3.209 \(\int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (a \cos (c+d x)+b)}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 267, 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 = {3872, 2837, 12, 948} \[ -\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}+\frac {\left (-9 a^2 b^2+3 a^4+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (a \cos (c+d x)+b)}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]

[Out]

-(((a^2 - 7*b^2)*(a^2 - b^2)^2*Cos[c + d*x])/(a^8*d)) - (3*b*(a^2 - b^2)^2*Cos[c + d*x]^2)/(a^7*d) + ((3*a^4 -
9*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(3*a^6*d) + (b*(3*a^2 - 2*b^2)*Cos[c + d*x]^4)/(2*a^5*d) - (3*(a^2 - b^2)*
Cos[c + d*x]^5)/(5*a^4*d) - (b*Cos[c + d*x]^6)/(3*a^3*d) + Cos[c + d*x]^7/(7*a^2*d) + (b^2*(a^2 - b^2)^3)/(a^9
*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]])/(a^9*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 948

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 2837

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*x)/b)^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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^7(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^3}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^3}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2-\frac {b^2 \left (-a^2+b^2\right )^3}{(b-x)^2}+\frac {2 b \left (-a^2+b^2\right )^2 \left (-a^2+4 b^2\right )}{b-x}-6 b \left (-a^2+b^2\right )^2 x-\left (3 a^4-9 a^2 b^2+5 b^4\right ) x^2-2 b \left (-3 a^2+2 b^2\right ) x^3+3 \left (a^2-b^2\right ) x^4-2 b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d}\\ \end {align*}

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Mathematica [A]  time = 3.61, size = 417, normalized size = 1.56 \[ \frac {588 a^8 \cos (4 (c+d x))-132 a^8 \cos (6 (c+d x))+15 a^8 \cos (8 (c+d x))-3675 a^8-3780 a^7 b \cos (3 (c+d x))+476 a^7 b \cos (5 (c+d x))-40 a^7 b \cos (7 (c+d x))-1848 a^6 b^2 \cos (4 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))+26880 a^6 b^2 \log (a \cos (c+d x)+b)+61320 a^6 b^2+8400 a^5 b^3 \cos (3 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))-161280 a^4 b^4 \log (a \cos (c+d x)+b)-132720 a^4 b^4-4480 a^3 b^5 \cos (3 (c+d x))+241920 a^2 b^6 \log (a \cos (c+d x)+b)+87360 a^2 b^6+1680 a b \cos (c+d x) \left (-8 a^6+67 a^4 b^2-116 a^2 b^4+16 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)+56 b^6\right )-140 \left (21 a^8-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6\right ) \cos (2 (c+d x))-107520 b^8 \log (a \cos (c+d x)+b)-13440 b^8}{13440 a^9 d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]

[Out]

(-3675*a^8 + 61320*a^6*b^2 - 132720*a^4*b^4 + 87360*a^2*b^6 - 13440*b^8 - 140*(21*a^8 - 228*a^6*b^2 + 400*a^4*
b^4 - 192*a^2*b^6)*Cos[2*(c + d*x)] - 3780*a^7*b*Cos[3*(c + d*x)] + 8400*a^5*b^3*Cos[3*(c + d*x)] - 4480*a^3*b
^5*Cos[3*(c + d*x)] + 588*a^8*Cos[4*(c + d*x)] - 1848*a^6*b^2*Cos[4*(c + d*x)] + 1120*a^4*b^4*Cos[4*(c + d*x)]
+ 476*a^7*b*Cos[5*(c + d*x)] - 336*a^5*b^3*Cos[5*(c + d*x)] - 132*a^8*Cos[6*(c + d*x)] + 112*a^6*b^2*Cos[6*(c
+ d*x)] - 40*a^7*b*Cos[7*(c + d*x)] + 15*a^8*Cos[8*(c + d*x)] + 26880*a^6*b^2*Log[b + a*Cos[c + d*x]] - 16128
0*a^4*b^4*Log[b + a*Cos[c + d*x]] + 241920*a^2*b^6*Log[b + a*Cos[c + d*x]] - 107520*b^8*Log[b + a*Cos[c + d*x]
] + 1680*a*b*Cos[c + d*x]*(-8*a^6 + 67*a^4*b^2 - 116*a^2*b^4 + 56*b^6 + 16*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b +
a*Cos[c + d*x]]))/(13440*a^9*d*(b + a*Cos[c + d*x]))

