Integrand size = 23, antiderivative size = 80 \[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\frac {6 \sqrt {2} \operatorname {AppellF1}\left (\frac {7}{6},-\frac {1}{2},2,\frac {13}{6},\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{5/3}}{7 a^2 f} \]
6/7*AppellF1(7/6,2,-1/2,13/6,1+sin(f*x+e),1/2+1/2*sin(f*x+e))*sec(f*x+e)*( a+a*sin(f*x+e))^(5/3)*2^(1/2)*(1-sin(f*x+e))^(1/2)/a^2/f
\[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx \]
Time = 0.36 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3198, 149, 1013, 27, 1012}
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.
\(\displaystyle \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a \sin (e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (e+f x)^2 \sqrt [3]{a \sin (e+f x)+a}}dx\) |
\(\Big \downarrow \) 3198 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {\csc ^2(e+f x) \sqrt {a-a \sin (e+f x)} \sqrt [6]{\sin (e+f x) a+a}}{a^2}d(a \sin (e+f x))}{a f}\) |
\(\Big \downarrow \) 149 |
\(\displaystyle \frac {6 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {a^6 \sin ^6(e+f x) \sqrt {2 a-a^6 \sin ^6(e+f x)}}{\left (a-a^6 \sin ^6(e+f x)\right )^2}d\sqrt [6]{\sin (e+f x) a+a}}{a f}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {6 \sqrt {2} \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \sqrt {2 a-a^6 \sin ^6(e+f x)} \int \frac {a^6 \sin ^6(e+f x) \sqrt {2-a^5 \sin ^6(e+f x)}}{\sqrt {2} \left (a-a^6 \sin ^6(e+f x)\right )^2}d\sqrt [6]{\sin (e+f x) a+a}}{a f \sqrt {2-a^5 \sin ^6(e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \sqrt {2 a-a^6 \sin ^6(e+f x)} \int \frac {a^6 \sin ^6(e+f x) \sqrt {2-a^5 \sin ^6(e+f x)}}{\left (a-a^6 \sin ^6(e+f x)\right )^2}d\sqrt [6]{\sin (e+f x) a+a}}{a f \sqrt {2-a^5 \sin ^6(e+f x)}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {6 \sqrt {2} a^4 \sin ^6(e+f x) \tan (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \sqrt {2 a-a^6 \sin ^6(e+f x)} \operatorname {AppellF1}\left (\frac {7}{6},2,-\frac {1}{2},\frac {13}{6},a^5 \sin ^6(e+f x),\frac {1}{2} a^5 \sin ^6(e+f x)\right )}{7 f \sqrt {2-a^5 \sin ^6(e+f x)}}\) |
(6*Sqrt[2]*a^4*AppellF1[7/6, 2, -1/2, 13/6, a^5*Sin[e + f*x]^6, (a^5*Sin[e + f*x]^6)/2]*Sin[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f *x]]*Sqrt[2*a - a^6*Sin[e + f*x]^6]*Tan[e + f*x])/(7*f*Sqrt[2 - a^5*Sin[e + f*x]^6])
3.2.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1 ) - 1)*(c - a*(d/b) + d*(x^k/b))^n*(e - a*(f/b) + f*(x^k/b))^p, x], x, (a + b*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[2*n] && IntegerQ[p]
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] /; 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]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[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])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[a - b*Sin[e + f*x]]/(b* f*Cos[e + f*x])) Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/ 2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b ^2, 0] && !IntegerQ[m] && IntegerQ[p/2]
\[\int \frac {\cot ^{2}\left (f x +e \right )}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \]