\(\int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx\) [418]

Optimal result

Integrand size = 21, antiderivative size = 320 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}-\frac {a \left (13-\frac {3 a^2}{b^2}\right ) b^7 \sin ^4(c+d x)}{8 d}+\frac {\sec ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d} \]

[Out]

-1/16*(a+b)^6*(3*a^2-18*a*b+35*b^2)*ln(1-sin(d*x+c))/d+1/16*(a-b)^6*(3*a^2+18*a*b+35*b^2)*ln(1+sin(d*x+c))/d+5
/8*b^2*(6*a^6-35*a^4*b^2-84*a^2*b^4-7*b^6)*sin(d*x+c)/d+1/4*a*b^3*(15*a^4-77*a^2*b^2-48*b^4)*sin(d*x+c)^2/d+5/
24*b^4*(9*a^4-42*a^2*b^2-7*b^4)*sin(d*x+c)^3/d-1/8*a*(13-3*a^2/b^2)*b^7*sin(d*x+c)^4/d+1/4*sec(d*x+c)^4*(b+a*s
in(d*x+c))*(a+b*sin(d*x+c))^7/d-1/8*sec(d*x+c)^2*(a+b*sin(d*x+c))^5*(b*(a^2+7*b^2)-a*(3*a^2-11*b^2)*sin(d*x+c)
)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2747, 753, 833, 815, 647, 31} \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {(a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (b \left (a^2+7 b^2\right )-a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 d}-\frac {a b^7 \left (13-\frac {3 a^2}{b^2}\right ) \sin ^4(c+d x)}{8 d}+\frac {5 b^4 \left (9 a^4-42 a^2 b^2-7 b^4\right ) \sin ^3(c+d x)}{24 d}+\frac {a b^3 \left (15 a^4-77 a^2 b^2-48 b^4\right ) \sin ^2(c+d x)}{4 d}+\frac {5 b^2 \left (6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right ) \sin (c+d x)}{8 d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d} \]

[In]

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

[Out]

-1/16*((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/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 31

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

Rule 647

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 753

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 815

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 833

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 2747

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]

Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {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 \text {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 \text {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 \text {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 \text {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 ) \text {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 ) \text {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] (verified)

Time = 3.31 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.61 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {3 \left (a^2-b^2\right )^2 \left ((a+b)^6 \left (3 a^2-18 a b+35 b^2\right ) \log (1-\sin (c+d x))-(a-b)^6 \left (3 a^2+18 a b+35 b^2\right ) \log (1+\sin (c+d x))\right )+6 b^2 \left (-108 a^{10}+234 a^8 b^2-28 a^6 b^4-595 a^4 b^6+350 a^2 b^8+35 b^{10}\right ) \sin (c+d x)-24 a b^3 \left (63 a^8-21 a^6 b^2+88 a^4 b^4-8 a^2 b^6-24 b^8\right ) \sin ^2(c+d x)+14 b^4 \left (-162 a^8-144 a^6 b^2-85 a^4 b^4+50 a^2 b^6+5 b^8\right ) \sin ^3(c+d x)-12 a b^5 \left (189 a^6+333 a^4 b^2-8 a^2 b^4-24 b^6\right ) \sin ^4(c+d x)+42 b^6 \left (-36 a^6-87 a^4 b^2+10 a^2 b^4+b^6\right ) \sin ^5(c+d x)-24 a b^7 \left (27 a^4+79 a^2 b^2-8 b^4\right ) \sin ^6(c+d x)+6 b^8 \left (-27 a^4-90 a^2 b^2+5 b^4\right ) \sin ^7(c+d x)-6 a b^9 \left (3 a^2+11 b^2\right ) \sin ^8(c+d x)+12 \left (a^2-b^2\right ) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9+6 \sec ^2(c+d x) (a+b \sin (c+d x))^9 \left (9 a^2 b+5 b^3-a \left (3 a^2+11 b^2\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^2 d} \]

[In]

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

[Out]

-1/48*(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 + 3
5*b^10)*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*(-162*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]))/((a^2 - b^2)^2
*d)

