3.466 \(\int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=527 \[ -\frac {a b \left (3 a^2+13 b^2\right )}{6 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {b \left (7 a^2+9 b^2\right )}{14 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac {a b \left (a^4+20 a^2 b^2+11 b^4\right )}{2 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {b \left (5 a^4+50 a^2 b^2+9 b^4\right )}{10 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {a b \left (a^6+77 a^4 b^2+147 a^2 b^4+31 b^6\right )}{2 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {b \left (3 a^6+115 a^4 b^2+129 a^2 b^4+9 b^6\right )}{6 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}+\frac {8 a b^3 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^9}-\frac {b \left (a^8+196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+9 b^8\right )}{2 d \left (a^2-b^2\right )^8 (a+b \sin (c+d x))}-\frac {(a+9 b) \log (1-\sin (c+d x))}{4 d (a+b)^9}+\frac {(a-9 b) \log (\sin (c+d x)+1)}{4 d (a-b)^9} \]

[Out]

-1/4*(a+9*b)*ln(1-sin(d*x+c))/(a+b)^9/d+1/4*(a-9*b)*ln(1+sin(d*x+c))/(a-b)^9/d+8*a*b^3*(15*a^6+63*a^4*b^2+45*a
^2*b^4+5*b^6)*ln(a+b*sin(d*x+c))/(a^2-b^2)^9/d-1/14*b*(7*a^2+9*b^2)/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^7-1/2*sec(d
*x+c)^2*(b-a*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c))^7-1/6*a*b*(3*a^2+13*b^2)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^
6-1/10*b*(5*a^4+50*a^2*b^2+9*b^4)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^5-1/2*a*b*(a^4+20*a^2*b^2+11*b^4)/(a^2-b^2)^5
/d/(a+b*sin(d*x+c))^4-1/6*b*(3*a^6+115*a^4*b^2+129*a^2*b^4+9*b^6)/(a^2-b^2)^6/d/(a+b*sin(d*x+c))^3-1/2*a*b*(a^
6+77*a^4*b^2+147*a^2*b^4+31*b^6)/(a^2-b^2)^7/d/(a+b*sin(d*x+c))^2-1/2*b*(a^8+196*a^6*b^2+574*a^4*b^4+244*a^2*b
^6+9*b^8)/(a^2-b^2)^8/d/(a+b*sin(d*x+c))

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Rubi [A]  time = 0.74, antiderivative size = 527, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2668, 741, 801} \[ -\frac {b \left (196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+a^8+9 b^8\right )}{2 d \left (a^2-b^2\right )^8 (a+b \sin (c+d x))}-\frac {a b \left (77 a^4 b^2+147 a^2 b^4+a^6+31 b^6\right )}{2 d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {b \left (115 a^4 b^2+129 a^2 b^4+3 a^6+9 b^6\right )}{6 d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}-\frac {a b \left (20 a^2 b^2+a^4+11 b^4\right )}{2 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {b \left (50 a^2 b^2+5 a^4+9 b^4\right )}{10 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {a b \left (3 a^2+13 b^2\right )}{6 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {b \left (7 a^2+9 b^2\right )}{14 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}+\frac {8 a b^3 \left (63 a^4 b^2+45 a^2 b^4+15 a^6+5 b^6\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^9}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac {(a+9 b) \log (1-\sin (c+d x))}{4 d (a+b)^9}+\frac {(a-9 b) \log (\sin (c+d x)+1)}{4 d (a-b)^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + 9*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^9*d) + ((a - 9*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^9*d) + (8*a*
b^3*(15*a^6 + 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^9*d) - (b*(7*a^2 + 9*b^2)
)/(14*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^7) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*
Sin[c + d*x])^7) - (a*b*(3*a^2 + 13*b^2))/(6*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^6) - (b*(5*a^4 + 50*a^2*b^2
+ 9*b^4))/(10*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^5) - (a*b*(a^4 + 20*a^2*b^2 + 11*b^4))/(2*(a^2 - b^2)^5*d*(
a + b*Sin[c + d*x])^4) - (b*(3*a^6 + 115*a^4*b^2 + 129*a^2*b^4 + 9*b^6))/(6*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x
])^3) - (a*b*(a^6 + 77*a^4*b^2 + 147*a^2*b^4 + 31*b^6))/(2*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])^2) - (b*(a^8 +
196*a^6*b^2 + 574*a^4*b^4 + 244*a^2*b^6 + 9*b^8))/(2*(a^2 - b^2)^8*d*(a + b*Sin[c + d*x]))

Rule 741

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*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -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 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]

