3.417 \(\int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=284 \[ \frac{7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac{3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac{7 b^4 \left (20 a^2 b^2+15 a^4+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac{a b^3 \left (112 a^2 b^2+35 a^4+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac{7 b^2 \left (30 a^4 b^2+20 a^2 b^4+3 a^6+b^6\right ) \sin (c+d x)}{2 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d} \]

[Out]

-((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]])/(4*d) + (7*b^
2*(3*a^6 + 30*a^4*b^2 + 20*a^2*b^4 + b^6)*Sin[c + d*x])/(2*d) + (a*b^3*(35*a^4 + 112*a^2*b^2 + 24*b^4)*Sin[c +
d*x]^2)/(2*d) + (7*b^4*(15*a^4 + 20*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(6*d) + (3*a*b^5*(7*a^2 + 4*b^2)*Sin[c + d
*x]^4)/(2*d) + (7*b^6*(5*a^2 + b^2)*Sin[c + d*x]^5)/(10*d) + (a*b^7*Sin[c + d*x]^6)/(2*d) + (Sec[c + d*x]^2*(b
+ a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(2*d)

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Rubi [A]  time = 0.242127, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2668, 739, 801, 633, 31} \[ \frac{7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac{3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac{7 b^4 \left (20 a^2 b^2+15 a^4+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac{a b^3 \left (112 a^2 b^2+35 a^4+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac{7 b^2 \left (30 a^4 b^2+20 a^2 b^4+3 a^6+b^6\right ) \sin (c+d x)}{2 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]

[Out]

-((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]])/(4*d) + (7*b^
2*(3*a^6 + 30*a^4*b^2 + 20*a^2*b^4 + b^6)*Sin[c + d*x])/(2*d) + (a*b^3*(35*a^4 + 112*a^2*b^2 + 24*b^4)*Sin[c +
d*x]^2)/(2*d) + (7*b^4*(15*a^4 + 20*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(6*d) + (3*a*b^5*(7*a^2 + 4*b^2)*Sin[c + d
*x]^4)/(2*d) + (7*b^6*(5*a^2 + b^2)*Sin[c + d*x]^5)/(10*d) + (a*b^7*Sin[c + d*x]^6)/(2*d) + (Sec[c + d*x]^2*(b
+ a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(2*d)

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 739

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+x)^8}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}-\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^6 \left (-a^2+7 b^2+6 a x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac{\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}-\frac{b \operatorname{Subst}\left (\int \left (-7 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right )-2 a \left (35 a^4+112 a^2 b^2+24 b^4\right ) x-7 \left (15 a^4+20 a^2 b^2+b^4\right ) x^2-12 a \left (7 a^2+4 b^2\right ) x^3-7 \left (5 a^2+b^2\right ) x^4-6 a x^5-\frac{a^8-28 a^6 b^2-210 a^4 b^4-140 a^2 b^6-7 b^8-16 a b^2 \left (7 a^4+14 a^2 b^2+3 b^4\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac{7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac{a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac{7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac{3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac{7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{a^8-28 a^6 b^2-210 a^4 b^4-140 a^2 b^6-7 b^8-16 a b^2 \left (7 a^4+14 a^2 b^2+3 b^4\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac{7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac{a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac{7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac{3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac{7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}+\frac{\left ((a-7 b) (a+b)^7\right ) \operatorname{Subst}\left (\int \frac{1}{b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}-\frac{\left ((a-b)^7 (a+7 b)\right ) \operatorname{Subst}\left (\int \frac{1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{(a-b)^7 (a+7 b) \log (1+\sin (c+d x))}{4 d}+\frac{7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac{a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac{7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac{3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac{7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}\\ \end{align*}

Mathematica [A]  time = 2.35673, size = 366, normalized size = 1.29 \[ \frac{b^9 \left (b^2-9 a^2\right ) \sin ^7(c+d x)-4 a b^8 \left (9 a^2-2 b^2\right ) \sin ^6(c+d x)+\frac{7}{5} b^7 \left (19 a^2 b^2-60 a^4+b^4\right ) \sin ^5(c+d x)-2 a b^6 \left (-22 a^2 b^2+63 a^4-6 b^4\right ) \sin ^4(c+d x)+\frac{7}{3} b^5 \left (10 a^4 b^2+19 a^2 b^4-54 a^6+b^6\right ) \sin ^3(c+d x)-4 a b^4 \left (14 a^4 b^2-22 a^2 b^4+21 a^6-6 b^6\right ) \sin ^2(c+d x)+b^3 \left (-182 a^6 b^2+70 a^4 b^4+133 a^2 b^6-36 a^8+7 b^8\right ) \sin (c+d x)+\frac{1}{2} b \left (a^2-b^2\right ) \left ((a-7 b) (a+b)^7 \log (1-\sin (c+d x))-(a-b)^7 (a+7 b) \log (\sin (c+d x)+1)\right )-a b^{10} \sin ^8(c+d x)+b \sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9}{2 b d \left (b^2-a^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]

