3.1011 \(\int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx\)
Optimal. Leaf size=79 \[ -\frac {B (a-a \sin (c+d x))^6}{6 a^8 d}+\frac {(A+3 B) (a-a \sin (c+d x))^5}{5 a^7 d}-\frac {(A+B) (a-a \sin (c+d x))^4}{2 a^6 d} \]
[Out]
-1/2*(A+B)*(a-a*sin(d*x+c))^4/a^6/d+1/5*(A+3*B)*(a-a*sin(d*x+c))^5/a^7/d-1/6*B*(a-a*sin(d*x+c))^6/a^8/d
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Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used =
{2836, 77} \[ \frac {(A+3 B) (a-a \sin (c+d x))^5}{5 a^7 d}-\frac {(A+B) (a-a \sin (c+d x))^4}{2 a^6 d}-\frac {B (a-a \sin (c+d x))^6}{6 a^8 d} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]
[Out]
-((A + B)*(a - a*Sin[c + d*x])^4)/(2*a^6*d) + ((A + 3*B)*(a - a*Sin[c + d*x])^5)/(5*a^7*d) - (B*(a - a*Sin[c +
d*x])^6)/(6*a^8*d)
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 2836
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x) \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (A+B) (a-x)^3+(-A-3 B) (a-x)^4+\frac {B (a-x)^5}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(A+B) (a-a \sin (c+d x))^4}{2 a^6 d}+\frac {(A+3 B) (a-a \sin (c+d x))^5}{5 a^7 d}-\frac {B (a-a \sin (c+d x))^6}{6 a^8 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 52, normalized size = 0.66 \[ -\frac {(\sin (c+d x)-1)^4 \left ((6 A+8 B) \sin (c+d x)+9 A+5 B \sin ^2(c+d x)+2 B\right )}{30 a^2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]
[Out]
-1/30*((-1 + Sin[c + d*x])^4*(9*A + 2*B + (6*A + 8*B)*Sin[c + d*x] + 5*B*Sin[c + d*x]^2))/(a^2*d)
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fricas [A] time = 0.81, size = 82, normalized size = 1.04 \[ \frac {5 \, B \cos \left (d x + c\right )^{6} + 15 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, A - B\right )} \cos \left (d x + c\right )^{2} - 12 \, A + 4 \, B\right )} \sin \left (d x + c\right )}{30 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
[Out]
1/30*(5*B*cos(d*x + c)^6 + 15*(A - B)*cos(d*x + c)^4 - 2*(3*(A - 2*B)*cos(d*x + c)^4 - 2*(3*A - B)*cos(d*x + c
)^2 - 12*A + 4*B)*sin(d*x + c))/(a^2*d)
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giac [A] time = 0.23, size = 95, normalized size = 1.20 \[ -\frac {5 \, B \sin \left (d x + c\right )^{6} + 6 \, A \sin \left (d x + c\right )^{5} - 12 \, B \sin \left (d x + c\right )^{5} - 15 \, A \sin \left (d x + c\right )^{4} + 20 \, B \sin \left (d x + c\right )^{3} + 30 \, A \sin \left (d x + c\right )^{2} - 15 \, B \sin \left (d x + c\right )^{2} - 30 \, A \sin \left (d x + c\right )}{30 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c))^2,x, algorithm="giac")
[Out]
-1/30*(5*B*sin(d*x + c)^6 + 6*A*sin(d*x + c)^5 - 12*B*sin(d*x + c)^5 - 15*A*sin(d*x + c)^4 + 20*B*sin(d*x + c)
^3 + 30*A*sin(d*x + c)^2 - 15*B*sin(d*x + c)^2 - 30*A*sin(d*x + c))/(a^2*d)
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maple [A] time = 0.61, size = 82, normalized size = 1.04 \[ \frac {-\frac {B \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-A +2 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {A \left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 B \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-2 A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c))^2,x)
[Out]
1/d/a^2*(-1/6*B*sin(d*x+c)^6+1/5*(-A+2*B)*sin(d*x+c)^5+1/2*A*sin(d*x+c)^4-2/3*B*sin(d*x+c)^3+1/2*(-2*A+B)*sin(
d*x+c)^2+A*sin(d*x+c))
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maxima [A] time = 0.34, size = 83, normalized size = 1.05 \[ -\frac {5 \, B \sin \left (d x + c\right )^{6} + 6 \, {\left (A - 2 \, B\right )} \sin \left (d x + c\right )^{5} - 15 \, A \sin \left (d x + c\right )^{4} + 20 \, B \sin \left (d x + c\right )^{3} + 15 \, {\left (2 \, A - B\right )} \sin \left (d x + c\right )^{2} - 30 \, A \sin \left (d x + c\right )}{30 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
[Out]
-1/30*(5*B*sin(d*x + c)^6 + 6*(A - 2*B)*sin(d*x + c)^5 - 15*A*sin(d*x + c)^4 + 20*B*sin(d*x + c)^3 + 15*(2*A -
B)*sin(d*x + c)^2 - 30*A*sin(d*x + c))/(a^2*d)
