3.3.21 \(\int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [221]

Optimal. Leaf size=329 \[ -\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (b+a \cos (c+d x))^2}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (b+a \cos (c+d x))}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (b+a \cos (c+d x))}{a^{10} d} \]

[Out]

-(a^6-18*a^4*b^2+45*a^2*b^4-28*b^6)*cos(d*x+c)/a^9/d-3/2*b*(3*a^4-10*a^2*b^2+7*b^4)*cos(d*x+c)^2/a^8/d+(a^4-6*
a^2*b^2+5*b^4)*cos(d*x+c)^3/a^7/d+1/4*b*(9*a^2-10*b^2)*cos(d*x+c)^4/a^6/d-3/5*(a^2-2*b^2)*cos(d*x+c)^5/a^5/d-1
/2*b*cos(d*x+c)^6/a^4/d+1/7*cos(d*x+c)^7/a^3/d-1/2*b^3*(a^2-b^2)^3/a^10/d/(b+a*cos(d*x+c))^2+3*b^2*(a^2-3*b^2)
*(a^2-b^2)^2/a^10/d/(b+a*cos(d*x+c))+3*b*(a^2-b^2)*(a^4-9*a^2*b^2+12*b^4)*ln(b+a*cos(d*x+c))/a^10/d

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Rubi [A]
time = 0.36, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 962} \begin {gather*} -\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (a \cos (c+d x)+b)}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (a \cos (c+d x)+b)^2}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (a \cos (c+d x)+b)}{a^{10} d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}-\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]

[Out]

-(((a^6 - 18*a^4*b^2 + 45*a^2*b^4 - 28*b^6)*Cos[c + d*x])/(a^9*d)) - (3*b*(3*a^4 - 10*a^2*b^2 + 7*b^4)*Cos[c +
d*x]^2)/(2*a^8*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(a^7*d) + (b*(9*a^2 - 10*b^2)*Cos[c + d*x]^4)/
(4*a^6*d) - (3*(a^2 - 2*b^2)*Cos[c + d*x]^5)/(5*a^5*d) - (b*Cos[c + d*x]^6)/(2*a^4*d) + Cos[c + d*x]^7/(7*a^3*
d) - (b^3*(a^2 - b^2)^3)/(2*a^10*d*(b + a*Cos[c + d*x])^2) + (3*b^2*(a^2 - 3*b^2)*(a^2 - b^2)^2)/(a^10*d*(b +
a*Cos[c + d*x])) + (3*b*(a^2 - b^2)*(a^4 - 9*a^2*b^2 + 12*b^4)*Log[b + a*Cos[c + d*x]])/(a^10*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 962

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rule 2916

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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )^3}{a^3 (-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )^3}{(-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=\frac {\text {Subst}\left (\int \left (a^6 \left (1+\frac {-18 a^4 b^2+45 a^2 b^4-28 b^6}{a^6}\right )+\frac {b^3 \left (-a^2+b^2\right )^3}{(b-x)^3}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{(b-x)^2}+\frac {3 b \left (-a^6+10 a^4 b^2-21 a^2 b^4+12 b^6\right )}{b-x}-3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) x-3 \left (a^4-6 a^2 b^2+5 b^4\right ) x^2-b \left (-9 a^2+10 b^2\right ) x^3+3 \left (a^2-2 b^2\right ) x^4-3 b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=-\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (b+a \cos (c+d x))^2}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (b+a \cos (c+d x))}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (b+a \cos (c+d x))}{a^{10} d}\\ \end {align*}

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Mathematica [A]
time = 3.19, size = 550, normalized size = 1.67 \begin {gather*} \frac {-7945 a^8 b+164080 a^6 b^3-502320 a^4 b^5+425600 a^2 b^7-76160 b^9-784 a^9 \cos (3 (c+d x))+17528 a^7 b^2 \cos (3 (c+d x))-43680 a^5 b^4 \cos (3 (c+d x))+26880 a^3 b^6 \cos (3 (c+d x))-1456 a^8 b \cos (4 (c+d x))+4872 a^6 b^3 \cos (4 (c+d x))-3360 a^4 b^5 \cos (4 (c+d x))+152 a^9 \cos (5 (c+d x))-840 a^7 b^2 \cos (5 (c+d x))+672 a^5 b^4 \cos (5 (c+d x))+174 a^8 b \cos (6 (c+d x))-168 a^6 b^3 \cos (6 (c+d x))-39 a^9 \cos (7 (c+d x))+48 a^7 b^2 \cos (7 (c+d x))-15 a^8 b \cos (8 (c+d x))+5 a^9 \cos (9 (c+d x))+13440 a^8 b \log (b+a \cos (c+d x))-107520 a^6 b^3 \log (b+a \cos (c+d x))+13440 a^4 b^5 \log (b+a \cos (c+d x))+403200 a^2 b^7 \log (b+a \cos (c+d x))-322560 b^9 \log (b+a \cos (c+d x))+70 a^2 b \cos (2 (c+d x)) \left (-137 a^6+1896 a^4 b^2-4656 a^2 b^4+2912 b^6+192 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )-70 a \cos (c+d x) \left (49 a^8-1472 a^6 b^2+3216 a^4 b^4+576 a^2 b^6-2432 b^8-768 b^2 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )}{8960 a^{10} d (b+a \cos (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]

[Out]

(-7945*a^8*b + 164080*a^6*b^3 - 502320*a^4*b^5 + 425600*a^2*b^7 - 76160*b^9 - 784*a^9*Cos[3*(c + d*x)] + 17528
*a^7*b^2*Cos[3*(c + d*x)] - 43680*a^5*b^4*Cos[3*(c + d*x)] + 26880*a^3*b^6*Cos[3*(c + d*x)] - 1456*a^8*b*Cos[4
*(c + d*x)] + 4872*a^6*b^3*Cos[4*(c + d*x)] - 3360*a^4*b^5*Cos[4*(c + d*x)] + 152*a^9*Cos[5*(c + d*x)] - 840*a
^7*b^2*Cos[5*(c + d*x)] + 672*a^5*b^4*Cos[5*(c + d*x)] + 174*a^8*b*Cos[6*(c + d*x)] - 168*a^6*b^3*Cos[6*(c + d
*x)] - 39*a^9*Cos[7*(c + d*x)] + 48*a^7*b^2*Cos[7*(c + d*x)] - 15*a^8*b*Cos[8*(c + d*x)] + 5*a^9*Cos[9*(c + d*
x)] + 13440*a^8*b*Log[b + a*Cos[c + d*x]] - 107520*a^6*b^3*Log[b + a*Cos[c + d*x]] + 13440*a^4*b^5*Log[b + a*C
os[c + d*x]] + 403200*a^2*b^7*Log[b + a*Cos[c + d*x]] - 322560*b^9*Log[b + a*Cos[c + d*x]] + 70*a^2*b*Cos[2*(c
+ d*x)]*(-137*a^6 + 1896*a^4*b^2 - 4656*a^2*b^4 + 2912*b^6 + 192*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)*Log
[b + a*Cos[c + d*x]]) - 70*a*Cos[c + d*x]*(49*a^8 - 1472*a^6*b^2 + 3216*a^4*b^4 + 576*a^2*b^6 - 2432*b^8 - 768
*b^2*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)*Log[b + a*Cos[c + d*x]]))/(8960*a^10*d*(b + a*Cos[c + d*x])^2)

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Maple [A]
time = 0.15, size = 367, normalized size = 1.12 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/a^9*(1/7*cos(d*x+c)^7*a^6-1/2*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c)^5+6/5*a^4*b^2*cos(d*x+c)^5+9/4*a^5*
b*cos(d*x+c)^4-5/2*a^3*b^3*cos(d*x+c)^4+a^6*cos(d*x+c)^3-6*a^4*b^2*cos(d*x+c)^3+5*a^2*b^4*cos(d*x+c)^3-9/2*a^5
*b*cos(d*x+c)^2+15*a^3*b^3*cos(d*x+c)^2-21/2*a*b^5*cos(d*x+c)^2-a^6*cos(d*x+c)+18*a^4*b^2*cos(d*x+c)-45*a^2*b^
4*cos(d*x+c)+28*b^6*cos(d*x+c))+3*b^2/a^10*(a^6-5*a^4*b^2+7*a^2*b^4-3*b^6)/(b+a*cos(d*x+c))+3*b/a^10*(a^6-10*a
^4*b^2+21*a^2*b^4-12*b^6)*ln(b+a*cos(d*x+c))-1/2*b^3*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^10/(b+a*cos(d*x+c))^2)

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Maxima [A]
time = 0.27, size = 326, normalized size = 0.99 \begin {gather*} \frac {\frac {70 \, {\left (5 \, a^{6} b^{3} - 27 \, a^{4} b^{5} + 39 \, a^{2} b^{7} - 17 \, b^{9} + 6 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 7 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )\right )}}{a^{12} \cos \left (d x + c\right )^{2} + 2 \, a^{11} b \cos \left (d x + c\right ) + a^{10} b^{2}} + \frac {20 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (a^{6} - 2 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (9 \, a^{5} b - 10 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{6} - 6 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} - 140 \, {\left (a^{6} - 18 \, a^{4} b^{2} + 45 \, a^{2} b^{4} - 28 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{9}} + \frac {420 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - 12 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10}}}{140 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/140*(70*(5*a^6*b^3 - 27*a^4*b^5 + 39*a^2*b^7 - 17*b^9 + 6*(a^7*b^2 - 5*a^5*b^4 + 7*a^3*b^6 - 3*a*b^8)*cos(d*
x + c))/(a^12*cos(d*x + c)^2 + 2*a^11*b*cos(d*x + c) + a^10*b^2) + (20*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x +
c)^6 - 84*(a^6 - 2*a^4*b^2)*cos(d*x + c)^5 + 35*(9*a^5*b - 10*a^3*b^3)*cos(d*x + c)^4 + 140*(a^6 - 6*a^4*b^2
+ 5*a^2*b^4)*cos(d*x + c)^3 - 210*(3*a^5*b - 10*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^2 - 140*(a^6 - 18*a^4*b^2 + 45
*a^2*b^4 - 28*b^6)*cos(d*x + c))/a^9 + 420*(a^6*b - 10*a^4*b^3 + 21*a^2*b^5 - 12*b^7)*log(a*cos(d*x + c) + b)/
a^10)/d

