Integrand size = 23, antiderivative size = 114 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {1}{2}+m+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 m-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (1-2 m-p)} (a+a \sin (c+d x))^{-1+m}}{d e (1+p)} \]
-2^(1/2+m+1/2*p)*a*(e*cos(d*x+c))^(p+1)*hypergeom([1/2+1/2*p, 1/2-m-1/2*p] ,[3/2+1/2*p],1/2-1/2*sin(d*x+c))*(1+sin(d*x+c))^(1/2-m-1/2*p)*(a+a*sin(d*x +c))^(-1+m)/d/e/(p+1)
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {1}{2} (1+2 m+p)} \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 m-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-2 m-p)} (a (1+\sin (c+d x)))^m}{d (1+p)} \]
-((2^((1 + 2*m + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[( 1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d *x])^((-1 - 2*m - p)/2)*(a*(1 + Sin[c + d*x]))^m)/(d*(1 + p)))
Time = 0.32 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3168, 80, 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.
\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^pdx\) |
\(\Big \downarrow \) 3168 |
\(\displaystyle \frac {a^2 (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)} (a \sin (c+d x)+a)^{\frac {1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \int (a-a \sin (c+d x))^{\frac {p-1}{2}} (\sin (c+d x) a+a)^{m+\frac {p-1}{2}}d\sin (c+d x)}{d e}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a^2 2^{\frac {1}{2} (2 m+p-1)} (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)} (e \cos (c+d x))^{p+1} (\sin (c+d x)+1)^{\frac {1}{2} (-2 m-p+1)} (a \sin (c+d x)+a)^{\frac {1}{2} (2 m+p-1)+\frac {1}{2} (-p-1)} \int \left (\frac {1}{2} \sin (c+d x)+\frac {1}{2}\right )^{m+\frac {p-1}{2}} (a-a \sin (c+d x))^{\frac {p-1}{2}}d\sin (c+d x)}{d e}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {a 2^{\frac {1}{2} (2 m+p-1)+1} (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)+\frac {p+1}{2}} (e \cos (c+d x))^{p+1} (\sin (c+d x)+1)^{\frac {1}{2} (-2 m-p+1)} (a \sin (c+d x)+a)^{\frac {1}{2} (2 m+p-1)+\frac {1}{2} (-p-1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 m-p+1),\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1)}\) |
-((2^(1 + (-1 + 2*m + p)/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[( 1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d *x])^((1 - 2*m - p)/2)*(a - a*Sin[c + d*x])^((-1 - p)/2 + (1 + p)/2)*(a + a*Sin[c + d*x])^((-1 - p)/2 + (-1 + 2*m + p)/2))/(d*e*(1 + p)))
3.4.42.3.1 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[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin [e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))) Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
\[\int \left (e \cos \left (d x +c \right )\right )^{p} \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{p} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{p}\, dx \]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{p} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{p} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^p\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \]