3.1.1 \(\int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx\) [1]

Optimal. Leaf size=87 \[ \frac {8 (a+a \sin (c+d x))^5}{5 a^4 d}-\frac {2 (a+a \sin (c+d x))^6}{a^5 d}+\frac {6 (a+a \sin (c+d x))^7}{7 a^6 d}-\frac {(a+a \sin (c+d x))^8}{8 a^7 d} \]

[Out]

8/5*(a+a*sin(d*x+c))^5/a^4/d-2*(a+a*sin(d*x+c))^6/a^5/d+6/7*(a+a*sin(d*x+c))^7/a^6/d-1/8*(a+a*sin(d*x+c))^8/a^
7/d

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2746, 45} \begin {gather*} -\frac {(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac {8 (a \sin (c+d x)+a)^5}{5 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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 2746

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 79, normalized size = 0.91 \begin {gather*} -\frac {a \cos ^8(c+d x)}{8 d}+\frac {35 a \sin (c+d x)}{64 d}+\frac {7 a \sin (3 (c+d x))}{64 d}+\frac {7 a \sin (5 (c+d x))}{320 d}+\frac {a \sin (7 (c+d x))}{448 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(a*Cos[c + d*x]^8)/d + (35*a*Sin[c + d*x])/(64*d) + (7*a*Sin[3*(c + d*x)])/(64*d) + (7*a*Sin[5*(c + d*x)]
)/(320*d) + (a*Sin[7*(c + d*x)])/(448*d)

________________________________________________________________________________________

Maple [A]
time = 0.21, size = 56, normalized size = 0.64

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{8}+\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(56\)
default	\(\frac {-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{8}+\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(56\)
risch	\(\frac {35 a \sin \left (d x +c \right )}{64 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}-\frac {a \cos \left (6 d x +6 c \right )}{128 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}-\frac {7 a \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}-\frac {7 a \cos \left (2 d x +2 c \right )}{128 d}\)	\(119\)
norman	\(\frac {\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {6 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {106 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {1026 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {1026 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {106 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {6 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8*a*cos(d*x+c)^8+1/7*a*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 92, normalized size = 1.06 \begin {gather*} -\frac {35 \, a \sin \left (d x + c\right )^{8} + 40 \, a \sin \left (d x + c\right )^{7} - 140 \, a \sin \left (d x + c\right )^{6} - 168 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3} - 140 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right )}{280 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/280*(35*a*sin(d*x + c)^8 + 40*a*sin(d*x + c)^7 - 140*a*sin(d*x + c)^6 - 168*a*sin(d*x + c)^5 + 210*a*sin(d*
x + c)^4 + 280*a*sin(d*x + c)^3 - 140*a*sin(d*x + c)^2 - 280*a*sin(d*x + c))/d

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 62, normalized size = 0.71 \begin {gather*} -\frac {35 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{280 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/280*(35*a*cos(d*x + c)^8 - 8*(5*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a*cos(d*x + c)^2 + 16*a)*sin(d*x
+ c))/d

________________________________________________________________________________________

Sympy [A]
time = 0.88, size = 105, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*a*sin(c + d*x)**7/(35*d) + 8*a*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*a*sin(c + d*x)**3*cos(c
+ d*x)**4/d + a*sin(c + d*x)*cos(c + d*x)**6/d - a*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(a*sin(c) + a)*cos(c)
**7, True))

________________________________________________________________________________________

Giac [A]
time = 6.13, size = 118, normalized size = 1.36 \begin {gather*} -\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {35 \, a \sin \left (d x + c\right )}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*a*cos(8*d*x + 8*c)/d - 1/128*a*cos(6*d*x + 6*c)/d - 7/256*a*cos(4*d*x + 4*c)/d - 7/128*a*cos(2*d*x + 2
*c)/d + 1/448*a*sin(7*d*x + 7*c)/d + 7/320*a*sin(5*d*x + 5*c)/d + 7/64*a*sin(3*d*x + 3*c)/d + 35/64*a*sin(d*x
+ c)/d

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 90, normalized size = 1.03 \begin {gather*} \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}-a\,{\sin \left (c+d\,x\right )}^3+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(c + d*x) + (a*sin(c + d*x)^2)/2 - a*sin(c + d*x)^3 - (3*a*sin(c + d*x)^4)/4 + (3*a*sin(c + d*x)^5)/5 +
(a*sin(c + d*x)^6)/2 - (a*sin(c + d*x)^7)/7 - (a*sin(c + d*x)^8)/8)/d

________________________________________________________________________________________