3.424 \(\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=236 \[ \frac {128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}+\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{63 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d} \]

[Out]

128/315*a*b*(a^2-b^2)^3*sec(d*x+c)/d+64/315*a*(a^2-b^2)^2*sec(d*x+c)^3*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^2/d+1
6/105*a*(a^2-b^2)*sec(d*x+c)^5*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^4/d+1/9*sec(d*x+c)^9*(b+a*sin(d*x+c))*(a+b*si
n(d*x+c))^7/d+1/63*sec(d*x+c)^7*(a+b*sin(d*x+c))^6*(a*b+(8*a^2-7*b^2)*sin(d*x+c))/d+128/315*a^2*(a^2-b^2)^3*ta
n(d*x+c)/d

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Rubi [A]  time = 0.38, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2691, 2861, 12, 2669, 3767, 8} \[ \frac {128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}+\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{63 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]

[Out]

(128*a*b*(a^2 - b^2)^3*Sec[c + d*x])/(315*d) + (64*a*(a^2 - b^2)^2*Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*
Sin[c + d*x])^2)/(315*d) + (16*a*(a^2 - b^2)*Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^4)/(105*
d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(9*d) + (Sec[c + d*x]^7*(a + b*Sin[c + d*x])
^6*(a*b + (8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*d) + (128*a^2*(a^2 - b^2)^3*Tan[c + d*x])/(315*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2691

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[((g*C
os[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1
)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin
[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[
2*m, 2*p] || IntegerQ[m])

Rule 2861

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x]))/(f*
g*(p + 1)), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p +
2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
&& GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x
])

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}-\frac {1}{9} \int \sec ^8(c+d x) (a+b \sin (c+d x))^6 \left (-8 a^2+7 b^2-a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{63} \int 48 a \left (a^2-b^2\right ) \sec ^6(c+d x) (a+b \sin (c+d x))^5 \, dx\\ &=\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{21} \left (16 a \left (a^2-b^2\right )\right ) \int \sec ^6(c+d x) (a+b \sin (c+d x))^5 \, dx\\ &=\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac {1}{105} \left (16 a \left (a^2-b^2\right )\right ) \int \left (-4 a^2+4 b^2\right ) \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\\ &=\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{105} \left (64 a \left (a^2-b^2\right )^2\right ) \int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\\ &=\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac {1}{315} \left (64 a \left (a^2-b^2\right )^2\right ) \int \left (-2 a^2+2 b^2\right ) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{315} \left (128 a \left (a^2-b^2\right )^3\right ) \int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{315} \left (128 a^2 \left (a^2-b^2\right )^3\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac {\left (128 a^2 \left (a^2-b^2\right )^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{315 d}\\ &=\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}\\ \end {align*}

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Mathematica [A]  time = 4.46, size = 313, normalized size = 1.33 \[ \frac {\cos (c+d x) \left (\frac {a \left (8 (a-b) (1-\sin (c+d x)) \left ((a-b) (1-\sin (c+d x)) \left (2 (a-b) (1-\sin (c+d x)) \left ((a-b) (1-\sin (c+d x)) \left (35 (a+b \sin (c+d x))^4-4 (a-b) (1-\sin (c+d x)) \left ((a+b) (\sin (c+d x)+1) \left (\left (2 a^2+6 a b+7 b^2\right ) \sin ^2(c+d x)+6 \left (a^2+3 a b+b^2\right ) \sin (c+d x)+7 a^2+6 a b+2 b^2\right )+5 (a+b \sin (c+d x))^3\right )\right )+7 (a+b \sin (c+d x))^5\right )+7 (a+b \sin (c+d x))^6\right )+5 (a+b \sin (c+d x))^7\right )+35 (a+b \sin (c+d x))^8\right )}{35 (1-\sin (c+d x))^5 (\sin (c+d x)+1)^4}-\sec ^{10}(c+d x) (a+b \sin (c+d x))^9\right )}{9 d (a-b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]

[Out]