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Fricas [A]
time = 4.56, size = 447, normalized size = 1.36 \begin {gather*} \frac {80 \, a^{9} \cos \left (d x + c\right )^{9} - 120 \, a^{8} b \cos \left (d x + c\right )^{8} + 2275 \, a^{6} b^{3} - 11235 \, a^{4} b^{5} + 13860 \, a^{2} b^{7} - 4760 \, b^{9} - 48 \, {\left (7 \, a^{9} - 4 \, a^{7} b^{2}\right )} \cos \left (d x + c\right )^{7} + 84 \, {\left (7 \, a^{8} b - 4 \, a^{6} b^{3}\right )} \cos \left (d x + c\right )^{6} + 56 \, {\left (10 \, a^{9} - 21 \, a^{7} b^{2} + 12 \, a^{5} b^{4}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (10 \, a^{8} b - 21 \, a^{6} b^{3} + 12 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )^{4} - 560 \, {\left (a^{9} - 10 \, a^{7} b^{2} + 21 \, a^{5} b^{4} - 12 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 35 \, {\left (7 \, a^{8} b - 399 \, a^{6} b^{3} + 1116 \, a^{4} b^{5} - 728 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (41 \, a^{7} b^{2} - 81 \, a^{5} b^{4} - 108 \, a^{3} b^{6} + 152 \, a b^{8}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{3} - 10 \, a^{4} b^{5} + 21 \, a^{2} b^{7} - 12 \, b^{9} + {\left (a^{8} b - 10 \, a^{6} b^{3} + 21 \, a^{4} b^{5} - 12 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{2} - 10 \, a^{5} b^{4} + 21 \, a^{3} b^{6} - 12 \, a b^{8}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{560 \, {\left (a^{12} d \cos \left (d x + c\right )^{2} + 2 \, a^{11} b d \cos \left (d x + c\right ) + a^{10} b^{2} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/560*(80*a^9*cos(d*x + c)^9 - 120*a^8*b*cos(d*x + c)^8 + 2275*a^6*b^3 - 11235*a^4*b^5 + 13860*a^2*b^7 - 4760*
b^9 - 48*(7*a^9 - 4*a^7*b^2)*cos(d*x + c)^7 + 84*(7*a^8*b - 4*a^6*b^3)*cos(d*x + c)^6 + 56*(10*a^9 - 21*a^7*b^
2 + 12*a^5*b^4)*cos(d*x + c)^5 - 140*(10*a^8*b - 21*a^6*b^3 + 12*a^4*b^5)*cos(d*x + c)^4 - 560*(a^9 - 10*a^7*b
^2 + 21*a^5*b^4 - 12*a^3*b^6)*cos(d*x + c)^3 - 35*(7*a^8*b - 399*a^6*b^3 + 1116*a^4*b^5 - 728*a^2*b^7)*cos(d*x
+ c)^2 + 70*(41*a^7*b^2 - 81*a^5*b^4 - 108*a^3*b^6 + 152*a*b^8)*cos(d*x + c) + 1680*(a^6*b^3 - 10*a^4*b^5 + 2
1*a^2*b^7 - 12*b^9 + (a^8*b - 10*a^6*b^3 + 21*a^4*b^5 - 12*a^2*b^7)*cos(d*x + c)^2 + 2*(a^7*b^2 - 10*a^5*b^4 +
21*a^3*b^6 - 12*a*b^8)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^12*d*cos(d*x + c)^2 + 2*a^11*b*d*cos(d*x + c
) + a^10*b^2*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**7/(a+b*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2150 vs. \(2 (317) = 634\).
time = 0.62, size = 2150, normalized size = 6.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/140*(420*(a^7*b - a^6*b^2 - 10*a^5*b^3 + 10*a^4*b^4 + 21*a^3*b^5 - 21*a^2*b^6 - 12*a*b^7 + 12*b^8)*log(abs(a
+ b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^11 - a^10*b) - 4
20*(a^6*b - 10*a^4*b^3 + 21*a^2*b^5 - 12*b^7)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^10 - 70*(
9*a^8*b + 6*a^7*b^2 - 105*a^6*b^3 - 148*a^5*b^4 + 187*a^4*b^5 + 390*a^3*b^6 + 17*a^2*b^7 - 248*a*b^8 - 108*b^9
+ 18*a^8*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^7*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 202*a^6
*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 56*a^5*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 566*a^4*b^5*(c
os(d*x + c) - 1)/(cos(d*x + c) + 1) - 76*a^3*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 598*a^2*b^7*(cos(d*x
+ c) - 1)/(cos(d*x + c) + 1) + 32*a*b^8*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 216*b^9*(cos(d*x + c) - 1)/(co
s(d*x + c) + 1) + 9*a^8*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 18*a^7*b^2*(cos(d*x + c) - 1)^2/(cos(d*x
+ c) + 1)^2 - 81*a^6*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 180*a^5*b^4*(cos(d*x + c) - 1)^2/(cos(d*
x + c) + 1)^2 + 99*a^4*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 378*a^3*b^6*(cos(d*x + c) - 1)^2/(cos(d
*x + c) + 1)^2 + 81*a^2*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 216*a*b^8*(cos(d*x + c) - 1)^2/(cos(d*
x + c) + 1)^2 - 108*b^9*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c
) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2*a^10) + (128*a^7 - 1089*a^6*b - 3696*a^5*b^2 + 10890*a^4*b
^3 + 11200*a^3*b^4 - 22869*a^2*b^5 - 7840*a*b^6 + 13068*b^7 - 896*a^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) +
8463*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 24192*a^5*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 81830
*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 70000*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 165963*
a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 47040*a*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 91476*b^7*
(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2688*a^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 28749*a^6*b*(cos(
d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 64176*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 262290*a^4*b^
3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 176400*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 50964
9*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 117600*a*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 +
274428*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 4480*a^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 5
6035*a^6*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 80640*a^5*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3
- 453950*a^4*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 229600*a^3*b^4*(cos(d*x + c) - 1)^3/(cos(d*x + c
) + 1)^3 + 859215*a^2*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 156800*a*b^6*(cos(d*x + c) - 1)^3/(cos(d
*x + c) + 1)^3 - 457380*b^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 56035*a^6*b*(cos(d*x + c) - 1)^4/(cos(
d*x + c) + 1)^4 - 48720*a^5*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 453950*a^4*b^3*(cos(d*x + c) - 1)^
4/(cos(d*x + c) + 1)^4 + 162400*a^3*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 859215*a^2*b^5*(cos(d*x +
c) - 1)^4/(cos(d*x + c) + 1)^4 - 117600*a*b^6*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 457380*b^7*(cos(d*x
+ c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749*a^6*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 13440*a^5*b^2*(cos(
d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 262290*a^4*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 58800*a^3*b^
4*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 509649*a^2*b^5*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 47040
*a*b^6*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 274428*b^7*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 8463
*a^6*b*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 1680*a^5*b^2*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 81
830*a^4*b^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 8400*a^3*b^4*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6
- 165963*a^2*b^5*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 7840*a*b^6*(cos(d*x + c) - 1)^6/(cos(d*x + c) +
1)^6 + 91476*b^7*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*a^6*b*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1
)^7 - 10890*a^4*b^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 22869*a^2*b^5*(cos(d*x + c) - 1)^7/(cos(d*x +
c) + 1)^7 - 13068*b^7*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/(a^10*((cos(d*x + c) - 1)/(cos(d*x + c) + 1)
- 1)^7))/d

