Integrand size = 21, antiderivative size = 109 \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} m \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f (1+m)} \]
-cot(f*x+e)*(a+a*csc(f*x+e))^m/f/(1+m)-2^(1/2+m)*m*cot(f*x+e)*(1+csc(f*x+e ))^(-1/2-m)*(a+a*csc(f*x+e))^m*hypergeom([1/2, 1/2-m],[3/2],1/2-1/2*csc(f* x+e))/f/(1+m)
Time = 0.97 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.16 \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {(a (1+\csc (e+f x)))^m \left ((-1+m) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-1-m,-2 m,-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1-m,-2 m,2-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right ) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )^{-2 m}}{2 f (-1+m) (1+m)} \]
-1/2*((a*(1 + Csc[e + f*x]))^m*((-1 + m)*Cot[(e + f*x)/2]^2*Hypergeometric 2F1[-1 - m, -2*m, -m, -Tan[(e + f*x)/2]] + (1 + m)*Hypergeometric2F1[1 - m , -2*m, 2 - m, -Tan[(e + f*x)/2]])*Tan[(e + f*x)/2])/(f*(-1 + m)*(1 + m)*( 1 + Tan[(e + f*x)/2])^(2*m))
Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4285, 3042, 4315, 3042, 4314, 79}
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.
\(\displaystyle \int \csc ^2(e+f x) (a \csc (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^2 (a \csc (e+f x)+a)^mdx\) |
\(\Big \downarrow \) 4285 |
\(\displaystyle \frac {m \int \csc (e+f x) (\csc (e+f x) a+a)^mdx}{m+1}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {m \int \csc (e+f x) (\csc (e+f x) a+a)^mdx}{m+1}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
\(\Big \downarrow \) 4315 |
\(\displaystyle \frac {m (\csc (e+f x)+1)^{-m} (a \csc (e+f x)+a)^m \int \csc (e+f x) (\csc (e+f x)+1)^mdx}{m+1}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {m (\csc (e+f x)+1)^{-m} (a \csc (e+f x)+a)^m \int \csc (e+f x) (\csc (e+f x)+1)^mdx}{m+1}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
\(\Big \downarrow \) 4314 |
\(\displaystyle \frac {m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \int \frac {(\csc (e+f x)+1)^{m-\frac {1}{2}}}{\sqrt {1-\csc (e+f x)}}d\csc (e+f x)}{f (m+1) \sqrt {1-\csc (e+f x)}}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{m+\frac {1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f (m+1)}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}\) |
-((Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*m*Cot[ e + f*x]*(1 + Csc[e + f*x])^(-1/2 - m)*(a + a*Csc[e + f*x])^m*Hypergeometr ic2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/(f*(1 + m))
3.1.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[a*(m/(b*(m + 1))) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*d*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x ]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(d*x)^(n - 1)*((a + b*x)^(m - 1/2 )/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Csc[e + f*x])^FracPart[m ]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Csc[e + f*x])^m*(d *Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
\[\int \csc \left (f x +e \right )^{2} \left (a +a \csc \left (f x +e \right )\right )^{m}d x\]
\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \csc ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int \frac {{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m}{{\sin \left (e+f\,x\right )}^2} \,d x \]