(Cos[c + d*x]*(-(Sec[c + d*x]^10*(a + b*Sin[c + d*x])^9) + (a*(35*(a + b*Sin[c + d*x])^8 + 8*(a - b)*(1 - Sin[
c + d*x])*(5*(a + b*Sin[c + d*x])^7 + (a - b)*(1 - Sin[c + d*x])*(7*(a + b*Sin[c + d*x])^6 + 2*(a - b)*(1 - Si
n[c + d*x])*(7*(a + b*Sin[c + d*x])^5 + (a - b)*(1 - Sin[c + d*x])*(35*(a + b*Sin[c + d*x])^4 - 4*(a - b)*(1 -
Sin[c + d*x])*(5*(a + b*Sin[c + d*x])^3 + (a + b)*(1 + Sin[c + d*x])*(7*a^2 + 6*a*b + 2*b^2 + 6*(a^2 + 3*a*b
+ b^2)*Sin[c + d*x] + (2*a^2 + 6*a*b + 7*b^2)*Sin[c + d*x]^2))))))))/(35*(1 - Sin[c + d*x])^5*(1 + Sin[c + d*x
])^4)))/(9*(a - b)*d)

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fricas [A]  time = 0.51, size = 336, normalized size = 1.42 \[ -\frac {840 \, a b^{7} \cos \left (d x + c\right )^{6} - 280 \, a^{7} b - 1960 \, a^{5} b^{3} - 1960 \, a^{3} b^{5} - 280 \, a b^{7} - 504 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 360 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (128 \, a^{8} - 448 \, a^{6} b^{2} + 560 \, a^{4} b^{4} - 280 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 35 \, a^{8} + 980 \, a^{6} b^{2} + 2450 \, a^{4} b^{4} + 980 \, a^{2} b^{6} + 35 \, b^{8} + 4 \, {\left (16 \, a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4} - 35 \, a^{2} b^{6} - 35 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (8 \, a^{8} - 28 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 350 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{8} - 7 \, a^{6} b^{2} - 175 \, a^{4} b^{4} - 133 \, a^{2} b^{6} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

-1/315*(840*a*b^7*cos(d*x + c)^6 - 280*a^7*b - 1960*a^5*b^3 - 1960*a^3*b^5 - 280*a*b^7 - 504*(7*a^3*b^5 + 3*a*
b^7)*cos(d*x + c)^4 + 360*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 - ((128*a^8 - 448*a^6*b^2 + 560*a^
4*b^4 - 280*a^2*b^6 + 35*b^8)*cos(d*x + c)^8 + 35*a^8 + 980*a^6*b^2 + 2450*a^4*b^4 + 980*a^2*b^6 + 35*b^8 + 4*
(16*a^8 - 56*a^6*b^2 + 70*a^4*b^4 - 35*a^2*b^6 - 35*b^8)*cos(d*x + c)^6 + 6*(8*a^8 - 28*a^6*b^2 + 35*a^4*b^4 +
350*a^2*b^6 + 35*b^8)*cos(d*x + c)^4 + 20*(2*a^8 - 7*a^6*b^2 - 175*a^4*b^4 - 133*a^2*b^6 - 7*b^8)*cos(d*x + c
)^2)*sin(d*x + c))/(d*cos(d*x + c)^9)

