3.418 \(\int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=320 \[ -\frac{a b^7 \left (13-\frac{3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac{5 b^4 \left (-42 a^2 b^2+9 a^4-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac{a b^3 \left (-77 a^2 b^2+15 a^4-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac{5 b^2 \left (-35 a^4 b^2-84 a^2 b^4+6 a^6-7 b^6\right ) \sin (c+d x)}{8 d}-\frac{(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]

[Out]

-((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + ((a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*Log
[1 + Sin[c + d*x]])/(16*d) + (5*b^2*(6*a^6 - 35*a^4*b^2 - 84*a^2*b^4 - 7*b^6)*Sin[c + d*x])/(8*d) + (a*b^3*(15
*a^4 - 77*a^2*b^2 - 48*b^4)*Sin[c + d*x]^2)/(4*d) + (5*b^4*(9*a^4 - 42*a^2*b^2 - 7*b^4)*Sin[c + d*x]^3)/(24*d)
- (a*(13 - (3*a^2)/b^2)*b^7*Sin[c + d*x]^4)/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])
^7)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^5*(b*(a^2 + 7*b^2) - a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*d)

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Rubi [A]  time = 0.303829, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2668, 739, 819, 801, 633, 31} \[ -\frac{a b^7 \left (13-\frac{3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac{5 b^4 \left (-42 a^2 b^2+9 a^4-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac{a b^3 \left (-77 a^2 b^2+15 a^4-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac{5 b^2 \left (-35 a^4 b^2-84 a^2 b^4+6 a^6-7 b^6\right ) \sin (c+d x)}{8 d}-\frac{(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]

[Out]

-((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + ((a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*Log
[1 + Sin[c + d*x]])/(16*d) + (5*b^2*(6*a^6 - 35*a^4*b^2 - 84*a^2*b^4 - 7*b^6)*Sin[c + d*x])/(8*d) + (a*b^3*(15
*a^4 - 77*a^2*b^2 - 48*b^4)*Sin[c + d*x]^2)/(4*d) + (5*b^4*(9*a^4 - 42*a^2*b^2 - 7*b^4)*Sin[c + d*x]^3)/(24*d)
- (a*(13 - (3*a^2)/b^2)*b^7*Sin[c + d*x]^4)/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])
^7)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^5*(b*(a^2 + 7*b^2) - a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*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 819

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

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 ^5(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{(a+x)^8}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+x)^6 \left (-3 a^2+7 b^2+4 a x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^4 \left (3 a^4+2 a^2 b^2+35 b^4-4 a \left (3 a^2-13 b^2\right ) x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{b \operatorname{Subst}\left (\int \left (5 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right )+4 a \left (15 a^4-77 a^2 b^2-48 b^4\right ) x+5 \left (9 a^4-42 a^2 b^2-7 b^4\right ) x^2+4 a \left (3 a^2-13 b^2\right ) x^3+\frac{3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac{a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac{5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac{a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^8-28 a^6 b^2+210 a^4 b^4+420 a^2 b^6+35 b^8+64 a b^4 \left (7 a^2+3 b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac{a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac{5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac{a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}+\frac{\left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}-\frac{\left ((a-b)^6 \left (3 a^2+18 a b+35 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=-\frac{(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac{5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac{a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac{5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac{a b^5 \left (3 a^2-13 b^2\right ) \sin ^4(c+d x)}{8 d}+\frac{\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}\\ \end{align*}

Mathematica [A]  time = 4.07519, size = 514, normalized size = 1.61 \[ -\frac{-6 a b^9 \left (3 a^2+11 b^2\right ) \sin ^8(c+d x)+6 b^8 \left (-90 a^2 b^2-27 a^4+5 b^4\right ) \sin ^7(c+d x)-24 a b^7 \left (79 a^2 b^2+27 a^4-8 b^4\right ) \sin ^6(c+d x)+42 b^6 \left (-87 a^4 b^2+10 a^2 b^4-36 a^6+b^6\right ) \sin ^5(c+d x)-12 a b^5 \left (333 a^4 b^2-8 a^2 b^4+189 a^6-24 b^6\right ) \sin ^4(c+d x)+14 b^4 \left (-144 a^6 b^2-85 a^4 b^4+50 a^2 b^6-162 a^8+5 b^8\right ) \sin ^3(c+d x)-24 a b^3 \left (-21 a^6 b^2+88 a^4 b^4-8 a^2 b^6+63 a^8-24 b^8\right ) \sin ^2(c+d x)+6 b^2 \left (234 a^8 b^2-28 a^6 b^4-595 a^4 b^6+350 a^2 b^8-108 a^{10}+35 b^{10}\right ) \sin (c+d x)+3 \left (a^2-b^2\right )^2 \left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))-(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)\right )+12 \left (a^2-b^2\right ) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9+6 \sec ^2(c+d x) (a+b \sin (c+d x))^9 \left (-a \left (3 a^2+11 b^2\right ) \sin (c+d x)+9 a^2 b+5 b^3\right )}{48 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]