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.36

method	result	size
parallelrisch	\(\frac {5376 \left (a^{2}+\frac {3 b^{2}}{7}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \,b^{5} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (a^{2}-6 a b +\frac {35}{3} b^{2}\right ) \left (a +b \right )^{6} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{6} \left (a^{2}+6 a b +\frac {35}{3} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-192 \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+\frac {9}{8} b^{6}\right ) a b \cos \left (2 d x +2 c \right )+\left (-48 a^{7} b +336 a^{5} b^{3}+1008 a^{3} b^{5}+288 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (18 a^{8}-168 a^{6} b^{2}-2100 a^{4} b^{4}-2520 a^{2} b^{6}-189 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-336 a^{2} b^{6}-35 b^{8}\right ) \sin \left (5 d x +5 c \right )+24 \cos \left (6 d x +6 c \right ) a \,b^{7}+\sin \left (7 d x +7 c \right ) b^{8}+\left (66 a^{8}+1176 a^{6} b^{2}+1260 a^{4} b^{4}-840 a^{2} b^{6}-105 b^{8}\right ) \sin \left (d x +c \right )+240 a^{7} b +1008 a^{5} b^{3}+336 a^{3} b^{5}-96 a \,b^{7}}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(434\)
derivativedivides	\(\frac {a^{8} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 a^{7} b}{\cos \left (d x +c \right )^{4}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {14 a^{5} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 a^{3} b^{5} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(544\)
default	\(\frac {a^{8} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 a^{7} b}{\cos \left (d x +c \right )^{4}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {14 a^{5} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 a^{3} b^{5} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(544\)
risch	\(\text {Expression too large to display}\)	\(1022\)

[In]