[Out]

((b*(a^2 - b^2)*((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]] - (a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]]))/2 + b
^3*(-36*a^8 - 182*a^6*b^2 + 70*a^4*b^4 + 133*a^2*b^6 + 7*b^8)*Sin[c + d*x] - 4*a*b^4*(21*a^6 + 14*a^4*b^2 - 22
*a^2*b^4 - 6*b^6)*Sin[c + d*x]^2 + (7*b^5*(-54*a^6 + 10*a^4*b^2 + 19*a^2*b^4 + b^6)*Sin[c + d*x]^3)/3 - 2*a*b^
6*(63*a^4 - 22*a^2*b^2 - 6*b^4)*Sin[c + d*x]^4 + (7*b^7*(-60*a^4 + 19*a^2*b^2 + b^4)*Sin[c + d*x]^5)/5 - 4*a*b
^8*(9*a^2 - 2*b^2)*Sin[c + d*x]^6 + b^9*(-9*a^2 + b^2)*Sin[c + d*x]^7 - a*b^10*Sin[c + d*x]^8 + b*Sec[c + d*x]
^2*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^9)/(2*b*(-a^2 + b^2)*d)

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Maple [B]  time = 0.127, size = 645, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x)

[Out]

4/d*a^7*b/cos(d*x+c)^2+28/d*a^5*b^3*tan(d*x+c)^2+1/2/d*b^8*sin(d*x+c)^9/cos(d*x+c)^2+1/2/d*a^8*ln(sec(d*x+c)+t
an(d*x+c))+7/2/d*b^8*sin(d*x+c)+7/6/d*b^8*sin(d*x+c)^3-7/2/d*b^8*ln(sec(d*x+c)+tan(d*x+c))+7/10/d*b^8*sin(d*x+
c)^5+1/2*b^8*sin(d*x+c)^7/d+105/d*a^4*b^4*sin(d*x+c)-105/d*a^4*b^4*ln(sec(d*x+c)+tan(d*x+c))+28/d*a^3*b^5*sin(
d*x+c)^4+56/d*a^3*b^5*sin(d*x+c)^2+112/d*a^3*b^5*ln(cos(d*x+c))+14/d*a^2*b^6*sin(d*x+c)^5+70/3/d*a^2*b^6*sin(d
*x+c)^3+70/d*a^2*b^6*sin(d*x+c)-70/d*a^2*b^6*ln(sec(d*x+c)+tan(d*x+c))+6/d*a*b^7*sin(d*x+c)^4+12/d*a*b^7*sin(d
*x+c)^2+24/d*a*b^7*ln(cos(d*x+c))+14/d*a^6*b^2*sin(d*x+c)-14/d*a^6*b^2*ln(sec(d*x+c)+tan(d*x+c))+56/d*a^5*b^3*
ln(cos(d*x+c))+35/d*a^4*b^4*sin(d*x+c)^3+14/d*a^2*b^6*sin(d*x+c)^7/cos(d*x+c)^2+4/d*a*b^7*sin(d*x+c)^8/cos(d*x
+c)^2+14/d*a^6*b^2*sin(d*x+c)^3/cos(d*x+c)^2+35/d*a^4*b^4*sin(d*x+c)^5/cos(d*x+c)^2+28/d*a^3*b^5*sin(d*x+c)^6/
cos(d*x+c)^2+1/2/d*a^8*sec(d*x+c)*tan(d*x+c)+4*a*b^7*sin(d*x+c)^6/d

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Maxima [A]  time = 0.97204, size = 436, normalized size = 1.54 \begin{align*} \frac{12 \, b^{8} \sin \left (d x + c\right )^{5} + 120 \, a b^{7} \sin \left (d x + c\right )^{4} + 40 \,{\left (14 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 240 \,{\left (7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 60 \,{\left (70 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac{30 \,{\left (8 \, a^{7} b + 56 \, a^{5} b^{3} + 56 \, a^{3} b^{5} + 8 \, a b^{7} +{\left (a^{8} + 28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/60*(12*b^8*sin(d*x + c)^5 + 120*a*b^7*sin(d*x + c)^4 + 40*(14*a^2*b^6 + b^8)*sin(d*x + c)^3 + 240*(7*a^3*b^5
+ 2*a*b^7)*sin(d*x + c)^2 + 15*(a^8 - 28*a^6*b^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2*b^6 + 48
*a*b^7 - 7*b^8)*log(sin(d*x + c) + 1) - 15*(a^8 - 28*a^6*b^2 - 112*a^5*b^3 - 210*a^4*b^4 - 224*a^3*b^5 - 140*a
^2*b^6 - 48*a*b^7 - 7*b^8)*log(sin(d*x + c) - 1) + 60*(70*a^4*b^4 + 56*a^2*b^6 + 3*b^8)*sin(d*x + c) - 30*(8*a
^7*b + 56*a^5*b^3 + 56*a^3*b^5 + 8*a*b^7 + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*sin(d*x + c))/(s
in(d*x + c)^2 - 1))/d