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{(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))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac {b \operatorname {Subst}\left (\int \frac {a^2-9 b^2+8 a x}{(a+x)^8 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac {b \operatorname {Subst}\left (\int \left (\frac {(a-b) (a+9 b)}{2 b (a+b)^8 (b-x)}+\frac {7 a^2+9 b^2}{(a-b) (a+b) (a+x)^8}+\frac {2 \left (3 a^3+13 a b^2\right )}{(a-b)^2 (a+b)^2 (a+x)^7}+\frac {5 a^4+50 a^2 b^2+9 b^4}{(a-b)^3 (a+b)^3 (a+x)^6}+\frac {4 \left (a^5+20 a^3 b^2+11 a b^4\right )}{(a-b)^4 (a+b)^4 (a+x)^5}+\frac {3 a^6+115 a^4 b^2+129 a^2 b^4+9 b^6}{(a-b)^5 (a+b)^5 (a+x)^4}+\frac {2 \left (a^7+77 a^5 b^2+147 a^3 b^4+31 a b^6\right )}{(a-b)^6 (a+b)^6 (a+x)^3}+\frac {a^8+196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+9 b^8}{(a-b)^7 (a+b)^7 (a+x)^2}+\frac {16 \left (15 a^7 b^2+63 a^5 b^4+45 a^3 b^6+5 a b^8\right )}{(a-b)^8 (a+b)^8 (a+x)}+\frac {(a-9 b) (a+b)}{2 (a-b)^8 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {(a+9 b) \log (1-\sin (c+d x))}{4 (a+b)^9 d}+\frac {(a-9 b) \log (1+\sin (c+d x))}{4 (a-b)^9 d}+\frac {8 a b^3 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^9 d}-\frac {b \left (7 a^2+9 b^2\right )}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^7}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}-\frac {a b \left (3 a^2+13 b^2\right )}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^6}-\frac {b \left (5 a^4+50 a^2 b^2+9 b^4\right )}{10 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^5}-\frac {a b \left (a^4+20 a^2 b^2+11 b^4\right )}{2 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^4}-\frac {b \left (3 a^6+115 a^4 b^2+129 a^2 b^4+9 b^6\right )}{6 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^3}-\frac {a b \left (a^6+77 a^4 b^2+147 a^2 b^4+31 b^6\right )}{2 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))^2}-\frac {b \left (a^8+196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+9 b^8\right )}{2 \left (a^2-b^2\right )^8 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.75, size = 770, normalized size = 1.46 \[ \frac {b^3 \left (\frac {\sec ^2(c+d x) \left (b^2-a b \sin (c+d x)\right )}{2 b^4 \left (b^2-a^2\right ) (a+b \sin (c+d x))^7}-\frac {8 a \left (\frac {2 a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))}+\frac {4 a \left (a^2+b^2\right )}{3 (a-b)^4 (a+b)^4 (a+b \sin (c+d x))^3}+\frac {3 a^2+b^2}{4 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^4}+\frac {1}{6 \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac {5 a^4+10 a^2 b^2+b^4}{2 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^2}-\frac {\left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right ) \log (a+b \sin (c+d x))}{(a-b)^7 (a+b)^7}+\frac {2 a}{5 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^5}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^7}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^7}\right )+\left (-7 a^2-9 b^2\right ) \left (\frac {a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac {a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac {3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac {1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac {7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}+\frac {a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^8}\right )}{2 b^2 \left (b^2-a^2\right )}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*((Sec[c + d*x]^2*(b^2 - a*b*Sin[c + d*x]))/(2*b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])^7) - (8*a*(-1/2*Log[1
- Sin[c + d*x]]/(b*(a + b)^7) + Log[1 + Sin[c + d*x]]/(2*(a - b)^7*b) - ((7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b
^6)*Log[a + b*Sin[c + d*x]])/((a - b)^7*(a + b)^7) + 1/(6*(a^2 - b^2)*(a + b*Sin[c + d*x])^6) + (2*a)/(5*(a -
b)^2*(a + b)^2*(a + b*Sin[c + d*x])^5) + (3*a^2 + b^2)/(4*(a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^4) + (4*a*(
a^2 + b^2))/(3*(a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^3) + (5*a^4 + 10*a^2*b^2 + b^4)/(2*(a - b)^5*(a + b)^5
*(a + b*Sin[c + d*x])^2) + (2*a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x]))) + (-7
*a^2 - 9*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^8) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*b) - (8*a*(a^2 +
b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a - b)^8*(a + b)^8) + 1/(7*(a^2 - b^2)*(a + b*Sin[c +
d*x])^7) + a/(3*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^6) + (3*a^2 + b^2)/(5*(a - b)^3*(a + b)^3*(a + b*Sin[
c + d*x])^5) + (a*(a^2 + b^2))/((a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^4) + (5*a^4 + 10*a^2*b^2 + b^4)/(3*(a
- b)^5*(a + b)^5*(a + b*Sin[c + d*x])^3) + (a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c
+ d*x])^2) + (7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)/((a - b)^7*(a + b)^7*(a + b*Sin[c + d*x]))))/(2*b^2*(-a^2
+ b^2))))/d

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fricas [B]  time = 8.07, size = 3678, normalized size = 6.98 \[ \text {result too large to display} \]

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]

