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

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

[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))

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Rubi [A]  time = 0.0857299, antiderivative size = 109, normalized size of antiderivative = 1., 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 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])

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])

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*}

Mathematica [A]  time = 0.698101, 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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Maple [F]  time = 3.554, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{7} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 3.09511, size = 408, normalized size = 3.74 \begin{align*} \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} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.10298, size = 698, normalized size = 6.4 \begin{align*} -\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{7}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} - 6 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{6}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{3} + 12 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{3} - 8 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{3} + 15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{7}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 96 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{6}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 204 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} - 144 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{2} + 74 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{7}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 498 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{6}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 1128 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m - 856 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m + 120 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{7}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 840 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{6}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 2016 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} - 1680 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3}}{{\left (a^{6} m^{4} + 22 \, a^{6} m^{3} + 179 \, a^{6} m^{2} + 638 \, a^{6} m + 840 \, a^{6}\right )} a d} \end{align*}

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]

-((a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m^3 - 6*(a*sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m*a*m^3 +
12*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2*m^3 - 8*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a^3
*m^3 + 15*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m^2 - 96*(a*sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m
*a*m^2 + 204*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2*m^2 - 144*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c
) + a)^m*a^3*m^2 + 74*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m - 498*(a*sin(d*x + c) + a)^6*(a*sin(d*x
+ c) + a)^m*a*m + 1128*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2*m - 856*(a*sin(d*x + c) + a)^4*(a*sin
(d*x + c) + a)^m*a^3*m + 120*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m - 840*(a*sin(d*x + c) + a)^6*(a*sin
(d*x + c) + a)^m*a + 2016*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2 - 1680*(a*sin(d*x + c) + a)^4*(a*s
in(d*x + c) + a)^m*a^3)/((a^6*m^4 + 22*a^6*m^3 + 179*a^6*m^2 + 638*a^6*m + 840*a^6)*a*d)