3.343 \(\int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=109 \[ -\frac {(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)}+\frac {6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac {12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac {8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)} \]

[Out]

8*(a+a*sin(d*x+c))^(4+m)/a^4/d/(4+m)-12*(a+a*sin(d*x+c))^(5+m)/a^5/d/(5+m)+6*(a+a*sin(d*x+c))^(6+m)/a^6/d/(6+m
)-(a+a*sin(d*x+c))^(7+m)/a^7/d/(7+m)

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Rubi [A]  time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac {12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac {6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac {(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]

[Out]

(8*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) - (12*(a + a*Sin[c + d*x])^(5 + m))/(a^5*d*(5 + m)) + (6*(a +
a*Sin[c + d*x])^(6 + m))/(a^6*d*(6 + m)) - (a + a*Sin[c + d*x])^(7 + m)/(a^7*d*(7 + m))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{3+m} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{3+m}-12 a^2 (a+x)^{4+m}+6 a (a+x)^{5+m}-(a+x)^{6+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {8 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}-\frac {12 (a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}+\frac {6 (a+a \sin (c+d x))^{6+m}}{a^6 d (6+m)}-\frac {(a+a \sin (c+d x))^{7+m}}{a^7 d (7+m)}\\ \end {align*}

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Mathematica [A]  time = 0.70, size = 89, normalized size = 0.82 \[ \frac {(a (\sin (c+d x)+1))^{m+4} \left (\frac {6 a^3 (\sin (c+d x)+1)^2}{m+6}-\frac {12 a^3 (\sin (c+d x)+1)}{m+5}+\frac {8 a^3}{m+4}-\frac {(a \sin (c+d x)+a)^3}{m+7}\right )}{a^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]

[Out]

((a*(1 + Sin[c + d*x]))^(4 + m)*((8*a^3)/(4 + m) - (12*a^3*(1 + Sin[c + d*x]))/(5 + m) + (6*a^3*(1 + Sin[c + d
*x])^2)/(6 + m) - (a + a*Sin[c + d*x])^3/(7 + m)))/(a^7*d)

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fricas [A]  time = 0.49, size = 153, normalized size = 1.40 \[ \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 20 \, m\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 3 \, m\right )} \cos \left (d x + c\right )^{4} + 96 \, m \cos \left (d x + c\right )^{2} + {\left ({\left (m^{3} + 15 \, m^{2} + 74 \, m + 120\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 7 \, m + 12\right )} \cos \left (d x + c\right )^{4} + 96 \, {\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 384\right )} \sin \left (d x + c\right ) + 384\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} + 22 \, d m^{3} + 179 \, d m^{2} + 638 \, d m + 840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

((m^3 + 9*m^2 + 20*m)*cos(d*x + c)^6 + 12*(m^2 + 3*m)*cos(d*x + c)^4 + 96*m*cos(d*x + c)^2 + ((m^3 + 15*m^2 +
74*m + 120)*cos(d*x + c)^6 + 12*(m^2 + 7*m + 12)*cos(d*x + c)^4 + 96*(m + 2)*cos(d*x + c)^2 + 384)*sin(d*x + c
) + 384)*(a*sin(d*x + c) + a)^m/(d*m^4 + 22*d*m^3 + 179*d*m^2 + 638*d*m + 840*d)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 6.77, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x)

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maxima [B]  time = 0.71, size = 520, normalized size = 4.77 \[ -\frac {\frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} a^{m} \sin \left (d x + c\right )^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a^{m} \sin \left (d x + c\right )^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (d x + c\right )^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 360 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 720 \, a^{m} m \sin \left (d x + c\right ) + 720 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} - \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 2 \, a^{m} m \sin \left (d x + c\right ) + 2 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} - \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a {\left (m + 1\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