1/420*(210*a^16*b - 1680*a^14*b^3 + 5880*a^12*b^5 - 11760*a^10*b^7 + 14700*a^8*b^9 - 11760*a^6*b^11 + 5880*a^4
*b^13 - 1680*a^2*b^15 + 210*b^17 - 210*(a^10*b^7 + 195*a^8*b^9 + 378*a^6*b^11 - 330*a^4*b^13 - 235*a^2*b^15 -
9*b^17)*cos(d*x + c)^8 + 70*(63*a^12*b^5 + 10015*a^10*b^7 + 18468*a^8*b^9 - 14274*a^6*b^11 - 12025*a^4*b^13 -
2157*a^2*b^15 - 90*b^17)*cos(d*x + c)^6 - 14*(525*a^14*b^3 + 59730*a^12*b^5 + 174637*a^10*b^7 + 77130*a^8*b^9
- 194265*a^6*b^11 - 106450*a^4*b^13 - 10785*a^2*b^15 - 522*b^17)*cos(d*x + c)^4 + 2*(735*a^16*b + 37165*a^14*b
^3 + 437199*a^12*b^5 + 836549*a^10*b^7 - 111195*a^8*b^9 - 812385*a^6*b^11 - 362915*a^4*b^13 - 23569*a^2*b^15 -
1584*b^17)*cos(d*x + c)^2 + 3360*(7*(15*a^8*b^9 + 63*a^6*b^11 + 45*a^4*b^13 + 5*a^2*b^15)*cos(d*x + c)^8 - 7*
(75*a^10*b^7 + 360*a^8*b^9 + 414*a^6*b^11 + 160*a^4*b^13 + 15*a^2*b^15)*cos(d*x + c)^6 + 7*(45*a^12*b^5 + 339*
a^10*b^7 + 810*a^8*b^9 + 654*a^6*b^11 + 185*a^4*b^13 + 15*a^2*b^15)*cos(d*x + c)^4 - (15*a^14*b^3 + 378*a^12*b
^5 + 1893*a^10*b^7 + 3260*a^8*b^9 + 2121*a^6*b^11 + 490*a^4*b^13 + 35*a^2*b^15)*cos(d*x + c)^2 + ((15*a^7*b^10
+ 63*a^5*b^12 + 45*a^3*b^14 + 5*a*b^16)*cos(d*x + c)^8 - 3*(105*a^9*b^8 + 456*a^7*b^10 + 378*a^5*b^12 + 80*a^
3*b^14 + 5*a*b^16)*cos(d*x + c)^6 + (525*a^11*b^6 + 2835*a^9*b^8 + 4266*a^7*b^10 + 2254*a^5*b^12 + 345*a^3*b^1
4 + 15*a*b^16)*cos(d*x + c)^4 - (105*a^13*b^4 + 966*a^11*b^6 + 2835*a^9*b^8 + 2948*a^7*b^10 + 1183*a^5*b^12 +
150*a^3*b^14 + 5*a*b^16)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 105*(7*(a^11*b^6 - 45*a^9*b^8
- 240*a^8*b^9 - 630*a^7*b^10 - 1008*a^6*b^11 - 1050*a^5*b^12 - 720*a^4*b^13 - 315*a^3*b^14 - 80*a^2*b^15 - 9*
a*b^16)*cos(d*x + c)^8 - 7*(5*a^13*b^4 - 222*a^11*b^6 - 1200*a^10*b^7 - 3285*a^9*b^8 - 5760*a^8*b^9 - 7140*a^7
*b^10 - 6624*a^6*b^11 - 4725*a^5*b^12 - 2560*a^4*b^13 - 990*a^3*b^14 - 240*a^2*b^15 - 27*a*b^16)*cos(d*x + c)^
6 + 7*(3*a^15*b^2 - 125*a^13*b^4 - 720*a^12*b^5 - 2337*a^11*b^6 - 5424*a^10*b^7 - 9585*a^9*b^8 - 12960*a^8*b^9
- 13335*a^7*b^10 - 10464*a^6*b^11 - 6327*a^5*b^12 - 2960*a^4*b^13 - 1035*a^3*b^14 - 240*a^2*b^15 - 27*a*b^16)
*cos(d*x + c)^4 - (a^17 - 24*a^15*b^2 - 240*a^14*b^3 - 1540*a^13*b^4 - 6048*a^12*b^5 - 15848*a^11*b^6 - 30288*
a^10*b^7 - 44730*a^9*b^8 - 52160*a^8*b^9 - 47784*a^7*b^10 - 33936*a^6*b^11 - 18564*a^5*b^12 - 7840*a^4*b^13 -
2520*a^3*b^14 - 560*a^2*b^15 - 63*a*b^16)*cos(d*x + c)^2 + ((a^10*b^7 - 45*a^8*b^9 - 240*a^7*b^10 - 630*a^6*b^
11 - 1008*a^5*b^12 - 1050*a^4*b^13 - 720*a^3*b^14 - 315*a^2*b^15 - 80*a*b^16 - 9*b^17)*cos(d*x + c)^8 - 3*(7*a
^12*b^5 - 314*a^10*b^7 - 1680*a^9*b^8 - 4455*a^8*b^9 - 7296*a^7*b^10 - 7980*a^6*b^11 - 6048*a^5*b^12 - 3255*a^
4*b^13 - 1280*a^3*b^14 - 378*a^2*b^15 - 80*a*b^16 - 9*b^17)*cos(d*x + c)^6 + (35*a^14*b^3 - 1533*a^12*b^5 - 84
00*a^11*b^6 - 23937*a^10*b^7 - 45360*a^9*b^8 - 63345*a^8*b^9 - 68256*a^7*b^10 - 57015*a^6*b^11 - 36064*a^5*b^1
2 - 16695*a^4*b^13 - 5520*a^3*b^14 - 1323*a^2*b^15 - 240*a*b^16 - 27*b^17)*cos(d*x + c)^4 - (7*a^16*b - 280*a^
14*b^3 - 1680*a^13*b^4 - 5964*a^12*b^5 - 15456*a^11*b^6 - 30344*a^10*b^7 - 45360*a^9*b^8 - 52230*a^8*b^9 - 471
68*a^7*b^10 - 33768*a^6*b^11 - 18928*a^5*b^12 - 7980*a^4*b^13 - 2400*a^3*b^14 - 504*a^2*b^15 - 80*a*b^16 - 9*b
^17)*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 105*(7*(a^11*b^6 - 45*a^9*b^8 + 240*a^8*b^9 - 630*a
^7*b^10 + 1008*a^6*b^11 - 1050*a^5*b^12 + 720*a^4*b^13 - 315*a^3*b^14 + 80*a^2*b^15 - 9*a*b^16)*cos(d*x + c)^8
- 7*(5*a^13*b^4 - 222*a^11*b^6 + 1200*a^10*b^7 - 3285*a^9*b^8 + 5760*a^8*b^9 - 7140*a^7*b^10 + 6624*a^6*b^11
