Integrand size = 12, antiderivative size = 64 \[ \int (-3-4 \sin (c+d x))^n \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2},\frac {3}{2},4 (1+\sin (c+d x)),\frac {1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1-\sin (c+d x)}} \] Output:
2^(1/2)*AppellF1(1/2,-n,1/2,3/2,4+4*sin(d*x+c),1/2+1/2*sin(d*x+c))*cos(d*x +c)/d/(1-sin(d*x+c))^(1/2)
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.31 \[ \int (-3-4 \sin (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (1+n,\frac {1}{2},\frac {1}{2},2+n,-3-4 \sin (c+d x),\frac {1}{7} (3+4 \sin (c+d x))\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x) (-3-4 \sin (c+d x))^{1+n}}{\sqrt {7} d (1+n)} \] Input:
Integrate[(-3 - 4*Sin[c + d*x])^n,x]
Output:
-((AppellF1[1 + n, 1/2, 1/2, 2 + n, -3 - 4*Sin[c + d*x], (3 + 4*Sin[c + d* x])/7]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*(-3 - 4*Sin[c + d*x])^(1 + n))/(S qrt[7]*d*(1 + n)))
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3144, 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.
\(\displaystyle \int (-4 \sin (c+d x)-3)^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (-4 \sin (c+d x)-3)^ndx\) |
\(\Big \downarrow \) 3144 |
\(\displaystyle \frac {\cos (c+d x) \int \frac {(-4 \sin (c+d x)-3)^n}{\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle \frac {\sqrt {2} \cos (c+d x) \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2},\frac {3}{2},4 (\sin (c+d x)+1),\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}}\) |
Input:
Int[(-3 - 4*Sin[c + d*x])^n,x]
Output:
(Sqrt[2]*AppellF1[1/2, -n, 1/2, 3/2, 4*(1 + Sin[c + d*x]), (1 + Sin[c + d* x])/2]*Cos[c + d*x])/(d*Sqrt[1 - Sin[c + d*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_.)*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]
\[\int \left (-3-4 \sin \left (d x +c \right )\right )^{n}d x\]
Input:
int((-3-4*sin(d*x+c))^n,x)
Output:
int((-3-4*sin(d*x+c))^n,x)
\[ \int (-3-4 \sin (c+d x))^n \, dx=\int { {\left (-4 \, \sin \left (d x + c\right ) - 3\right )}^{n} \,d x } \] Input:
integrate((-3-4*sin(d*x+c))^n,x, algorithm="fricas")
Output:
integral((-4*sin(d*x + c) - 3)^n, x)
\[ \int (-3-4 \sin (c+d x))^n \, dx=\int \left (- 4 \sin {\left (c + d x \right )} - 3\right )^{n}\, dx \] Input:
integrate((-3-4*sin(d*x+c))**n,x)
Output:
Integral((-4*sin(c + d*x) - 3)**n, x)
\[ \int (-3-4 \sin (c+d x))^n \, dx=\int { {\left (-4 \, \sin \left (d x + c\right ) - 3\right )}^{n} \,d x } \] Input:
integrate((-3-4*sin(d*x+c))^n,x, algorithm="maxima")
Output:
integrate((-4*sin(d*x + c) - 3)^n, x)
\[ \int (-3-4 \sin (c+d x))^n \, dx=\int { {\left (-4 \, \sin \left (d x + c\right ) - 3\right )}^{n} \,d x } \] Input:
integrate((-3-4*sin(d*x+c))^n,x, algorithm="giac")
Output:
integrate((-4*sin(d*x + c) - 3)^n, x)
Timed out. \[ \int (-3-4 \sin (c+d x))^n \, dx=\int {\left (-4\,\sin \left (c+d\,x\right )-3\right )}^n \,d x \] Input:
int((- 4*sin(c + d*x) - 3)^n,x)
Output:
int((- 4*sin(c + d*x) - 3)^n, x)
\[ \int (-3-4 \sin (c+d x))^n \, dx=\int \left (-4 \sin \left (d x +c \right )-3\right )^{n}d x \] Input:
int((-3-4*sin(d*x+c))^n,x)
Output:
int(( - 4*sin(c + d*x) - 3)**n,x)