-(((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*a^m*sin(d*x + c)^7 + (m^6 + 15*m^5 + 85*m^4 +
225*m^3 + 274*m^2 + 120*m)*a^m*sin(d*x + c)^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^m*sin(d*x + c)^5 +
30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c)^4 - 120*(m^3 + 3*m^2 + 2*m)*a^m*sin(d*x + c)^3 + 360*(m^2 +
m)*a^m*sin(d*x + c)^2 - 720*a^m*m*sin(d*x + c) + 720*a^m)*(sin(d*x + c) + 1)^m/(m^7 + 28*m^6 + 322*m^5 + 1960*
m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) - 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*a^m*sin(d*x + c)^5 + (m^
4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c)^4 - 4*(m^3 + 3*m^2 + 2*m)*a^m*sin(d*x + c)^3 + 12*(m^2 + m)*a^m*sin
(d*x + c)^2 - 24*a^m*m*sin(d*x + c) + 24*a^m)*(sin(d*x + c) + 1)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m +
120) + 3*((m^2 + 3*m + 2)*a^m*sin(d*x + c)^3 + (m^2 + m)*a^m*sin(d*x + c)^2 - 2*a^m*m*sin(d*x + c) + 2*a^m)*(s
in(d*x + c) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) - (a*sin(d*x + c) + a)^(m + 1)/(a*(m + 1)))/d

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mupad [B]  time = 10.48, size = 555, normalized size = 5.09 \[ {\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m\,\left (\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\left (m^3\,40{}\mathrm {i}+m^2\,936{}\mathrm {i}+m\,8672{}\mathrm {i}+49152{}\mathrm {i}\right )}{128\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,\left (m^3\,30{}\mathrm {i}+m^2\,654{}\mathrm {i}+m\,4824{}\mathrm {i}\right )}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (5\,c+5\,d\,x\right )\,\left (5\,m^3+123\,m^2+706\,m+1176\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,\left (9\,m^3+279\,m^2+3210\,m+5880\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (7\,c+7\,d\,x\right )\,\left (m^3+15\,m^2+74\,m+120\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,\left (5\,m^3+171\,m^2+2578\,m+29400\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {m\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (6\,c+6\,d\,x\right )\,\left (m^2\,1{}\mathrm {i}+m\,9{}\mathrm {i}+20{}\mathrm {i}\right )}{32\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {3\,m\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,\left (m^2\,1{}\mathrm {i}+m\,17{}\mathrm {i}+44{}\mathrm {i}\right )}{16\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(a + a*sin(c + d*x))^m,x)

[Out]

exp(- c*7i - d*x*7i)*(a + a*sin(c + d*x))^m*((exp(c*7i + d*x*7i)*(m*8672i + m^2*936i + m^3*40i + 49152i))/(128
*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*cos(2*c + 2*d*x)*(m*4824i + m^2*654i +
m^3*30i))/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(5*c + 5*d*x)*(706*m
+ 123*m^2 + 5*m^3 + 1176)*1i)/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(3
*c + 3*d*x)*(3210*m + 279*m^2 + 9*m^3 + 5880)*1i)/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(
c*7i + d*x*7i)*sin(7*c + 7*d*x)*(74*m + 15*m^2 + m^3 + 120)*1i)/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i +
840i)) + (exp(c*7i + d*x*7i)*sin(c + d*x)*(2578*m + 171*m^2 + 5*m^3 + 29400)*1i)/(64*d*(m*638i + m^2*179i + m^
3*22i + m^4*1i + 840i)) + (m*exp(c*7i + d*x*7i)*cos(6*c + 6*d*x)*(m*9i + m^2*1i + 20i))/(32*d*(m*638i + m^2*17
9i + m^3*22i + m^4*1i + 840i)) + (3*m*exp(c*7i + d*x*7i)*cos(4*c + 4*d*x)*(m*17i + m^2*1i + 44i))/(16*d*(m*638
i + m^2*179i + m^3*22i + m^4*1i + 840i)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+a*sin(d*x+c))**m,x)

[Out]

Timed out

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