- 4725*a^5*b^12 + 2560*a^4*b^13 - 990*a^3*b^14 + 240*a^2*b^15 - 27*a*b^16)*cos(d*x + c)^6 + 7*(3*a^15*b^2 - 12
5*a^13*b^4 + 720*a^12*b^5 - 2337*a^11*b^6 + 5424*a^10*b^7 - 9585*a^9*b^8 + 12960*a^8*b^9 - 13335*a^7*b^10 + 10
464*a^6*b^11 - 6327*a^5*b^12 + 2960*a^4*b^13 - 1035*a^3*b^14 + 240*a^2*b^15 - 27*a*b^16)*cos(d*x + c)^4 - (a^1
7 - 24*a^15*b^2 + 240*a^14*b^3 - 1540*a^13*b^4 + 6048*a^12*b^5 - 15848*a^11*b^6 + 30288*a^10*b^7 - 44730*a^9*b
^8 + 52160*a^8*b^9 - 47784*a^7*b^10 + 33936*a^6*b^11 - 18564*a^5*b^12 + 7840*a^4*b^13 - 2520*a^3*b^14 + 560*a^
2*b^15 - 63*a*b^16)*cos(d*x + c)^2 + ((a^10*b^7 - 45*a^8*b^9 + 240*a^7*b^10 - 630*a^6*b^11 + 1008*a^5*b^12 - 1
050*a^4*b^13 + 720*a^3*b^14 - 315*a^2*b^15 + 80*a*b^16 - 9*b^17)*cos(d*x + c)^8 - 3*(7*a^12*b^5 - 314*a^10*b^7
+ 1680*a^9*b^8 - 4455*a^8*b^9 + 7296*a^7*b^10 - 7980*a^6*b^11 + 6048*a^5*b^12 - 3255*a^4*b^13 + 1280*a^3*b^14
- 378*a^2*b^15 + 80*a*b^16 - 9*b^17)*cos(d*x + c)^6 + (35*a^14*b^3 - 1533*a^12*b^5 + 8400*a^11*b^6 - 23937*a^
10*b^7 + 45360*a^9*b^8 - 63345*a^8*b^9 + 68256*a^7*b^10 - 57015*a^6*b^11 + 36064*a^5*b^12 - 16695*a^4*b^13 + 5
520*a^3*b^14 - 1323*a^2*b^15 + 240*a*b^16 - 27*b^17)*cos(d*x + c)^4 - (7*a^16*b - 280*a^14*b^3 + 1680*a^13*b^4
- 5964*a^12*b^5 + 15456*a^11*b^6 - 30344*a^10*b^7 + 45360*a^9*b^8 - 52230*a^8*b^9 + 47168*a^7*b^10 - 33768*a^
6*b^11 + 18928*a^5*b^12 - 7980*a^4*b^13 + 2400*a^3*b^14 - 504*a^2*b^15 + 80*a*b^16 - 9*b^17)*cos(d*x + c)^2)*s
in(d*x + c))*log(-sin(d*x + c) + 1) - 14*(15*a^17 - 120*a^15*b^2 + 420*a^13*b^4 - 840*a^11*b^6 + 1050*a^9*b^8
- 840*a^7*b^10 + 420*a^5*b^12 - 120*a^3*b^14 + 15*a*b^16 - 15*(7*a^11*b^6 + 1245*a^9*b^8 + 2262*a^7*b^10 - 216
6*a^5*b^12 - 1325*a^3*b^14 - 23*a*b^16)*cos(d*x + c)^6 + 5*(105*a^13*b^4 + 14464*a^11*b^6 + 28953*a^9*b^8 - 11
736*a^7*b^10 - 23605*a^5*b^12 - 8040*a^3*b^14 - 141*a*b^16)*cos(d*x + c)^4 - (315*a^15*b^2 + 26665*a^13*b^4 +
97499*a^11*b^6 + 88065*a^9*b^8 - 106455*a^7*b^10 - 85325*a^5*b^12 - 20415*a^3*b^14 - 349*a*b^16)*cos(d*x + c)^
2)*sin(d*x + c))/(7*(a^19*b^6 - 9*a^17*b^8 + 36*a^15*b^10 - 84*a^13*b^12 + 126*a^11*b^14 - 126*a^9*b^16 + 84*a
^7*b^18 - 36*a^5*b^20 + 9*a^3*b^22 - a*b^24)*d*cos(d*x + c)^8 - 7*(5*a^21*b^4 - 42*a^19*b^6 + 153*a^17*b^8 - 3
12*a^15*b^10 + 378*a^13*b^12 - 252*a^11*b^14 + 42*a^9*b^16 + 72*a^7*b^18 - 63*a^5*b^20 + 22*a^3*b^22 - 3*a*b^2
4)*d*cos(d*x + c)^6 + 7*(3*a^23*b^2 - 17*a^21*b^4 + 21*a^19*b^6 + 81*a^17*b^8 - 354*a^15*b^10 + 630*a^13*b^12
- 630*a^11*b^14 + 354*a^9*b^16 - 81*a^7*b^18 - 21*a^5*b^20 + 17*a^3*b^22 - 3*a*b^24)*d*cos(d*x + c)^4 - (a^25
+ 12*a^23*b^2 - 118*a^21*b^4 + 364*a^19*b^6 - 441*a^17*b^8 - 168*a^15*b^10 + 1260*a^13*b^12 - 1800*a^11*b^14 +
1311*a^9*b^16 - 484*a^7*b^18 + 42*a^5*b^20 + 28*a^3*b^22 - 7*a*b^24)*d*cos(d*x + c)^2 + ((a^18*b^7 - 9*a^16*b
^9 + 36*a^14*b^11 - 84*a^12*b^13 + 126*a^10*b^15 - 126*a^8*b^17 + 84*a^6*b^19 - 36*a^4*b^21 + 9*a^2*b^23 - b^2
5)*d*cos(d*x + c)^8 - 3*(7*a^20*b^5 - 62*a^18*b^7 + 243*a^16*b^9 - 552*a^14*b^11 + 798*a^12*b^13 - 756*a^10*b^
15 + 462*a^8*b^17 - 168*a^6*b^19 + 27*a^4*b^21 + 2*a^2*b^23 - b^25)*d*cos(d*x + c)^6 + (35*a^22*b^3 - 273*a^20
*b^5 + 885*a^18*b^7 - 1455*a^16*b^9 + 990*a^14*b^11 + 630*a^12*b^13 - 1974*a^10*b^15 + 1890*a^8*b^17 - 945*a^6
*b^19 + 235*a^4*b^21 - 15*a^2*b^23 - 3*b^25)*d*cos(d*x + c)^4 - (7*a^24*b - 28*a^22*b^3 - 42*a^20*b^5 + 484*a^
18*b^7 - 1311*a^16*b^9 + 1800*a^14*b^11 - 1260*a^12*b^13 + 168*a^10*b^15 + 441*a^8*b^17 - 364*a^6*b^19 + 118*a
^4*b^21 - 12*a^2*b^23 - b^25)*d*cos(d*x + c)^2)*sin(d*x + c))

