Integrand size = 21, antiderivative size = 302 \[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {a^4 \left (55+29 m+4 m^2\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m) (4+m)}+\frac {\cos ^{1+m}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{d (4+m)}+\frac {2 (5+m) \cos ^{1+m}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{d (3+m) (4+m)}-\frac {a^4 \left (35+40 m+8 m^2\right ) \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) (2+m) (4+m) \sqrt {\sin ^2(c+d x)}}-\frac {4 a^4 (5+2 m) \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) (3+m) \sqrt {\sin ^2(c+d x)}} \]
a^4*(4*m^2+29*m+55)*cos(d*x+c)^(1+m)*sin(d*x+c)/d/(4+m)/(m^2+5*m+6)+cos(d* x+c)^(1+m)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/(4+m)+2*(5+m)*cos(d*x+c)^(1 +m)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/(3+m)/(4+m)-a^4*(8*m^2+40*m+35)*cos( d*x+c)^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],cos(d*x+c)^2)*sin(d*x+ c)/d/(m^3+7*m^2+14*m+8)/(sin(d*x+c)^2)^(1/2)-4*a^4*(5+2*m)*cos(d*x+c)^(2+m )*hypergeom([1/2, 1+1/2*m],[2+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(2+m)/(3+m )/(sin(d*x+c)^2)^(1/2)
Time = 1.64 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.93 \[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {a^2 \cos ^{1+m}(c+d x) \csc (c+d x) \left ((a+a \cos (c+d x))^2 \sin ^2(c+d x)-\frac {a^2 (5+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{1+m}-\frac {2 a^2 (10+3 m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{2+m}-\frac {a^2 (25+6 m) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{3+m}-\frac {2 a^2 (5+m) \cos ^3(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{4+m}\right )}{d (4+m)} \]
(a^2*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*((a + a*Cos[c + d*x])^2*Sin[c + d*x ]^2 - (a^2*(5 + 2*m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(1 + m) - (2*a^2*(10 + 3*m)*Cos[c + d*x]*Hyp ergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x ]^2])/(2 + m) - (a^2*(25 + 6*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(3 + m) - (2*a^2*( 5 + m)*Cos[c + d*x]^3*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(4 + m)))/(d*(4 + m))
Time = 1.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3242, 3042, 3455, 3042, 3447, 3042, 3502, 3042, 3227, 3042, 3122}
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.
| \(\displaystyle \int (a \cos (c+d x)+a)^4 \cos ^m(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \sin \left (c+d x+\frac {\pi }{2}\right )^mdx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {\int \cos ^m(c+d x) (\cos (c+d x) a+a)^2 \left ((2 m+5) a^2+2 (m+5) \cos (c+d x) a^2\right )dx}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((2 m+5) a^2+2 (m+5) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {\int \cos ^m(c+d x) (\cos (c+d x) a+a) \left (\left (4 m^2+23 m+25\right ) a^3+\left (4 m^2+29 m+55\right ) \cos (c+d x) a^3\right )dx}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (\left (4 m^2+23 m+25\right ) a^3+\left (4 m^2+29 m+55\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int \cos ^m(c+d x) \left (\left (4 m^2+29 m+55\right ) \cos ^2(c+d x) a^4+\left (4 m^2+23 m+25\right ) a^4+\left (\left (4 m^2+23 m+25\right ) a^4+\left (4 m^2+29 m+55\right ) a^4\right ) \cos (c+d x)\right )dx}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (\left (4 m^2+29 m+55\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+\left (4 m^2+23 m+25\right ) a^4+\left (\left (4 m^2+23 m+25\right ) a^4+\left (4 m^2+29 m+55\right ) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {\int \cos ^m(c+d x) \left ((m+3) \left (8 m^2+40 m+35\right ) a^4+4 (m+2) (m+4) (2 m+5) \cos (c+d x) a^4\right )dx}{m+2}+\frac {a^4 \left (4 m^2+29 m+55\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left ((m+3) \left (8 m^2+40 m+35\right ) a^4+4 (m+2) (m+4) (2 m+5) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx}{m+2}+\frac {a^4 \left (4 m^2+29 m+55\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {a^4 (m+3) \left (8 m^2+40 m+35\right ) \int \cos ^m(c+d x)dx+4 a^4 (m+2) (m+4) (2 m+5) \int \cos ^{m+1}(c+d x)dx}{m+2}+\frac {a^4 \left (4 m^2+29 m+55\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^4 (m+3) \left (8 m^2+40 m+35\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^mdx+4 a^4 (m+2) (m+4) (2 m+5) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+1}dx}{m+2}+\frac {a^4 \left (4 m^2+29 m+55\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
| \(\Big \downarrow \) 3122 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a^4 (m+3) \left (8 m^2+40 m+35\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(c+d x)\right )}{d (m+1) \sqrt {\sin ^2(c+d x)}}-\frac {4 a^4 (m+4) (2 m+5) \sin (c+d x) \cos ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)}}}{m+2}+\frac {a^4 \left (4 m^2+29 m+55\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}}{m+3}+\frac {2 (m+5) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right ) \cos ^{m+1}(c+d x)}{d (m+3)}}{m+4}+\frac {\sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2 \cos ^{m+1}(c+d x)}{d (m+4)}\) |
(Cos[c + d*x]^(1 + m)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) + ((2*(5 + m)*Cos[c + d*x]^(1 + m)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x]) /(d*(3 + m)) + ((a^4*(55 + 29*m + 4*m^2)*Cos[c + d*x]^(1 + m)*Sin[c + d*x] )/(d*(2 + m)) + (-((a^4*(3 + m)*(35 + 40*m + 8*m^2)*Cos[c + d*x]^(1 + m)*H ypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/ (d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (4*a^4*(4 + m)*(5 + 2*m)*Cos[c + d*x]^ (2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]))/(2 + m))/(3 + m))/(4 + m)
3.4.97.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (a +\cos \left (d x +c \right ) a \right )^{4}d x\]
\[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{m} \,d x } \]
integral((a^4*cos(d*x + c)^4 + 4*a^4*cos(d*x + c)^3 + 6*a^4*cos(d*x + c)^2 + 4*a^4*cos(d*x + c) + a^4)*cos(d*x + c)^m, x)
Timed out. \[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{m} \,d x } \]
\[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{m} \,d x } \]
Timed out. \[ \int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx=\int {\cos \left (c+d\,x\right )}^m\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]