(d*x+c)^2/a)^p)

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Rubi [A]  time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3196, 511, 510} \[ -\frac {\sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p,x]

[Out]

-(AppellF1[-3/2, -3/2, -p, -1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^3*
Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(3*d*(1 + (b*Sin[c + d*x]^2)/a)^p)

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 3196

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

Rubi steps

\begin {align*} \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx &=\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (1+\frac {b x^2}{a}\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d}\\ \end {align*}

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Mathematica [A]  time = 3.36, size = 102, normalized size = 1.01 \[ -\frac {\sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p} F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p,x]

[Out]

-1/3*(AppellF1[-3/2, -3/2, -p, -1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x
]^3*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(d*((a + b*Sin[c + d*x]^2)/a)^p)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \left (d x + c\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

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maple [F]  time = 1.11, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{4}\left (d x +c \right )\right ) \left (a +b \left (\sin ^{2}\left (d x +c \right )\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x)

[Out]

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^2)^p,x)

[Out]

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^2)^p, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*sin(d*x+c)**2)**p,x)

[Out]

Timed out

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