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giac [B]  time = 4.38, size = 1327, normalized size = 2.52 \[ \text {result too large to display} \]

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]

1/420*(3360*(15*a^7*b^4 + 63*a^5*b^6 + 45*a^3*b^8 + 5*a*b^10)*log(abs(b*sin(d*x + c) + a))/(a^18*b - 9*a^16*b^
3 + 36*a^14*b^5 - 84*a^12*b^7 + 126*a^10*b^9 - 126*a^8*b^11 + 84*a^6*b^13 - 36*a^4*b^15 + 9*a^2*b^17 - b^19) +
105*(a - 9*b)*log(abs(sin(d*x + c) + 1))/(a^9 - 9*a^8*b + 36*a^7*b^2 - 84*a^6*b^3 + 126*a^5*b^4 - 126*a^4*b^5
+ 84*a^3*b^6 - 36*a^2*b^7 + 9*a*b^8 - b^9) - 105*(a + 9*b)*log(abs(sin(d*x + c) - 1))/(a^9 + 9*a^8*b + 36*a^7
*b^2 + 84*a^6*b^3 + 126*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9) + 210*(120*a^7*b^3*si
n(d*x + c)^2 + 504*a^5*b^5*sin(d*x + c)^2 + 360*a^3*b^7*sin(d*x + c)^2 + 40*a*b^9*sin(d*x + c)^2 - a^10*sin(d*
x + c) - 27*a^8*b^2*sin(d*x + c) - 42*a^6*b^4*sin(d*x + c) + 42*a^4*b^6*sin(d*x + c) + 27*a^2*b^8*sin(d*x + c)
+ b^10*sin(d*x + c) + 8*a^9*b - 72*a^7*b^3 - 504*a^5*b^5 - 408*a^3*b^7 - 48*a*b^9)/((a^18 - 9*a^16*b^2 + 36*a
^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*(sin(d*x
+ c)^2 - 1)) - 4*(32670*a^7*b^10*sin(d*x + c)^7 + 137214*a^5*b^12*sin(d*x + c)^7 + 98010*a^3*b^14*sin(d*x + c)
^7 + 10890*a*b^16*sin(d*x + c)^7 + 237510*a^8*b^9*sin(d*x + c)^6 + 978138*a^6*b^11*sin(d*x + c)^6 + 670950*a^4
*b^13*sin(d*x + c)^6 + 65310*a^2*b^15*sin(d*x + c)^6 - 420*b^17*sin(d*x + c)^6 + 741930*a^9*b^8*sin(d*x + c)^5
+ 2987334*a^7*b^10*sin(d*x + c)^5 + 1959930*a^5*b^12*sin(d*x + c)^5 + 166530*a^3*b^14*sin(d*x + c)^5 - 1260*a
*b^16*sin(d*x + c)^5 + 1291675*a^10*b^7*sin(d*x + c)^4 + 5064885*a^8*b^9*sin(d*x + c)^4 + 3165120*a^6*b^11*sin
(d*x + c)^4 + 237020*a^4*b^13*sin(d*x + c)^4 - 1155*a^2*b^15*sin(d*x + c)^4 - 105*b^17*sin(d*x + c)^4 + 135467
5*a^11*b^6*sin(d*x + c)^3 + 5144685*a^9*b^8*sin(d*x + c)^3 + 3051720*a^7*b^10*sin(d*x + c)^3 + 207620*a^5*b^12
*sin(d*x + c)^3 - 1155*a^3*b^14*sin(d*x + c)^3 - 105*a*b^16*sin(d*x + c)^3 + 856905*a^12*b^5*sin(d*x + c)^2 +
3126501*a^10*b^7*sin(d*x + c)^2 + 1759590*a^8*b^9*sin(d*x + c)^2 + 113400*a^6*b^11*sin(d*x + c)^2 - 2205*a^4*b
^13*sin(d*x + c)^2 + 315*a^2*b^15*sin(d*x + c)^2 - 42*b^17*sin(d*x + c)^2 + 303275*a^13*b^4*sin(d*x + c) + 104
9727*a^11*b^6*sin(d*x + c) + 565530*a^9*b^8*sin(d*x + c) + 33600*a^7*b^10*sin(d*x + c) - 735*a^5*b^12*sin(d*x
+ c) + 105*a^3*b^14*sin(d*x + c) - 14*a*b^16*sin(d*x + c) + 46475*a^14*b^3 + 149331*a^12*b^5 + 79845*a^10*b^7
+ 2385*a^8*b^9 + 1155*a^6*b^11 - 525*a^4*b^13 + 133*a^2*b^15 - 15*b^17)/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84
*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*(b*sin(d*x + c) + a)^
7))/d

