Integrand size = 25, antiderivative size = 100 \[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=-\frac {2^{\frac {1}{2}+m} a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2}-m,3,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-1+m}}{(c+d)^3 f} \] Output:
-2^(1/2+m)*a*AppellF1(1/2,3,1/2-m,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f *x+e))*cos(f*x+e)*(1+sin(f*x+e))^(1/2-m)*(a+a*sin(f*x+e))^(-1+m)/(c+d)^3/f
\[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx \] Input:
Integrate[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3,x]
Output:
Integrate[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x]
Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3267, 154, 27, 153}
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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m}{(c+d \sin (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m}{(c+d \sin (e+f x))^3}dx\) |
\(\Big \downarrow \) 3267 |
\(\displaystyle \frac {a^2 \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {a-a \sin (e+f x)} (c+d \sin (e+f x))^3}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 154 |
\(\displaystyle \frac {a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) \int \frac {\sqrt {2} (\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^3}d\sin (e+f x)}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^3}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle \frac {\sqrt {2} a \sqrt {1-\sin (e+f x)} \cos (e+f x) (a \sin (e+f x)+a)^m \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},3,m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+1) (c-d)^3 (a-a \sin (e+f x))}\) |
Input:
Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3,x]
Output:
(Sqrt[2]*a*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*( 1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Si n[e + f*x])^m)/((c - d)^3*f*(1 + 2*m)*(a - a*Sin[e + f*x]))
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_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( b*c - a*d)], 0] && !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[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*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && !G tQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d* x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m , n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[m]
\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{3}}d x\]
Input:
int((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x)
Output:
int((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x)
\[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x, algorithm="fricas")
Output:
integral(-(a*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**m/(c+d*sin(f*x+e))**3,x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c)^3, x)
\[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c)^3, x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:
int((a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^3,x)
Output:
int((a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^3, x)
\[ \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \] Input:
int((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x)
Output:
int((sin(e + f*x)*a + a)**m/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d* *2 + 3*sin(e + f*x)*c**2*d + c**3),x)