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giac [B]  time = 1.92, size = 892, normalized size = 3.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-2/315*(315*a^8*tan(1/2*d*x + 1/2*c)^17 + 2520*a^7*b*tan(1/2*d*x + 1/2*c)^16 - 840*a^8*tan(1/2*d*x + 1/2*c)^15
+ 11760*a^6*b^2*tan(1/2*d*x + 1/2*c)^15 + 35280*a^5*b^3*tan(1/2*d*x + 1/2*c)^14 + 4788*a^8*tan(1/2*d*x + 1/2*
c)^13 + 14112*a^6*b^2*tan(1/2*d*x + 1/2*c)^13 + 70560*a^4*b^4*tan(1/2*d*x + 1/2*c)^13 + 23520*a^7*b*tan(1/2*d*
x + 1/2*c)^12 + 58800*a^5*b^3*tan(1/2*d*x + 1/2*c)^12 + 94080*a^3*b^5*tan(1/2*d*x + 1/2*c)^12 - 5112*a^8*tan(1
/2*d*x + 1/2*c)^11 + 79632*a^6*b^2*tan(1/2*d*x + 1/2*c)^11 + 120960*a^4*b^4*tan(1/2*d*x + 1/2*c)^11 + 80640*a^
2*b^6*tan(1/2*d*x + 1/2*c)^11 + 176400*a^5*b^3*tan(1/2*d*x + 1/2*c)^10 + 141120*a^3*b^5*tan(1/2*d*x + 1/2*c)^1
0 + 40320*a*b^7*tan(1/2*d*x + 1/2*c)^10 + 10658*a^8*tan(1/2*d*x + 1/2*c)^9 + 39872*a^6*b^2*tan(1/2*d*x + 1/2*c
)^9 + 244160*a^4*b^4*tan(1/2*d*x + 1/2*c)^9 + 89600*a^2*b^6*tan(1/2*d*x + 1/2*c)^9 + 8960*b^8*tan(1/2*d*x + 1/
2*c)^9 + 35280*a^7*b*tan(1/2*d*x + 1/2*c)^8 + 105840*a^5*b^3*tan(1/2*d*x + 1/2*c)^8 + 197568*a^3*b^5*tan(1/2*d
*x + 1/2*c)^8 + 24192*a*b^7*tan(1/2*d*x + 1/2*c)^8 - 5112*a^8*tan(1/2*d*x + 1/2*c)^7 + 79632*a^6*b^2*tan(1/2*d
*x + 1/2*c)^7 + 120960*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 80640*a^2*b^6*tan(1/2*d*x + 1/2*c)^7 + 105840*a^5*b^3*
tan(1/2*d*x + 1/2*c)^6 + 56448*a^3*b^5*tan(1/2*d*x + 1/2*c)^6 + 10752*a*b^7*tan(1/2*d*x + 1/2*c)^6 + 4788*a^8*
tan(1/2*d*x + 1/2*c)^5 + 14112*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 70560*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 10080*a
^7*b*tan(1/2*d*x + 1/2*c)^4 + 15120*a^5*b^3*tan(1/2*d*x + 1/2*c)^4 + 16128*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 46
08*a*b^7*tan(1/2*d*x + 1/2*c)^4 - 840*a^8*tan(1/2*d*x + 1/2*c)^3 + 11760*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 5040
*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 4032*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 + 1152*a*b^7*tan(1/2*d*x + 1/2*c)^2 + 31
5*a^8*tan(1/2*d*x + 1/2*c) + 280*a^7*b - 560*a^5*b^3 + 448*a^3*b^5 - 128*a*b^7)/((tan(1/2*d*x + 1/2*c)^2 - 1)^
9*d)