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maple [A]  time = 0.48, size = 804, normalized size = 1.53 \[ -\frac {b^{7}}{d \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) a}{4 d \left (a +b \right )^{9}}-\frac {9 \ln \left (\sin \left (d x +c \right )-1\right ) b}{4 d \left (a +b \right )^{9}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) a}{4 d \left (a -b \right )^{9}}-\frac {9 \ln \left (1+\sin \left (d x +c \right )\right ) b}{4 d \left (a -b \right )^{9}}-\frac {2 b^{5}}{5 d \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {4 b^{9}}{d \left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{3}}{7 d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {5 b^{3} a^{3}}{d \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {3 b^{5} a}{d \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {28 b^{3} a^{5}}{d \left (a +b \right )^{7} \left (a -b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {56 b^{5} a^{3}}{d \left (a +b \right )^{7} \left (a -b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {12 b^{7} a}{d \left (a +b \right )^{7} \left (a -b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {120 b^{3} a^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{9} \left (a -b \right )^{9}}+\frac {504 b^{5} a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{9} \left (a -b \right )^{9}}+\frac {360 b^{7} a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{9} \left (a -b \right )^{9}}-\frac {1}{4 d \left (a +b \right )^{8} \left (\sin \left (d x +c \right )-1\right )}-\frac {1}{4 d \left (a -b \right )^{8} \left (1+\sin \left (d x +c \right )\right )}+\frac {40 b^{9} a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{9} \left (a -b \right )^{9}}-\frac {35 b^{3} a^{4}}{3 d \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {14 b^{5} a^{2}}{d \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {2 a \,b^{3}}{3 d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {2 b^{3} a^{2}}{d \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {84 b^{3} a^{6}}{d \left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {252 b^{5} a^{4}}{d \left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {108 b^{7} a^{2}}{d \left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )} \]

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]

-1/d*b^7/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^3-1/4/d/(a+b)^9*ln(sin(d*x+c)-1)*a-9/4/d/(a+b)^9*ln(sin(d*x+c)-1)*b+
1/4/d/(a-b)^9*ln(1+sin(d*x+c))*a-9/4/d/(a-b)^9*ln(1+sin(d*x+c))*b-2/5/d*b^5/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^5
-4/d*b^9/(a+b)^8/(a-b)^8/(a+b*sin(d*x+c))-1/7/d*b^3/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^7-5/d*b^3*a^3/(a+b)^5/(a-
b)^5/(a+b*sin(d*x+c))^4-3/d*b^5*a/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^4-28/d*b^3*a^5/(a+b)^7/(a-b)^7/(a+b*sin(d*x
+c))^2-56/d*b^5*a^3/(a+b)^7/(a-b)^7/(a+b*sin(d*x+c))^2-12/d*b^7*a/(a+b)^7/(a-b)^7/(a+b*sin(d*x+c))^2+120/d*b^3
*a^7/(a+b)^9/(a-b)^9*ln(a+b*sin(d*x+c))+504/d*b^5*a^5/(a+b)^9/(a-b)^9*ln(a+b*sin(d*x+c))+360/d*b^7*a^3/(a+b)^9
/(a-b)^9*ln(a+b*sin(d*x+c))-1/4/d/(a+b)^8/(sin(d*x+c)-1)-1/4/d/(a-b)^8/(1+sin(d*x+c))+40/d*b^9*a/(a+b)^9/(a-b)
^9*ln(a+b*sin(d*x+c))-35/3/d*b^3/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^3*a^4-14/d*b^5/(a+b)^6/(a-b)^6/(a+b*sin(d*x+
c))^3*a^2-2/3/d*a*b^3/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))^6-2/d*b^3/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^5*a^2-84/d*b
^3/(a+b)^8/(a-b)^8/(a+b*sin(d*x+c))*a^6-252/d*b^5/(a+b)^8/(a-b)^8/(a+b*sin(d*x+c))*a^4-108/d*b^7/(a+b)^8/(a-b)
^8/(a+b*sin(d*x+c))*a^2