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fricas [A]  time = 0.82, size = 344, normalized size = 1.29 \[ \frac {120 \, a^{8} \cos \left (d x + c\right )^{8} - 160 \, a^{7} b \cos \left (d x + c\right )^{7} + 1715 \, a^{6} b^{2} - 4725 \, a^{4} b^{4} + 3780 \, a^{2} b^{6} - 840 \, b^{8} - 56 \, {\left (9 \, a^{8} - 4 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{6} + 84 \, {\left (9 \, a^{7} b - 4 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (6 \, a^{8} - 9 \, a^{6} b^{2} + 4 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 280 \, {\left (6 \, a^{7} b - 9 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 840 \, {\left (a^{8} - 6 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (a^{7} b + 153 \, a^{5} b^{3} - 324 \, a^{3} b^{5} + 168 \, a b^{7}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{2} - 6 \, a^{4} b^{4} + 9 \, a^{2} b^{6} - 4 \, b^{8} + {\left (a^{7} b - 6 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{840 \, {\left (a^{10} d \cos \left (d x + c\right ) + a^{9} b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/840*(120*a^8*cos(d*x + c)^8 - 160*a^7*b*cos(d*x + c)^7 + 1715*a^6*b^2 - 4725*a^4*b^4 + 3780*a^2*b^6 - 840*b^
8 - 56*(9*a^8 - 4*a^6*b^2)*cos(d*x + c)^6 + 84*(9*a^7*b - 4*a^5*b^3)*cos(d*x + c)^5 + 140*(6*a^8 - 9*a^6*b^2 +
4*a^4*b^4)*cos(d*x + c)^4 - 280*(6*a^7*b - 9*a^5*b^3 + 4*a^3*b^5)*cos(d*x + c)^3 - 840*(a^8 - 6*a^6*b^2 + 9*a
^4*b^4 - 4*a^2*b^6)*cos(d*x + c)^2 + 35*(a^7*b + 153*a^5*b^3 - 324*a^3*b^5 + 168*a*b^7)*cos(d*x + c) + 1680*(a
^6*b^2 - 6*a^4*b^4 + 9*a^2*b^6 - 4*b^8 + (a^7*b - 6*a^5*b^3 + 9*a^3*b^5 - 4*a*b^7)*cos(d*x + c))*log(a*cos(d*x
+ c) + b))/(a^10*d*cos(d*x + c) + a^9*b*d)

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giac [B]  time = 0.77, size = 1861, normalized size = 6.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/210*(420*(a^7*b - a^6*b^2 - 6*a^5*b^3 + 6*a^4*b^4 + 9*a^3*b^5 - 9*a^2*b^6 - 4*a*b^7 + 4*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^10 - a^9*b) - 420*(a^6
*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^9 - 420*(a^7*b - 7*
a^5*b^3 - 4*a^4*b^4 + 11*a^3*b^5 + 8*a^2*b^6 - 5*a*b^7 - 4*b^8 + a^7*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) -
a^6*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*a^5*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^4*b^4*(
cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a^3*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 9*a^2*b^6*(cos(d*x +
c) - 1)/(cos(d*x + c) + 1) - 4*a*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b^8*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1))/((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^9)
+ (192*a^7 - 1089*a^6*b - 2772*a^5*b^2 + 6534*a^4*b^3 + 5600*a^3*b^4 - 9801*a^2*b^5 - 2940*a*b^6 + 4356*b^7 -
1344*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) + 18144*a^5*
b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 49098*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 35000*a^3*b^
4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 71127*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 17640*a*b^6*(c
os(d*x + c) - 1)/(cos(d*x + c) + 1) - 30492*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4032*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 - 48132*a^5*b^2*(cos(d*x
+ c) - 1)^2/(cos(d*x + c) + 1)^2 + 157374*a^4*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 88200*a^3*b^4*(c
os(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 218421*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 44100*a*b
^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 91476*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 6720*a^7*
(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*a^6*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 60480*a^5*
b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 272370*a^4*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 114
800*a^3*b^4*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 368235*a^2*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)
^3 + 58800*a*b^6*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 152460*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 - 36540*a^5*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c)
+ 1)^4 + 272370*a^4*b^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 81200*a^3*b^4*(cos(d*x + c) - 1)^4/(cos(d
*x + c) + 1)^4 - 368235*a^2*b^5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 44100*a*b^6*(cos(d*x + c) - 1)^4/(
cos(d*x + c) + 1)^4 + 152460*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 + 10080*a^5*b^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 157374*a^4*b^3*(cos(d*x + c)
- 1)^5/(cos(d*x + c) + 1)^5 - 29400*a^3*b^4*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 218421*a^2*b^5*(cos(d*
x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 17640*a*b^6*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 91476*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 - 1260*a^5*b^2*(cos(
d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 49098*a^4*b^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 4200*a^3*b^4*
(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 71127*a^2*b^5*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 2940*a*b
^6*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 30492*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 - 6534*a^4*b^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 9801*a^
2*b^5*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 - 4356*b^7*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/(a^9*((c
os(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^7))/d