[Out]

-(3*(a^2 - b^2)^2*((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]] - (a - b)^6*(3*a^2 + 18*a*b + 35*
b^2)*Log[1 + Sin[c + d*x]]) + 6*b^2*(-108*a^10 + 234*a^8*b^2 - 28*a^6*b^4 - 595*a^4*b^6 + 350*a^2*b^8 + 35*b^1
0)*Sin[c + d*x] - 24*a*b^3*(63*a^8 - 21*a^6*b^2 + 88*a^4*b^4 - 8*a^2*b^6 - 24*b^8)*Sin[c + d*x]^2 + 14*b^4*(-1
62*a^8 - 144*a^6*b^2 - 85*a^4*b^4 + 50*a^2*b^6 + 5*b^8)*Sin[c + d*x]^3 - 12*a*b^5*(189*a^6 + 333*a^4*b^2 - 8*a
^2*b^4 - 24*b^6)*Sin[c + d*x]^4 + 42*b^6*(-36*a^6 - 87*a^4*b^2 + 10*a^2*b^4 + b^6)*Sin[c + d*x]^5 - 24*a*b^7*(
27*a^4 + 79*a^2*b^2 - 8*b^4)*Sin[c + d*x]^6 + 6*b^8*(-27*a^4 - 90*a^2*b^2 + 5*b^4)*Sin[c + d*x]^7 - 6*a*b^9*(3
*a^2 + 11*b^2)*Sin[c + d*x]^8 + 12*(a^2 - b^2)*Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^9 + 6*
Sec[c + d*x]^2*(a + b*Sin[c + d*x])^9*(9*a^2*b + 5*b^3 - a*(3*a^2 + 11*b^2)*Sin[c + d*x]))/(48*(a^2 - b^2)^2*d
)

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Maple [B]  time = 0.139, size = 760, normalized size = 2.4 \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)^5*(a+b*sin(d*x+c))^8,x)

[Out]

-5/8/d*b^8*sin(d*x+c)^9/cos(d*x+c)^2+1/4/d*a^8*tan(d*x+c)*sec(d*x+c)^3+3/8/d*a^8*ln(sec(d*x+c)+tan(d*x+c))-35/
8/d*b^8*sin(d*x+c)-35/24/d*b^8*sin(d*x+c)^3+35/8/d*b^8*ln(sec(d*x+c)+tan(d*x+c))-7/8/d*b^8*sin(d*x+c)^5-5/8*b^
8*sin(d*x+c)^7/d+7/d*a^6*b^2*sin(d*x+c)^3/cos(d*x+c)^4-105/4/d*a^4*b^4*sin(d*x+c)+105/4/d*a^4*b^4*ln(sec(d*x+c
)+tan(d*x+c))-56/d*a^3*b^5*ln(cos(d*x+c))-21/2/d*a^2*b^6*sin(d*x+c)^5-35/2/d*a^2*b^6*sin(d*x+c)^3-105/2/d*a^2*
b^6*sin(d*x+c)+105/2/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))+7/2/d*a^6*b^2*sin(d*x+c)-7/2/d*a^6*b^2*ln(sec(d*x+c)+tan(d*x+c))-35/4/d*a^4*b^4*sin(d*x+c)
^3-21/2/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+7/2/d*a^6*b^2*sin(d*x+c)^3/cos
(d*x+c)^2-35/4/d*a^4*b^4*sin(d*x+c)^5/cos(d*x+c)^2+3/8/d*a^8*sec(d*x+c)*tan(d*x+c)+1/4/d*b^8*sin(d*x+c)^9/cos(
d*x+c)^4+2/d*a^7*b/cos(d*x+c)^4+14/d*a^3*b^5*tan(d*x+c)^4-28/d*a^3*b^5*tan(d*x+c)^2-4*a*b^7*sin(d*x+c)^6/d+35/
2/d*a^4*b^4*sin(d*x+c)^5/cos(d*x+c)^4+7/d*a^2*b^6*sin(d*x+c)^7/cos(d*x+c)^4+2/d*a*b^7*sin(d*x+c)^8/cos(d*x+c)^
4+14/d*a^5*b^3*sin(d*x+c)^4/cos(d*x+c)^4

