\(\int \cot ^2(c+d x) (a+b \sin ^2(c+d x))^p \, dx\) [550]

Optimal result

Integrand size = 23, antiderivative size = 97 \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3275, 525, 524} \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{d} \]

[In]

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

[Out]

-((AppellF1[-1/2, -1/2, -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]*Se
c[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(d*(1 + (b*Sin[c + d*x]^2)/a)^p))

Rule 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 3275

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*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (a+b x^2\right )^p}{x^2} \, 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 ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{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 (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}}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \]

[In]

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

[Out]

-((AppellF1[-1/2, -1/2, -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]*Se
c[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(d*((a + b*Sin[c + d*x]^2)/a)^p))

Maple [F]

\[\int \left (\cot ^{2}\left (d x +c \right )\right ) {\left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^2*(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)^2, x)

Sympy [F]

\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int \left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{p} \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^2*(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)^2, x)

Giac [F]

\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^2*(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)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]

[In]

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

[Out]

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