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maxima [B]  time = 0.44, size = 1670, normalized size = 3.17 \[ \text {result too large to display} \]

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]

1/420*(3360*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9)*log(b*sin(d*x + c) + a)/(a^18 - 9*a^16*b^2 + 36*a
^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18) + 105*(a
- 9*b)*log(sin(d*x + c) + 1)/(a^9 - 9*a^8*b + 36*a^7*b^2 - 84*a^6*b^3 + 126*a^5*b^4 - 126*a^4*b^5 + 84*a^3*b^6
- 36*a^2*b^7 + 9*a*b^8 - b^9) - 105*(a + 9*b)*log(sin(d*x + c) - 1)/(a^9 + 9*a^8*b + 36*a^7*b^2 + 84*a^6*b^3
+ 126*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9) - 2*(840*a^14*b + 33490*a^12*b^3 + 5772
4*a^10*b^5 + 16354*a^8*b^7 - 1496*a^6*b^9 + 814*a^4*b^11 - 236*a^2*b^13 + 30*b^15 - 105*(a^8*b^7 + 196*a^6*b^9
+ 574*a^4*b^11 + 244*a^2*b^13 + 9*b^15)*sin(d*x + c)^8 - 105*(7*a^9*b^6 + 1252*a^7*b^8 + 3514*a^5*b^10 + 1348
*a^3*b^12 + 23*a*b^14)*sin(d*x + c)^7 - 35*(63*a^10*b^5 + 10066*a^8*b^7 + 26194*a^6*b^9 + 7384*a^4*b^11 - 681*
a^2*b^13 - 18*b^15)*sin(d*x + c)^6 - 35*(105*a^11*b^4 + 14506*a^9*b^6 + 32254*a^7*b^8 + 160*a^5*b^10 - 3951*a^
3*b^12 - 66*a*b^14)*sin(d*x + c)^5 - 7*(525*a^12*b^3 + 59310*a^10*b^5 + 83812*a^8*b^7 - 98528*a^6*b^9 - 44663*
a^4*b^11 - 438*a^2*b^13 - 18*b^15)*sin(d*x + c)^4 - 7*(315*a^13*b^2 + 25930*a^11*b^4 - 20896*a^9*b^6 - 166336*
a^7*b^8 - 53641*a^5*b^10 - 386*a^3*b^12 - 26*a*b^14)*sin(d*x + c)^3 - (735*a^14*b + 30550*a^12*b^3 - 361856*a^
10*b^5 - 919070*a^8*b^7 - 252845*a^6*b^9 - 3050*a^4*b^11 + 310*a^2*b^13 - 54*b^15)*sin(d*x + c)^2 - 7*(15*a^15
- 420*a^13*b^2 - 26140*a^11*b^4 - 52264*a^9*b^6 - 13189*a^7*b^8 - 184*a^5*b^10 + 26*a^3*b^12 - 4*a*b^14)*sin(
d*x + c))/(a^23 - 8*a^21*b^2 + 28*a^19*b^4 - 56*a^17*b^6 + 70*a^15*b^8 - 56*a^13*b^10 + 28*a^11*b^12 - 8*a^9*b
^14 + a^7*b^16 - (a^16*b^7 - 8*a^14*b^9 + 28*a^12*b^11 - 56*a^10*b^13 + 70*a^8*b^15 - 56*a^6*b^17 + 28*a^4*b^1
9 - 8*a^2*b^21 + b^23)*sin(d*x + c)^9 - 7*(a^17*b^6 - 8*a^15*b^8 + 28*a^13*b^10 - 56*a^11*b^12 + 70*a^9*b^14 -
56*a^7*b^16 + 28*a^5*b^18 - 8*a^3*b^20 + a*b^22)*sin(d*x + c)^8 - (21*a^18*b^5 - 169*a^16*b^7 + 596*a^14*b^9
- 1204*a^12*b^11 + 1526*a^10*b^13 - 1246*a^8*b^15 + 644*a^6*b^17 - 196*a^4*b^19 + 29*a^2*b^21 - b^23)*sin(d*x
+ c)^7 - 7*(5*a^19*b^4 - 41*a^17*b^6 + 148*a^15*b^8 - 308*a^13*b^10 + 406*a^11*b^12 - 350*a^9*b^14 + 196*a^7*b
^16 - 68*a^5*b^18 + 13*a^3*b^20 - a*b^22)*sin(d*x + c)^6 - 7*(5*a^20*b^3 - 43*a^18*b^5 + 164*a^16*b^7 - 364*a^
14*b^9 + 518*a^12*b^11 - 490*a^10*b^13 + 308*a^8*b^15 - 124*a^6*b^17 + 29*a^4*b^19 - 3*a^2*b^21)*sin(d*x + c)^
5 - 7*(3*a^21*b^2 - 29*a^19*b^4 + 124*a^17*b^6 - 308*a^15*b^8 + 490*a^13*b^10 - 518*a^11*b^12 + 364*a^9*b^14 -
164*a^7*b^16 + 43*a^5*b^18 - 5*a^3*b^20)*sin(d*x + c)^4 - 7*(a^22*b - 13*a^20*b^3 + 68*a^18*b^5 - 196*a^16*b^
7 + 350*a^14*b^9 - 406*a^12*b^11 + 308*a^10*b^13 - 148*a^8*b^15 + 41*a^6*b^17 - 5*a^4*b^19)*sin(d*x + c)^3 - (
a^23 - 29*a^21*b^2 + 196*a^19*b^4 - 644*a^17*b^6 + 1246*a^15*b^8 - 1526*a^13*b^10 + 1204*a^11*b^12 - 596*a^9*b
^14 + 169*a^7*b^16 - 21*a^5*b^18)*sin(d*x + c)^2 + 7*(a^22*b - 8*a^20*b^3 + 28*a^18*b^5 - 56*a^16*b^7 + 70*a^1
4*b^9 - 56*a^12*b^11 + 28*a^10*b^13 - 8*a^8*b^15 + a^6*b^17)*sin(d*x + c)))/d