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maple [B]  time = 0.40, size = 662, normalized size = 2.81 \[ \frac {-a^{8} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{9 \cos \left (d x +c \right )^{9}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin ^{6}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{105}\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin ^{8}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {b^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9 \cos \left (d x +c \right )^{9}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(-a^8*(-128/315-1/9*sec(d*x+c)^8-8/63*sec(d*x+c)^6-16/105*sec(d*x+c)^4-64/315*sec(d*x+c)^2)*tan(d*x+c)+8/9
*a^7*b/cos(d*x+c)^9+28*a^6*b^2*(1/9*sin(d*x+c)^3/cos(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105*sin(d*x+c)^
3/cos(d*x+c)^5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)+56*a^5*b^3*(1/9*sin(d*x+c)^4/cos(d*x+c)^9+5/63*sin(d*x+c)^4/c
os(d*x+c)^7+1/21*sin(d*x+c)^4/cos(d*x+c)^5+1/63*sin(d*x+c)^4/cos(d*x+c)^3-1/63*sin(d*x+c)^4/cos(d*x+c)-1/63*(2
+sin(d*x+c)^2)*cos(d*x+c))+70*a^4*b^4*(1/9*sin(d*x+c)^5/cos(d*x+c)^9+4/63*sin(d*x+c)^5/cos(d*x+c)^7+8/315*sin(
d*x+c)^5/cos(d*x+c)^5)+56*a^3*b^5*(1/9*sin(d*x+c)^6/cos(d*x+c)^9+1/21*sin(d*x+c)^6/cos(d*x+c)^7+1/105*sin(d*x+
c)^6/cos(d*x+c)^5-1/315*sin(d*x+c)^6/cos(d*x+c)^3+1/105*sin(d*x+c)^6/cos(d*x+c)+1/105*(8/3+sin(d*x+c)^4+4/3*si
n(d*x+c)^2)*cos(d*x+c))+28*a^2*b^6*(1/9*sin(d*x+c)^7/cos(d*x+c)^9+2/63*sin(d*x+c)^7/cos(d*x+c)^7)+8*a*b^7*(1/9
*sin(d*x+c)^8/cos(d*x+c)^9+1/63*sin(d*x+c)^8/cos(d*x+c)^7-1/315*sin(d*x+c)^8/cos(d*x+c)^5+1/315*sin(d*x+c)^8/c
os(d*x+c)^3-1/63*sin(d*x+c)^8/cos(d*x+c)-1/63*(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c)
)+1/9*b^8*sin(d*x+c)^9/cos(d*x+c)^9)

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maxima [A]  time = 0.33, size = 315, normalized size = 1.33 \[ \frac {35 \, b^{8} \tan \left (d x + c\right )^{9} + {\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 70 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + 140 \, {\left (7 \, \tan \left (d x + c\right )^{9} + 9 \, \tan \left (d x + c\right )^{7}\right )} a^{2} b^{6} - \frac {280 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac {56 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{9}} - \frac {8 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 189 \, \cos \left (d x + c\right )^{4} + 135 \, \cos \left (d x + c\right )^{2} - 35\right )} a b^{7}}{\cos \left (d x + c\right )^{9}} + \frac {280 \, a^{7} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/315*(35*b^8*tan(d*x + c)^9 + (35*tan(d*x + c)^9 + 180*tan(d*x + c)^7 + 378*tan(d*x + c)^5 + 420*tan(d*x + c)
^3 + 315*tan(d*x + c))*a^8 + 28*(35*tan(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c
)^3)*a^6*b^2 + 70*(35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5)*a^4*b^4 + 140*(7*tan(d*x + c)^9
+ 9*tan(d*x + c)^7)*a^2*b^6 - 280*(9*cos(d*x + c)^2 - 7)*a^5*b^3/cos(d*x + c)^9 + 56*(63*cos(d*x + c)^4 - 90*c
os(d*x + c)^2 + 35)*a^3*b^5/cos(d*x + c)^9 - 8*(105*cos(d*x + c)^6 - 189*cos(d*x + c)^4 + 135*cos(d*x + c)^2 -
35)*a*b^7/cos(d*x + c)^9 + 280*a^7*b/cos(d*x + c)^9)/d