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maple [A]  time = 0.59, size = 456, normalized size = 1.71 \[ \frac {\cos ^{7}\left (d x +c \right )}{7 a^{2} d}-\frac {b \left (\cos ^{6}\left (d x +c \right )\right )}{3 a^{3} d}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{5 a^{2} d}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right ) b^{2}}{5 d \,a^{4}}+\frac {3 b \left (\cos ^{4}\left (d x +c \right )\right )}{2 a^{3} d}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) b^{3}}{d \,a^{5}}+\frac {\cos ^{3}\left (d x +c \right )}{a^{2} d}-\frac {3 \left (\cos ^{3}\left (d x +c \right )\right ) b^{2}}{d \,a^{4}}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) b^{4}}{3 d \,a^{6}}-\frac {3 b \left (\cos ^{2}\left (d x +c \right )\right )}{a^{3} d}+\frac {6 \left (\cos ^{2}\left (d x +c \right )\right ) b^{3}}{d \,a^{5}}-\frac {3 \left (\cos ^{2}\left (d x +c \right )\right ) b^{5}}{d \,a^{7}}-\frac {\cos \left (d x +c \right )}{a^{2} d}+\frac {9 \cos \left (d x +c \right ) b^{2}}{d \,a^{4}}-\frac {15 \cos \left (d x +c \right ) b^{4}}{d \,a^{6}}+\frac {7 \cos \left (d x +c \right ) b^{6}}{d \,a^{8}}+\frac {2 b \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{3} d}-\frac {12 b^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{5}}+\frac {18 b^{5} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{7}}-\frac {8 b^{7} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{9}}+\frac {b^{2}}{a^{3} d \left (b +a \cos \left (d x +c \right )\right )}-\frac {3 b^{4}}{d \,a^{5} \left (b +a \cos \left (d x +c \right )\right )}+\frac {3 b^{6}}{d \,a^{7} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{8}}{d \,a^{9} \left (b +a \cos \left (d x +c \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a+b*sec(d*x+c))^2,x)

[Out]

1/7*cos(d*x+c)^7/a^2/d-1/3*b*cos(d*x+c)^6/a^3/d-3/5*cos(d*x+c)^5/a^2/d+3/5/d/a^4*cos(d*x+c)^5*b^2+3/2*b*cos(d*
x+c)^4/a^3/d-1/d/a^5*cos(d*x+c)^4*b^3+cos(d*x+c)^3/a^2/d-3/d/a^4*cos(d*x+c)^3*b^2+5/3/d/a^6*cos(d*x+c)^3*b^4-3
*b*cos(d*x+c)^2/a^3/d+6/d/a^5*cos(d*x+c)^2*b^3-3/d/a^7*cos(d*x+c)^2*b^5-cos(d*x+c)/a^2/d+9/d/a^4*cos(d*x+c)*b^
2-15/d/a^6*cos(d*x+c)*b^4+7/d/a^8*cos(d*x+c)*b^6+2*b*ln(b+a*cos(d*x+c))/a^3/d-12/d/a^5*b^3*ln(b+a*cos(d*x+c))+
18/d/a^7*b^5*ln(b+a*cos(d*x+c))-8/d/a^9*b^7*ln(b+a*cos(d*x+c))+b^2/a^3/d/(b+a*cos(d*x+c))-3/d*b^4/a^5/(b+a*cos
(d*x+c))+3/d*b^6/a^7/(b+a*cos(d*x+c))-1/d*b^8/a^9/(b+a*cos(d*x+c))

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maxima [A]  time = 0.79, size = 271, normalized size = 1.01 \[ \frac {\frac {210 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )}}{a^{10} \cos \left (d x + c\right ) + a^{9} b} + \frac {30 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 126 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - 2 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (3 \, a^{6} - 9 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 210 \, {\left (a^{6} - 9 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{8}} + \frac {420 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{9}}}{210 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

1/210*(210*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)/(a^10*cos(d*x + c) + a^9*b) + (30*a^6*cos(d*x + c)^7 - 70*a
^5*b*cos(d*x + c)^6 - 126*(a^6 - a^4*b^2)*cos(d*x + c)^5 + 105*(3*a^5*b - 2*a^3*b^3)*cos(d*x + c)^4 + 70*(3*a^
6 - 9*a^4*b^2 + 5*a^2*b^4)*cos(d*x + c)^3 - 630*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c)^2 - 210*(a^6 - 9*a^4*
b^2 + 15*a^2*b^4 - 7*b^6)*cos(d*x + c))/a^8 + 420*(a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(a*cos(d*x + c) +
b)/a^9)/d

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mupad [B]  time = 0.19, size = 588, normalized size = 2.20 \[ \frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {b^3}{2\,a^5}+\frac {b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{2\,a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{2\,a^2}+\frac {b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^2}-\frac {3\,b^2}{5\,a^4}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a^2}-\frac {2\,b\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a^2}+\frac {2\,b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{3\,a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{3\,a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{3\,a^3\,d}-\frac {-a^6\,b^2+3\,a^4\,b^4-3\,a^2\,b^6+b^8}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^9+b\,a^8\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^6\,b-12\,a^4\,b^3+18\,a^2\,b^5-8\,b^7\right )}{a^9\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^7/(a + b/cos(c + d*x))^2,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**7/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

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