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mupad [B] time = 0.08, size = 98, normalized size = 1.24 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^5\,\left (A-2\,B\right )}{5\,a^2}-\frac {A\,{\sin \left (c+d\,x\right )}^4}{2\,a^2}+\frac {2\,B\,{\sin \left (c+d\,x\right )}^3}{3\,a^2}+\frac {B\,{\sin \left (c+d\,x\right )}^6}{6\,a^2}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (2\,A-B\right )}{2\,a^2}-\frac {A\,\sin \left (c+d\,x\right )}{a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2,x)
[Out]
-((sin(c + d*x)^5*(A - 2*B))/(5*a^2) - (A*sin(c + d*x)^4)/(2*a^2) + (2*B*sin(c + d*x)^3)/(3*a^2) + (B*sin(c +
d*x)^6)/(6*a^2) + (sin(c + d*x)^2*(2*A - B))/(2*a^2) - (A*sin(c + d*x))/a^2)/d
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sympy [A] time = 132.04, size = 2705, normalized size = 34.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c))**2,x)
[Out]
Piecewise((30*A*tan(c/2 + d*x/2)**11/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a*
*2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2
+ d*x/2)**2 + 15*a**2*d) - 60*A*tan(c/2 + d*x/2)**10/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*
x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 +
90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 150*A*tan(c/2 + d*x/2)**9/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a*
*2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c
/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 120*A*tan(c/2 + d*x/2)**8/(15*a**2*d*tan(c/2 + d
*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 +
225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 204*A*tan(c/2 + d*x/2)**7/(15*a
**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(
c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 120*A*tan(c/2
+ d*x/2)**6/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8
+ 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d)
+ 204*A*tan(c/2 + d*x/2)**5/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan
(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2
)**2 + 15*a**2*d) - 120*A*tan(c/2 + d*x/2)**4/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10
+ 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*
d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 150*A*tan(c/2 + d*x/2)**3/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan
(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x
/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 60*A*tan(c/2 + d*x/2)**2/(15*a**2*d*tan(c/2 + d*x/2)**12
+ 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2
*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 30*A*tan(c/2 + d*x/2)/(15*a**2*d*tan(c/2
+ d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)*
*6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 30*B*tan(c/2 + d*x/2)**10/(
15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*
tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 80*B*tan(c
/2 + d*x/2)**9/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)*
*8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2
*d) + 120*B*tan(c/2 + d*x/2)**8/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*
tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*
x/2)**2 + 15*a**2*d) - 48*B*tan(c/2 + d*x/2)**7/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**
10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**
2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 20*B*tan(c/2 + d*x/2)**6/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*ta
n(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*
x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 48*B*tan(c/2 + d*x/2)**5/(15*a**2*d*tan(c/2 + d*x/2)**1
2 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**
2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 120*B*tan(c/2 + d*x/2)**4/(15*a**2*d*ta
n(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**2*d*tan(c/2 + d*
x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) - 80*B*tan(c/2 + d*x/2)*
*3/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x/2)**8 + 300*a**
2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*a**2*d) + 30*B*t
an(c/2 + d*x/2)**2/(15*a**2*d*tan(c/2 + d*x/2)**12 + 90*a**2*d*tan(c/2 + d*x/2)**10 + 225*a**2*d*tan(c/2 + d*x
/2)**8 + 300*a**2*d*tan(c/2 + d*x/2)**6 + 225*a**2*d*tan(c/2 + d*x/2)**4 + 90*a**2*d*tan(c/2 + d*x/2)**2 + 15*
a**2*d), Ne(d, 0)), (x*(A + B*sin(c))*cos(c)**7/(a*sin(c) + a)**2, True))
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