\(\int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\) [682]

Optimal result

Integrand size = 27, antiderivative size = 73 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^6(c+d x)}{6 a d}-\frac {\sin ^3(c+d x)}{3 a d}+\frac {2 \sin ^5(c+d x)}{5 a d}-\frac {\sin ^7(c+d x)}{7 a d} \]

[Out]

-1/6*cos(d*x+c)^6/a/d-1/3*sin(d*x+c)^3/a/d+2/5*sin(d*x+c)^5/a/d-1/7*sin(d*x+c)^7/a/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2914, 2645, 30, 2644, 276} \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin ^7(c+d x)}{7 a d}+\frac {2 \sin ^5(c+d x)}{5 a d}-\frac {\sin ^3(c+d x)}{3 a d}-\frac {\cos ^6(c+d x)}{6 a d} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^5(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^5(c+d x) \sin ^2(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^5 \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {\cos ^6(c+d x)}{6 a d}-\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {\cos ^6(c+d x)}{6 a d}-\frac {\sin ^3(c+d x)}{3 a d}+\frac {2 \sin ^5(c+d x)}{5 a d}-\frac {\sin ^7(c+d x)}{7 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^2(c+d x) \left (105-70 \sin (c+d x)-105 \sin ^2(c+d x)+84 \sin ^3(c+d x)+35 \sin ^4(c+d x)-30 \sin ^5(c+d x)\right )}{210 a d} \]

[In]

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

[Out]

(Sin[c + d*x]^2*(105 - 70*Sin[c + d*x] - 105*Sin[c + d*x]^2 + 84*Sin[c + d*x]^3 + 35*Sin[c + d*x]^4 - 30*Sin[c
+ d*x]^5))/(210*a*d)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}}{a d}\)	\(70\)
default	\(-\frac {\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}}{a d}\)	\(70\)
parallelrisch	\(\frac {15 \sin \left (7 d x +7 c \right )+35 \sin \left (3 d x +3 c \right )+63 \sin \left (5 d x +5 c \right )-525 \sin \left (d x +c \right )-525 \cos \left (2 d x +2 c \right )-35 \cos \left (6 d x +6 c \right )-210 \cos \left (4 d x +4 c \right )+770}{6720 a d}\)	\(85\)
risch	\(-\frac {5 \sin \left (d x +c \right )}{64 a d}+\frac {\sin \left (7 d x +7 c \right )}{448 d a}-\frac {\cos \left (6 d x +6 c \right )}{192 a d}+\frac {3 \sin \left (5 d x +5 c \right )}{320 d a}-\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {\sin \left (3 d x +3 c \right )}{192 d a}-\frac {5 \cos \left (2 d x +2 c \right )}{64 a d}\)	\(118\)
norman	\(\frac {\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {4 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {52 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}+\frac {52 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}+\frac {122 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}+\frac {122 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}+\frac {74 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {74 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {712 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d a}+\frac {712 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(297\)

[In]

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

[Out]

-1/a/d*(1/7*sin(d*x+c)^7-1/6*sin(d*x+c)^6-2/5*sin(d*x+c)^5+1/2*sin(d*x+c)^4+1/3*sin(d*x+c)^3-1/2*sin(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {35 \, \cos \left (d x + c\right )^{6} - 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{210 \, a d} \]

[In]

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

[Out]

-1/210*(35*cos(d*x + c)^6 - 2*(15*cos(d*x + c)^6 - 3*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 8)*sin(d*x + c))/(a*d
)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1530 vs. \(2 (54) = 108\).

Time = 32.28 (sec) , antiderivative size = 1530, normalized size of antiderivative = 20.96 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((210*tan(c/2 + d*x/2)**12/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*ta
n(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**
4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 280*tan(c/2 + d*x/2)**11/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*
tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2
)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 210*tan(c/2 + d*x/2)**10/(105*a
*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*
x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d)
+ 224*tan(c/2 + d*x/2)**9/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d
*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a
*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 700*tan(c/2 + d*x/2)**8/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 +
d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 220
5*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 912*tan(c/2 + d*x/2)**7/(105*a*d*tan(c/2
+ d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3
675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 700*tan(
c/2 + d*x/2)**6/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 +
3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2
+ d*x/2)**2 + 105*a*d) + 224*tan(c/2 + d*x/2)**5/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12
+ 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c
/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 210*tan(c/2 + d*x/2)**4/(105*a*d*tan(c/2 + d*x/2)**1
4 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan
(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 280*tan(c/2 + d*x/2
)**3/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*t
an(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2
+ 105*a*d) + 210*tan(c/2 + d*x/2)**2/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*
tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)
**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d), Ne(d, 0)), (x*sin(c)*cos(c)**7/(a*sin(c) + a), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \sin \left (d x + c\right )^{7} - 35 \, \sin \left (d x + c\right )^{6} - 84 \, \sin \left (d x + c\right )^{5} + 105 \, \sin \left (d x + c\right )^{4} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2}}{210 \, a d} \]

[In]

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

[Out]

-1/210*(30*sin(d*x + c)^7 - 35*sin(d*x + c)^6 - 84*sin(d*x + c)^5 + 105*sin(d*x + c)^4 + 70*sin(d*x + c)^3 - 1
05*sin(d*x + c)^2)/(a*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \sin \left (d x + c\right )^{7} - 35 \, \sin \left (d x + c\right )^{6} - 84 \, \sin \left (d x + c\right )^{5} + 105 \, \sin \left (d x + c\right )^{4} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2}}{210 \, a d} \]

[In]

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

[Out]

-1/210*(30*sin(d*x + c)^7 - 35*sin(d*x + c)^6 - 84*sin(d*x + c)^5 + 105*sin(d*x + c)^4 + 70*sin(d*x + c)^3 - 1
05*sin(d*x + c)^2)/(a*d)

Mupad [B] (verification not implemented)

Time = 10.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^4}{2\,a}+\frac {2\,{\sin \left (c+d\,x\right )}^5}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}-\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}}{d} \]

[In]

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

[Out]

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