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Mupad [B]
time = 0.24, size = 762, normalized size = 2.32 \begin {gather*} \frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {2\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{4\,a}\right )}{d}-\frac {\frac {-5\,a^6\,b^3+27\,a^4\,b^5-39\,a^2\,b^7+17\,b^9}{2\,a}+\cos \left (c+d\,x\right )\,\left (-3\,a^6\,b^2+15\,a^4\,b^4-21\,a^2\,b^6+9\,b^8\right )}{d\,\left (a^{11}\,{\cos \left (c+d\,x\right )}^2+2\,a^{10}\,b\,\cos \left (c+d\,x\right )+a^9\,b^2\right )}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^3}-\frac {6\,b^2}{5\,a^5}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,b^2\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{2\,a^2}-\frac {b^3\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{2\,a^3}+\frac {3\,b\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{2\,a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^3\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^3}+\frac {b^3\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a^3}+\frac {3\,b^2\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{a^2}-\frac {3\,b\,\left (\frac {3\,b^2\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a^2}-\frac {b^3\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^3}+\frac {3\,b\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^3}+\frac {b^4}{a^7}+\frac {b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{2\,a^4\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^6\,b-30\,a^4\,b^3+63\,a^2\,b^5-36\,b^7\right )}{a^{10}\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^7/(a + b/cos(c + d*x))^3,x)

[Out]

(cos(c + d*x)^4*((2*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/(4*a)))/d - ((17*b^9 - 39*a^2*b^7 + 27*a^4*b^5 - 5*
a^6*b^3)/(2*a) + cos(c + d*x)*(9*b^8 - 21*a^2*b^6 + 15*a^4*b^4 - 3*a^6*b^2))/(d*(a^11*cos(c + d*x)^2 + a^9*b^2
+ 2*a^10*b*cos(c + d*x))) - (cos(c + d*x)^5*(3/(5*a^3) - (6*b^2)/(5*a^5)))/d - (cos(c + d*x)^2*((3*b^2*((8*b^
3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/(2*a^2) - (b^3*(3/a^3 - (6*b^2)/a^5))/(2*a^3) + (3*b*(3/a^3 + (3*b^4)
/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a))/(2*a)))/d +
cos(c + d*x)^7/(7*a^3*d) - (cos(c + d*x)*(1/a^3 + (b^3*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a^3 + (
3*b^2*(3/a^3 + (3*b^4)/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5
))/a))/a))/a^2 - (3*b*((3*b^2*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a^2 - (b^3*(3/a^3 - (6*b^2)/a^5))
/a^3 + (3*b*(3/a^3 + (3*b^4)/a^7 + (3*b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (3*b*((8*b^3)/a^6 + (3*b*(3/a^3 - (6*b^
2)/a^5))/a))/a))/a))/a))/d + (cos(c + d*x)^3*(1/a^3 + b^4/a^7 + (b^2*(3/a^3 - (6*b^2)/a^5))/a^2 - (b*((8*b^3)/
a^6 + (3*b*(3/a^3 - (6*b^2)/a^5))/a))/a))/d - (b*cos(c + d*x)^6)/(2*a^4*d) + (log(b + a*cos(c + d*x))*(3*a^6*b
- 36*b^7 + 63*a^2*b^5 - 30*a^4*b^3))/(a^10*d)

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