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Fricas [A]  time = 3.36943, size = 894, normalized size = 3.15 \begin{align*} \frac{120 \, a b^{7} \cos \left (d x + c\right )^{6} + 240 \, a^{7} b + 1680 \, a^{5} b^{3} + 1680 \, a^{3} b^{5} + 240 \, a b^{7} - 240 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 105 \,{\left (8 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} - 8 \,{\left (35 \, a^{2} b^{6} + 4 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (525 \, a^{4} b^{4} + 490 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/60*(120*a*b^7*cos(d*x + c)^6 + 240*a^7*b + 1680*a^5*b^3 + 1680*a^3*b^5 + 240*a*b^7 - 240*(7*a^3*b^5 + 3*a*b^
7)*cos(d*x + c)^4 + 15*(a^8 - 28*a^6*b^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2*b^6 + 48*a*b^7 -
7*b^8)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 15*(a^8 - 28*a^6*b^2 - 112*a^5*b^3 - 210*a^4*b^4 - 224*a^3*b^5 -
140*a^2*b^6 - 48*a*b^7 - 7*b^8)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 105*(8*a^3*b^5 + 3*a*b^7)*cos(d*x + c
)^2 + 2*(6*b^8*cos(d*x + c)^6 + 15*a^8 + 420*a^6*b^2 + 1050*a^4*b^4 + 420*a^2*b^6 + 15*b^8 - 8*(35*a^2*b^6 + 4
*b^8)*cos(d*x + c)^4 + 4*(525*a^4*b^4 + 490*a^2*b^6 + 29*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3*(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.19351, size = 551, normalized size = 1.94 \begin{align*} \frac{12 \, b^{8} \sin \left (d x + c\right )^{5} + 120 \, a b^{7} \sin \left (d x + c\right )^{4} + 560 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 40 \, b^{8} \sin \left (d x + c\right )^{3} + 1680 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 480 \, a b^{7} \sin \left (d x + c\right )^{2} + 4200 \, a^{4} b^{4} \sin \left (d x + c\right ) + 3360 \, a^{2} b^{6} \sin \left (d x + c\right ) + 180 \, b^{8} \sin \left (d x + c\right ) + 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 15 \,{\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{30 \,{\left (56 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} + 112 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 24 \, a b^{7} \sin \left (d x + c\right )^{2} + a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) + 70 \, a^{4} b^{4} \sin \left (d x + c\right ) + 28 \, a^{2} b^{6} \sin \left (d x + c\right ) + b^{8} \sin \left (d x + c\right ) + 8 \, a^{7} b - 56 \, a^{3} b^{5} - 16 \, a b^{7}\right )}}{\sin \left (d x + c\right )^{2} - 1}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/60*(12*b^8*sin(d*x + c)^5 + 120*a*b^7*sin(d*x + c)^4 + 560*a^2*b^6*sin(d*x + c)^3 + 40*b^8*sin(d*x + c)^3 +
1680*a^3*b^5*sin(d*x + c)^2 + 480*a*b^7*sin(d*x + c)^2 + 4200*a^4*b^4*sin(d*x + c) + 3360*a^2*b^6*sin(d*x + c)
+ 180*b^8*sin(d*x + c) + 15*(a^8 - 28*a^6*b^2 + 112*a^5*b^3 - 210*a^4*b^4 + 224*a^3*b^5 - 140*a^2*b^6 + 48*a*
b^7 - 7*b^8)*log(abs(sin(d*x + c) + 1)) - 15*(a^8 - 28*a^6*b^2 - 112*a^5*b^3 - 210*a^4*b^4 - 224*a^3*b^5 - 140
*a^2*b^6 - 48*a*b^7 - 7*b^8)*log(abs(sin(d*x + c) - 1)) - 30*(56*a^5*b^3*sin(d*x + c)^2 + 112*a^3*b^5*sin(d*x
+ c)^2 + 24*a*b^7*sin(d*x + c)^2 + a^8*sin(d*x + c) + 28*a^6*b^2*sin(d*x + c) + 70*a^4*b^4*sin(d*x + c) + 28*a
^2*b^6*sin(d*x + c) + b^8*sin(d*x + c) + 8*a^7*b - 56*a^3*b^5 - 16*a*b^7)/(sin(d*x + c)^2 - 1))/d