Integrand size = 23, antiderivative size = 97 \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \]
-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 )
Time = 0.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \]
-((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]*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/( d*((a + b*Sin[c + d*x]^2)/a)^p))
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3675, 395, 394}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin (c+d x)^2\right )^p}{\tan (c+d x)^2}dx\) |
\(\Big \downarrow \) 3675 |
\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)} \left (b \sin ^2(c+d x)+a\right )^pd\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 395 |
\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \int \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)} \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^pd\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 394 |
\(\displaystyle -\frac {\sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-p,\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{d}\) |
-((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]*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/( d*(1 + (b*Sin[c + d*x]^2)/a)^p))
3.6.50.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[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]
\[\int \left (\cot ^{2}\left (d x +c \right )\right ) {\left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )}^{p}d x\]
\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]
\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int \left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{p} \cot ^{2}{\left (c + d x \right )}\, dx \]
\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]
\[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]