Integrand size = 25, antiderivative size = 102 \[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2^{-1+\frac {p}{2}} (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {4-p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{1-\frac {p}{2}}}{d e (1+p) (a+a \sin (c+d x))^{3/2}} \]
-2^(-1+1/2*p)*(e*cos(d*x+c))^(p+1)*hypergeom([2-1/2*p, 1/2+1/2*p],[3/2+1/2 *p],1/2-1/2*sin(d*x+c))*(1+sin(d*x+c))^(1-1/2*p)/d/e/(p+1)/(a+a*sin(d*x+c) )^(3/2)
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99 \[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2^{-1+\frac {p}{2}} \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (2-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{1-\frac {p}{2}}}{d (1+p) (a (1+\sin (c+d x)))^{3/2}} \]
-((2^(-1 + p/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[2 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1 - p/2)) /(d*(1 + p)*(a*(1 + Sin[c + d*x]))^(3/2)))
Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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.
\(\displaystyle \int \frac {(e \cos (c+d x))^p}{(a \sin (c+d x)+a)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \cos (c+d x))^p}{(a \sin (c+d x)+a)^{3/2}}dx\) |
\(\Big \downarrow \) 3168 |
\(\displaystyle \frac {a^2 (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)} (a \sin (c+d x)+a)^{\frac {1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \int (a-a \sin (c+d x))^{\frac {p-1}{2}} (\sin (c+d x) a+a)^{\frac {p-4}{2}}d\sin (c+d x)}{d e}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {2^{\frac {p}{2}-2} (\sin (c+d x)+1)^{-p/2} (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)} (a \sin (c+d x)+a)^{\frac {1}{2} (-p-1)+\frac {p}{2}} (e \cos (c+d x))^{p+1} \int \left (\frac {1}{2} \sin (c+d x)+\frac {1}{2}\right )^{\frac {p-4}{2}} (a-a \sin (c+d x))^{\frac {p-1}{2}}d\sin (c+d x)}{d e}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{\frac {p}{2}-1} (\sin (c+d x)+1)^{-p/2} (a-a \sin (c+d x))^{\frac {1}{2} (-p-1)+\frac {p+1}{2}} (a \sin (c+d x)+a)^{\frac {1}{2} (-p-1)+\frac {p}{2}} (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {4-p}{2},\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{a d e (p+1)}\) |
-((2^(-1 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(4 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(a - a*Sin[c + d*x])^((-1 - p)/2 + (1 + p)/2)*(a + a*Sin[c + d*x])^((-1 - p)/2 + p/2))/(a*d*e*(1 + p)*(1 + S in[c + d*x])^(p/2)))
3.4.40.3.1 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]
\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{p}}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{p}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(-sqrt(a*sin(d*x + c) + a)*(e*cos(d*x + c))^p/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)
\[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\left (e \cos {\left (c + d x \right )}\right )^{p}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{p}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^p}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]