int(sec(d*x+c)^5*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/24*(5376*(a^2+3/7*b^2)*(3/4+1/4*cos(4*d*x+4*c)+cos(2*d*x+2*c))*a*b^5*ln(sec(1/2*d*x+1/2*c)^2)-36*(a^2-6*a*b+
35/3*b^2)*(a+b)^6*(3/4+1/4*cos(4*d*x+4*c)+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)+36*(3/4+1/4*cos(4*d*x+4*c)+
cos(2*d*x+2*c))*(a-b)^6*(a^2+6*a*b+35/3*b^2)*ln(tan(1/2*d*x+1/2*c)+1)-192*(a^6+7*a^4*b^2+7*a^2*b^4+9/8*b^6)*a*
b*cos(2*d*x+2*c)+(-48*a^7*b+336*a^5*b^3+1008*a^3*b^5+288*a*b^7)*cos(4*d*x+4*c)+(18*a^8-168*a^6*b^2-2100*a^4*b^
4-2520*a^2*b^6-189*b^8)*sin(3*d*x+3*c)+(-336*a^2*b^6-35*b^8)*sin(5*d*x+5*c)+24*cos(6*d*x+6*c)*a*b^7+sin(7*d*x+
7*c)*b^8+(66*a^8+1176*a^6*b^2+1260*a^4*b^4-840*a^2*b^6-105*b^8)*sin(d*x+c)+240*a^7*b+1008*a^5*b^3+336*a^3*b^5-
96*a*b^7)/d/(cos(4*d*x+4*c)+4*cos(2*d*x+2*c)+3)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.14 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {192 \, a b^{7} \cos \left (d x + c\right )^{6} - 96 \, a b^{7} \cos \left (d x + c\right )^{4} + 96 \, a^{7} b + 672 \, a^{5} b^{3} + 672 \, a^{3} b^{5} + 96 \, a b^{7} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 6 \, a^{8} + 168 \, a^{6} b^{2} + 420 \, a^{4} b^{4} + 168 \, a^{2} b^{6} + 6 \, b^{8} - 16 \, {\left (42 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.09 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 48 \, {\left (28 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac {6 \, {\left (16 \, a^{7} b - 112 \, a^{5} b^{3} - 336 \, a^{3} b^{5} - 80 \, a b^{7} - {\left (3 \, a^{8} - 28 \, a^{6} b^{2} - 350 \, a^{4} b^{4} - 252 \, a^{2} b^{6} - 13 \, b^{8}\right )} \sin \left (d x + c\right )^{3} + 32 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + {\left (5 \, a^{8} + 28 \, a^{6} b^{2} - 210 \, a^{4} b^{4} - 196 \, a^{2} b^{6} - 11 \, b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.34 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {16 \, b^{8} \sin \left (d x + c\right )^{3} + 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 1344 \, a^{2} b^{6} \sin \left (d x + c\right ) + 144 \, b^{8} \sin \left (d x + c\right ) - 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} - 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} - 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (3 \, a^{8} - 28 \, a^{6} b^{2} + 210 \, a^{4} b^{4} + 448 \, a^{3} b^{5} + 420 \, a^{2} b^{6} + 192 \, a b^{7} + 35 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (336 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 144 \, a b^{7} \sin \left (d x + c\right )^{4} - 3 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} + 350 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 252 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 13 \, b^{8} \sin \left (d x + c\right )^{3} + 224 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} - 224 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} - 192 \, a b^{7} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) - 210 \, a^{4} b^{4} \sin \left (d x + c\right ) - 196 \, a^{2} b^{6} \sin \left (d x + c\right ) - 11 \, b^{8} \sin \left (d x + c\right ) + 16 \, a^{7} b - 112 \, a^{5} b^{3} + 64 \, a b^{7}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.95 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^6\,\left (3\,a^2+18\,a\,b+35\,b^2\right )}{16\,d}-\frac {b^8\,{\sin \left (c+d\,x\right )}^3}{3\,d}-\frac {\sin \left (c+d\,x\right )\,\left (28\,a^2\,b^6+3\,b^8\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (-\frac {5\,a^8}{8}-\frac {7\,a^6\,b^2}{2}+\frac {105\,a^4\,b^4}{4}+\frac {49\,a^2\,b^6}{2}+\frac {11\,b^8}{8}\right )-{\sin \left (c+d\,x\right )}^3\,\left (-\frac {3\,a^8}{8}+\frac {7\,a^6\,b^2}{2}+\frac {175\,a^4\,b^4}{4}+\frac {63\,a^2\,b^6}{2}+\frac {13\,b^8}{8}\right )+10\,a\,b^7-2\,a^7\,b-{\sin \left (c+d\,x\right )}^2\,\left (28\,a^5\,b^3+56\,a^3\,b^5+12\,a\,b^7\right )+42\,a^3\,b^5+14\,a^5\,b^3}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}-\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^2}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^6\,\left (3\,a^2-18\,a\,b+35\,b^2\right )}{16\,d} \]

[In]

int((a + b*sin(c + d*x))^8/cos(c + d*x)^5,x)

[Out]

(log(sin(c + d*x) + 1)*(a - b)^6*(18*a*b + 3*a^2 + 35*b^2))/(16*d) - (b^8*sin(c + d*x)^3)/(3*d) - (sin(c + d*x
)*(3*b^8 + 28*a^2*b^6))/d - (sin(c + d*x)*((11*b^8)/8 - (5*a^8)/8 + (49*a^2*b^6)/2 + (105*a^4*b^4)/4 - (7*a^6*
b^2)/2) - sin(c + d*x)^3*((13*b^8)/8 - (3*a^8)/8 + (63*a^2*b^6)/2 + (175*a^4*b^4)/4 + (7*a^6*b^2)/2) + 10*a*b^
7 - 2*a^7*b - sin(c + d*x)^2*(12*a*b^7 + 56*a^3*b^5 + 28*a^5*b^3) + 42*a^3*b^5 + 14*a^5*b^3)/(d*(sin(c + d*x)^
4 - 2*sin(c + d*x)^2 + 1)) - (4*a*b^7*sin(c + d*x)^2)/d - (log(sin(c + d*x) - 1)*(a + b)^6*(3*a^2 - 18*a*b + 3
5*b^2))/(16*d)