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Maxima [A]  time = 0.980675, size = 470, normalized size = 1.47 \begin{align*} -\frac{16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} - 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 48 \,{\left (28 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac{6 \,{\left (16 \, a^{7} b - 112 \, a^{5} b^{3} - 336 \, a^{3} b^{5} - 80 \, a b^{7} -{\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \sin \left (d x + c\right )^{3} + 32 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} +{\left (5 \, a^{8} + 28 \, a^{6} b^{2} - 210 \, a^{4} b^{4} - 196 \, a^{2} b^{6} - 11 \, b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/48*(16*b^8*sin(d*x + c)^3 + 192*a*b^7*sin(d*x + c)^2 - 3*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 - 448*a^3*b^5 +
420*a^2*b^6 - 192*a*b^7 + 35*b^8)*log(sin(d*x + c) + 1) + 3*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 + 448*a^3*b^5 +
420*a^2*b^6 + 192*a*b^7 + 35*b^8)*log(sin(d*x + c) - 1) + 48*(28*a^2*b^6 + 3*b^8)*sin(d*x + c) - 6*(16*a^7*b -
112*a^5*b^3 - 336*a^3*b^5 - 80*a*b^7 - (3*a^8 - 28*a^6*b^2 - 350*a^4*b^4 - 252*a^2*b^6 - 13*b^8)*sin(d*x + c)
^3 + 32*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*sin(d*x + c)^2 + (5*a^8 + 28*a^6*b^2 - 210*a^4*b^4 - 196*a^2*b^6 -
11*b^8)*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1))/d

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Fricas [A]  time = 3.37023, size = 879, normalized size = 2.75 \begin{align*} \frac{192 \, a b^{7} \cos \left (d x + c\right )^{6} - 96 \, a b^{7} \cos \left (d x + c\right )^{4} + 96 \, a^{7} b + 672 \, a^{5} b^{3} + 672 \, a^{3} b^{5} + 96 \, a b^{7} + 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 192 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 6 \, a^{8} + 168 \, a^{6} b^{2} + 420 \, a^{4} b^{4} + 168 \, a^{2} b^{6} + 6 \, b^{8} - 16 \,{\left (42 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/48*(192*a*b^7*cos(d*x + c)^6 - 96*a*b^7*cos(d*x + c)^4 + 96*a^7*b + 672*a^5*b^3 + 672*a^3*b^5 + 96*a*b^7 + 3
*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 - 448*a^3*b^5 + 420*a^2*b^6 - 192*a*b^7 + 35*b^8)*cos(d*x + c)^4*log(sin(d*
x + c) + 1) - 3*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 + 448*a^3*b^5 + 420*a^2*b^6 + 192*a*b^7 + 35*b^8)*cos(d*x +
c)^4*log(-sin(d*x + c) + 1) - 192*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + 2*(8*b^8*cos(d*x + c)^6
+ 6*a^8 + 168*a^6*b^2 + 420*a^4*b^4 + 168*a^2*b^6 + 6*b^8 - 16*(42*a^2*b^6 + 5*b^8)*cos(d*x + c)^4 + 3*(3*a^8
- 28*a^6*b^2 - 350*a^4*b^4 - 252*a^2*b^6 - 13*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)

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

[Out]

Timed out

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Giac [A]  time = 1.21373, size = 579, normalized size = 1.81 \begin{align*} -\frac{16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 1344 \, a^{2} b^{6} \sin \left (d x + c\right ) + 144 \, b^{8} \sin \left (d x + c\right ) - 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \,{\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{6 \,{\left (336 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 144 \, a b^{7} \sin \left (d x + c\right )^{4} - 3 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} + 350 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 252 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 13 \, b^{8} \sin \left (d x + c\right )^{3} + 224 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} - 224 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} - 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) - 210 \, a^{4} b^{4} \sin \left (d x + c\right ) - 196 \, a^{2} b^{6} \sin \left (d x + c\right ) - 11 \, b^{8} \sin \left (d x + c\right ) + 16 \, a^{7} b - 112 \, a^{5} b^{3} + 64 \, a b^{7}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/48*(16*b^8*sin(d*x + c)^3 + 192*a*b^7*sin(d*x + c)^2 + 1344*a^2*b^6*sin(d*x + c) + 144*b^8*sin(d*x + c) - 3
*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 - 448*a^3*b^5 + 420*a^2*b^6 - 192*a*b^7 + 35*b^8)*log(abs(sin(d*x + c) + 1)
) + 3*(3*a^8 - 28*a^6*b^2 + 210*a^4*b^4 + 448*a^3*b^5 + 420*a^2*b^6 + 192*a*b^7 + 35*b^8)*log(abs(sin(d*x + c)
- 1)) - 6*(336*a^3*b^5*sin(d*x + c)^4 + 144*a*b^7*sin(d*x + c)^4 - 3*a^8*sin(d*x + c)^3 + 28*a^6*b^2*sin(d*x
+ c)^3 + 350*a^4*b^4*sin(d*x + c)^3 + 252*a^2*b^6*sin(d*x + c)^3 + 13*b^8*sin(d*x + c)^3 + 224*a^5*b^3*sin(d*x
+ c)^2 - 224*a^3*b^5*sin(d*x + c)^2 - 192*a*b^7*sin(d*x + c)^2 + 5*a^8*sin(d*x + c) + 28*a^6*b^2*sin(d*x + c)
- 210*a^4*b^4*sin(d*x + c) - 196*a^2*b^6*sin(d*x + c) - 11*b^8*sin(d*x + c) + 16*a^7*b - 112*a^5*b^3 + 64*a*b
^7)/(sin(d*x + c)^2 - 1)^2)/d