3.1003 \(\int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx\)
Optimal. Leaf size=105 \[ \frac {B (a-a \sin (c+d x))^7}{7 a^8 d}-\frac {(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(A+B) (a-a \sin (c+d x))^4}{a^5 d} \]
[Out]
-(A+B)*(a-a*sin(d*x+c))^4/a^5/d+4/5*(A+2*B)*(a-a*sin(d*x+c))^5/a^6/d-1/6*(A+5*B)*(a-a*sin(d*x+c))^6/a^7/d+1/7*
B*(a-a*sin(d*x+c))^7/a^8/d
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Rubi [A] time = 0.15, antiderivative size = 105, 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} \[ -\frac {(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac {B (a-a \sin (c+d x))^7}{7 a^8 d} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
[Out]
-(((A + B)*(a - a*Sin[c + d*x])^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - ((A + 5*B)*(a -
a*Sin[c + d*x])^6)/(6*a^7*d) + (B*(a - a*Sin[c + d*x])^7)/(7*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)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^2 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (A+B) (a-x)^3-4 a (A+2 B) (a-x)^4+(A+5 B) (a-x)^5-\frac {B (a-x)^6}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac {B (a-a \sin (c+d x))^7}{7 a^8 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 69, normalized size = 0.66 \[ -\frac {(\sin (c+d x)-1)^4 \left (5 (7 A+17 B) \sin ^2(c+d x)+(98 A+76 B) \sin (c+d x)+77 A+30 B \sin ^3(c+d x)+19 B\right )}{210 a d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
[Out]
-1/210*((-1 + Sin[c + d*x])^4*(77*A + 19*B + (98*A + 76*B)*Sin[c + d*x] + 5*(7*A + 17*B)*Sin[c + d*x]^2 + 30*B
*Sin[c + d*x]^3))/(a*d)
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fricas [A] time = 0.75, size = 84, normalized size = 0.80 \[ \frac {35 \, {\left (A - B\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, B \cos \left (d x + c\right )^{6} + 3 \, {\left (7 \, A - B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (7 \, A - B\right )} \cos \left (d x + c\right )^{2} + 56 \, A - 8 \, B\right )} \sin \left (d x + c\right )}{210 \, a d} \]
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)),x, algorithm="fricas")
[Out]
1/210*(35*(A - B)*cos(d*x + c)^6 + 2*(15*B*cos(d*x + c)^6 + 3*(7*A - B)*cos(d*x + c)^4 + 4*(7*A - B)*cos(d*x +
c)^2 + 56*A - 8*B)*sin(d*x + c))/(a*d)
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giac [A] time = 0.21, size = 139, normalized size = 1.32 \[ -\frac {30 \, B \sin \left (d x + c\right )^{7} + 35 \, A \sin \left (d x + c\right )^{6} - 35 \, B \sin \left (d x + c\right )^{6} - 42 \, A \sin \left (d x + c\right )^{5} - 84 \, B \sin \left (d x + c\right )^{5} - 105 \, A \sin \left (d x + c\right )^{4} + 105 \, B \sin \left (d x + c\right )^{4} + 140 \, A \sin \left (d x + c\right )^{3} + 70 \, B \sin \left (d x + c\right )^{3} + 105 \, A \sin \left (d x + c\right )^{2} - 105 \, B \sin \left (d x + c\right )^{2} - 210 \, A \sin \left (d x + c\right )}{210 \, a d} \]
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)),x, algorithm="giac")
[Out]
-1/210*(30*B*sin(d*x + c)^7 + 35*A*sin(d*x + c)^6 - 35*B*sin(d*x + c)^6 - 42*A*sin(d*x + c)^5 - 84*B*sin(d*x +
c)^5 - 105*A*sin(d*x + c)^4 + 105*B*sin(d*x + c)^4 + 140*A*sin(d*x + c)^3 + 70*B*sin(d*x + c)^3 + 105*A*sin(d
*x + c)^2 - 105*B*sin(d*x + c)^2 - 210*A*sin(d*x + c))/(a*d)
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maple [A] time = 0.56, size = 107, normalized size = 1.02 \[ \frac {-\frac {B \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-A +B \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (A +2 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (2 A -2 B \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (-2 A -B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )}{d a} \]
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)),x)
[Out]
1/d/a*(-1/7*B*sin(d*x+c)^7+1/6*(-A+B)*sin(d*x+c)^6+1/5*(A+2*B)*sin(d*x+c)^5+1/4*(2*A-2*B)*sin(d*x+c)^4+1/3*(-2
*A-B)*sin(d*x+c)^3+1/2*(-A+B)*sin(d*x+c)^2+A*sin(d*x+c))
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maxima [A] time = 0.32, size = 104, normalized size = 0.99 \[ -\frac {30 \, B \sin \left (d x + c\right )^{7} + 35 \, {\left (A - B\right )} \sin \left (d x + c\right )^{6} - 42 \, {\left (A + 2 \, B\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (A - B\right )} \sin \left (d x + c\right )^{4} + 70 \, {\left (2 \, A + B\right )} \sin \left (d x + c\right )^{3} + 105 \, {\left (A - B\right )} \sin \left (d x + c\right )^{2} - 210 \, A \sin \left (d x + c\right )}{210 \, a d} \]
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)),x, algorithm="maxima")
[Out]
-1/210*(30*B*sin(d*x + c)^7 + 35*(A - B)*sin(d*x + c)^6 - 42*(A + 2*B)*sin(d*x + c)^5 - 105*(A - B)*sin(d*x +
c)^4 + 70*(2*A + B)*sin(d*x + c)^3 + 105*(A - B)*sin(d*x + c)^2 - 210*A*sin(d*x + c))/(a*d)
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mupad [B] time = 0.11, size = 124, normalized size = 1.18 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^2\,\left (A-B\right )}{2\,a}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (2\,A+B\right )}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (A+2\,B\right )}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^6\,\left (A-B\right )}{6\,a}+\frac {B\,{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (2\,A-2\,B\right )}{4\,a}-\frac {A\,\sin \left (c+d\,x\right )}{a}}{d} \]
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)),x)
[Out]
-((sin(c + d*x)^2*(A - B))/(2*a) + (sin(c + d*x)^3*(2*A + B))/(3*a) - (sin(c + d*x)^5*(A + 2*B))/(5*a) + (sin(
c + d*x)^6*(A - B))/(6*a) + (B*sin(c + d*x)^7)/(7*a) - (sin(c + d*x)^4*(2*A - 2*B))/(4*a) - (A*sin(c + d*x))/a
)/d
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sympy [A] time = 83.08, size = 3363, normalized size = 32.03 \[ \text {result too large to display} \]
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)),x)
[Out]
Piecewise((210*A*tan(c/2 + d*x/2)**13/(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*A*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*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*A*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 + 10
5*a*d) - 210*A*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*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) + 1582*A*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*A*tan(c/2 + d*x/2)**8/(10
5*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) + 2184*A*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 + 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*A*tan(c/2 + d*x/2)**6/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*ta
n(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) + 1582*A*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*A*tan(c/2 + d*x/2)**4/(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*A*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*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*A*tan(c/2 + d*x/2)**2/(105*a*d*ta
n(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) + 21
0*A*tan(c/2 + d*x/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) + 210*B*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*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*B*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 + 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) + 210*B*ta
n(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)**1
0 + 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*B*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*B*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 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 912*B*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 + 3
675*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*B*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*B*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*t
an(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*B*tan(c/2 + d
*x/2)**4/(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) - 280*B*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 + 220
5*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*B*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*(A + B*si
n(c))*cos(c)**7/(a*sin(c) + a), True))
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