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mupad [B]  time = 9.89, size = 1443, normalized size = 2.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^3*(a + b*sin(c + d*x))^8),x)

[Out]

((sin(c + d*x)^7*(23*a*b^14 + 1348*a^3*b^12 + 3514*a^5*b^10 + 1252*a^7*b^8 + 7*a^9*b^6))/(2*(a^16 + b^16 - 8*a
^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (420*a^14*b + 15
*b^15 - 118*a^2*b^13 + 407*a^4*b^11 - 748*a^6*b^9 + 8177*a^8*b^7 + 28862*a^10*b^5 + 16745*a^12*b^3)/(105*(a^2
- b^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c
+ d*x)^6*(7384*a^4*b^11 - 681*a^2*b^13 - 18*b^15 + 26194*a^6*b^9 + 10066*a^8*b^7 + 63*a^10*b^5))/(6*(a^16 + b
^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c
+ d*x)^8*(9*b^15 + 244*a^2*b^13 + 574*a^4*b^11 + 196*a^6*b^9 + a^8*b^7))/(2*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4
*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^5*(160*a^5*b^10 -
3951*a^3*b^12 - 66*a*b^14 + 32254*a^7*b^8 + 14506*a^9*b^6 + 105*a^11*b^4))/(6*(a^16 + b^16 - 8*a^2*b^14 + 28*a
^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^4*(18*b^13 + 456
*a^2*b^11 + 45119*a^4*b^9 + 143647*a^6*b^7 + 59835*a^8*b^5 + 525*a^10*b^3))/(30*(a^14 - b^14 + 7*a^2*b^12 - 21
*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) - (sin(c + d*x)^2*(54*b^15 - 735*a^14*b - 310
*a^2*b^13 + 3050*a^4*b^11 + 252845*a^6*b^9 + 919070*a^8*b^7 + 361856*a^10*b^5 - 30550*a^12*b^3))/(210*(a^2 - b
^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) - (sin(c +
d*x)^3*(26*a*b^14 + 386*a^3*b^12 + 53641*a^5*b^10 + 166336*a^7*b^8 + 20896*a^9*b^6 - 25930*a^11*b^4 - 315*a^13
*b^2))/(30*(a^2 - b^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^1
2*b^2)) - (a*sin(c + d*x)*(4*b^14 - 15*a^14 - 26*a^2*b^12 + 184*a^4*b^10 + 13189*a^6*b^8 + 52264*a^8*b^6 + 261
40*a^10*b^4 + 420*a^12*b^2))/(30*(a^2 - b^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6
+ 21*a^10*b^4 - 7*a^12*b^2)))/(d*(sin(c + d*x)^7*(b^7 - 21*a^2*b^5) - sin(c + d*x)^2*(a^7 - 21*a^5*b^2) + sin
(c + d*x)^4*(35*a^3*b^4 - 21*a^5*b^2) + sin(c + d*x)^5*(21*a^2*b^5 - 35*a^4*b^3) + a^7 - b^7*sin(c + d*x)^9 -
sin(c + d*x)^3*(7*a^6*b - 35*a^4*b^3) + sin(c + d*x)^6*(7*a*b^6 - 35*a^3*b^4) - 7*a*b^6*sin(c + d*x)^8 + 7*a^6
*b*sin(c + d*x))) - (log(sin(c + d*x) - 1)*((2*b)/(a + b)^9 + 1/(4*(a + b)^8)))/d + (log(a + b*sin(c + d*x))*(
(2*b)/(a + b)^9 + 1/(4*(a + b)^8) + (2*b)/(a - b)^9 - 1/(4*(a - b)^8)))/d + (log(sin(c + d*x) + 1)*(a - 9*b))/
(4*d*(a - b)^9)

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

[Out]

Timed out

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