Integrand size = 12, antiderivative size = 98 \[ \int (-4-3 \sin (c+d x))^n \, dx=-\frac {\sqrt {2} 7^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {3}{7} (1-\sin (c+d x))\right ) \cos (c+d x) (-4-3 \sin (c+d x))^n (4+3 \sin (c+d x))^{-n}}{d \sqrt {1+\sin (c+d x)}} \] Output:
-2^(1/2)*7^n*AppellF1(1/2,1/2,-n,3/2,1/2-1/2*sin(d*x+c),3/7-3/7*sin(d*x+c) )*cos(d*x+c)*(-4-3*sin(d*x+c))^n/d/(1+sin(d*x+c))^(1/2)/((4+3*sin(d*x+c))^ n)
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int (-4-3 \sin (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (1+n,\frac {1}{2},\frac {1}{2},2+n,4+3 \sin (c+d x),\frac {1}{7} (4+3 \sin (c+d x))\right ) \sec (c+d x) (-4-3 \sin (c+d x))^{1+n} \sqrt {-1-\sin (c+d x)} \sqrt {1-\sin (c+d x)}}{\sqrt {7} d (1+n)} \] Input:
Integrate[(-4 - 3*Sin[c + d*x])^n,x]
Output:
-((AppellF1[1 + n, 1/2, 1/2, 2 + n, 4 + 3*Sin[c + d*x], (4 + 3*Sin[c + d*x ])/7]*Sec[c + d*x]*(-4 - 3*Sin[c + d*x])^(1 + n)*Sqrt[-1 - Sin[c + d*x]]*S qrt[1 - Sin[c + d*x]])/(Sqrt[7]*d*(1 + n)))
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3144, 156, 27, 155}
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.
\(\displaystyle \int (-3 \sin (c+d x)-4)^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (-3 \sin (c+d x)-4)^ndx\) |
\(\Big \downarrow \) 3144 |
\(\displaystyle \frac {\cos (c+d x) \int \frac {(-3 \sin (c+d x)-4)^n}{\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle \frac {\sqrt {3} \sqrt {-\sin (c+d x)-1} \cos (c+d x) \int \frac {(-3 \sin (c+d x)-4)^n}{\sqrt {3} \sqrt {-\sin (c+d x)-1} \sqrt {1-\sin (c+d x)}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-\sin (c+d x)-1} \cos (c+d x) \int \frac {(-3 \sin (c+d x)-4)^n}{\sqrt {-\sin (c+d x)-1} \sqrt {1-\sin (c+d x)}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle -\frac {\sqrt {-\sin (c+d x)-1} \cos (c+d x) (-3 \sin (c+d x)-4)^{n+1} \operatorname {AppellF1}\left (n+1,\frac {1}{2},\frac {1}{2},n+2,3 \sin (c+d x)+4,\frac {1}{7} (3 \sin (c+d x)+4)\right )}{\sqrt {7} d (n+1) \sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)}\) |
Input:
Int[(-4 - 3*Sin[c + d*x])^n,x]
Output:
-((AppellF1[1 + n, 1/2, 1/2, 2 + n, 4 + 3*Sin[c + d*x], (4 + 3*Sin[c + d*x ])/7]*Cos[c + d*x]*(-4 - 3*Sin[c + d*x])^(1 + n)*Sqrt[-1 - Sin[c + d*x]])/ (Sqrt[7]*d*(1 + n)*Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*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[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*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, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]) Subst[Int[(a + b*x )^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
\[\int \left (-4-3 \sin \left (d x +c \right )\right )^{n}d x\]
Input:
int((-4-3*sin(d*x+c))^n,x)
Output:
int((-4-3*sin(d*x+c))^n,x)
\[ \int (-4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n} \,d x } \] Input:
integrate((-4-3*sin(d*x+c))^n,x, algorithm="fricas")
Output:
integral((-3*sin(d*x + c) - 4)^n, x)
\[ \int (-4-3 \sin (c+d x))^n \, dx=\int \left (- 3 \sin {\left (c + d x \right )} - 4\right )^{n}\, dx \] Input:
integrate((-4-3*sin(d*x+c))**n,x)
Output:
Integral((-3*sin(c + d*x) - 4)**n, x)
\[ \int (-4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n} \,d x } \] Input:
integrate((-4-3*sin(d*x+c))^n,x, algorithm="maxima")
Output:
integrate((-3*sin(d*x + c) - 4)^n, x)
\[ \int (-4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n} \,d x } \] Input:
integrate((-4-3*sin(d*x+c))^n,x, algorithm="giac")
Output:
integrate((-3*sin(d*x + c) - 4)^n, x)
Timed out. \[ \int (-4-3 \sin (c+d x))^n \, dx=\int {\left (-3\,\sin \left (c+d\,x\right )-4\right )}^n \,d x \] Input:
int((- 3*sin(c + d*x) - 4)^n,x)
Output:
int((- 3*sin(c + d*x) - 4)^n, x)
\[ \int (-4-3 \sin (c+d x))^n \, dx=\int \left (-3 \sin \left (d x +c \right )-4\right )^{n}d x \] Input:
int((-4-3*sin(d*x+c))^n,x)
Output:
int(( - 3*sin(c + d*x) - 4)**n,x)