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mupad [B]  time = 6.68, size = 659, normalized size = 2.79 \[ \frac {{\left (a-b\right )}^8}{2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^8}-\frac {{\left (a+b\right )}^8}{9\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^9}-\frac {{\left (a+b\right )}^8}{2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^8}-\frac {{\left (a-b\right )}^8}{9\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9}-\frac {{\left (a+b\right )}^7\,\left (37\,a+21\,b\right )}{28\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^7}-\frac {{\left (a+b\right )}^7\,\left (55\,a+7\,b\right )}{24\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^6}+\frac {{\left (a-b\right )}^5\,\left (187\,a^3+191\,a^2\,b+65\,a\,b^2+5\,b^3\right )}{128\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^2}+\frac {{\left (a-b\right )}^5\,\left (-463\,a^3-67\,a^2\,b+67\,a\,b^2+15\,b^3\right )}{192\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3}+\frac {{\left (a-b\right )}^6\,\left (95\,a^2+18\,a\,b-b^2\right )}{32\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^4}+\frac {{\left (a-b\right )}^6\,\left (-241\,a^2+114\,a\,b+15\,b^2\right )}{80\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5}-\frac {{\left (a-b\right )}^7\,\left (37\,a-21\,b\right )}{28\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7}+\frac {{\left (a-b\right )}^7\,\left (55\,a-7\,b\right )}{24\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^6}+\frac {{\left (a+b\right )}^6\,\left (-95\,a^2+18\,a\,b+b^2\right )}{32\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^4}-\frac {{\left (a+b\right )}^5\,\left (187\,a^3-191\,a^2\,b+65\,a\,b^2-5\,b^3\right )}{128\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^2}+\frac {{\left (a+b\right )}^5\,\left (-463\,a^3+67\,a^2\,b+67\,a\,b^2-15\,b^3\right )}{192\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3}-\frac {{\left (a+b\right )}^6\,\left (241\,a^2+114\,a\,b-15\,b^2\right )}{80\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^5}-\frac {a\,{\left (a+b\right )}^4\,\left (16\,a^3-29\,a^2\,b+20\,a\,b^2-5\,b^3\right )}{16\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}-\frac {a\,{\left (a-b\right )}^4\,\left (16\,a^3+29\,a^2\,b+20\,a\,b^2+5\,b^3\right )}{16\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x))^8/cos(c + d*x)^10,x)

[Out]

(a - b)^8/(2*d*(tan(c/2 + (d*x)/2) + 1)^8) - (a + b)^8/(9*d*(tan(c/2 + (d*x)/2) - 1)^9) - (a + b)^8/(2*d*(tan(
c/2 + (d*x)/2) - 1)^8) - (a - b)^8/(9*d*(tan(c/2 + (d*x)/2) + 1)^9) - ((a + b)^7*(37*a + 21*b))/(28*d*(tan(c/2
+ (d*x)/2) - 1)^7) - ((a + b)^7*(55*a + 7*b))/(24*d*(tan(c/2 + (d*x)/2) - 1)^6) + ((a - b)^5*(65*a*b^2 + 191*
a^2*b + 187*a^3 + 5*b^3))/(128*d*(tan(c/2 + (d*x)/2) + 1)^2) + ((a - b)^5*(67*a*b^2 - 67*a^2*b - 463*a^3 + 15*
b^3))/(192*d*(tan(c/2 + (d*x)/2) + 1)^3) + ((a - b)^6*(18*a*b + 95*a^2 - b^2))/(32*d*(tan(c/2 + (d*x)/2) + 1)^
4) + ((a - b)^6*(114*a*b - 241*a^2 + 15*b^2))/(80*d*(tan(c/2 + (d*x)/2) + 1)^5) - ((a - b)^7*(37*a - 21*b))/(2
8*d*(tan(c/2 + (d*x)/2) + 1)^7) + ((a - b)^7*(55*a - 7*b))/(24*d*(tan(c/2 + (d*x)/2) + 1)^6) + ((a + b)^6*(18*
a*b - 95*a^2 + b^2))/(32*d*(tan(c/2 + (d*x)/2) - 1)^4) - ((a + b)^5*(65*a*b^2 - 191*a^2*b + 187*a^3 - 5*b^3))/
(128*d*(tan(c/2 + (d*x)/2) - 1)^2) + ((a + b)^5*(67*a*b^2 + 67*a^2*b - 463*a^3 - 15*b^3))/(192*d*(tan(c/2 + (d
*x)/2) - 1)^3) - ((a + b)^6*(114*a*b + 241*a^2 - 15*b^2))/(80*d*(tan(c/2 + (d*x)/2) - 1)^5) - (a*(a + b)^4*(20
*a*b^2 - 29*a^2*b + 16*a^3 - 5*b^3))/(16*d*(tan(c/2 + (d*x)/2) - 1)) - (a*(a - b)^4*(20*a*b^2 + 29*a^2*b + 16*
a^3 + 5*b^3))/(16*d*(tan(c/2 + (d*x)/2) + 1))

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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(sec(d*x+c)**10*(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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