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

Optimal. Leaf size=385 \[ \frac{b \left (35 a^4 b^2+21 a^2 b^4+7 a^6+b^6\right )}{d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{a b \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (10 a^2 b^2+5 a^4+b^4\right )}{3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{a b \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (3 a^2+b^2\right )}{5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{a b}{3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{8 a b \left (a^2+b^2\right ) \left (6 a^2 b^2+a^4+b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^8}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^8} \]

[Out]

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

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Rubi [A]  time = 0.528099, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2668, 710, 801} \[ \frac{b \left (35 a^4 b^2+21 a^2 b^4+7 a^6+b^6\right )}{d \left (a^2-b^2\right )^7 (a+b \sin (c+d x))}+\frac{a b \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac{b \left (10 a^2 b^2+5 a^4+b^4\right )}{3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}+\frac{a b \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac{b \left (3 a^2+b^2\right )}{5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac{a b}{3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac{b}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{8 a b \left (a^2+b^2\right ) \left (6 a^2 b^2+a^4+b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^8}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 710

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

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]

Rubi steps

\begin{align*} \int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+x)^8 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{b \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^7 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=\frac{b}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{b \operatorname{Subst}\left (\int \left (\frac{a-b}{2 b (a+b)^7 (b-x)}-\frac{2 a}{(a-b) (a+b) (a+x)^7}+\frac{-3 a^2-b^2}{(a-b)^2 (a+b)^2 (a+x)^6}-\frac{4 a \left (a^2+b^2\right )}{(a-b)^3 (a+b)^3 (a+x)^5}+\frac{-5 a^4-10 a^2 b^2-b^4}{(a-b)^4 (a+b)^4 (a+x)^4}-\frac{2 \left (3 a^5+10 a^3 b^2+3 a b^4\right )}{(a-b)^5 (a+b)^5 (a+x)^3}+\frac{-7 a^6-35 a^4 b^2-21 a^2 b^4-b^6}{(a-b)^6 (a+b)^6 (a+x)^2}-\frac{8 a \left (a^6+7 a^4 b^2+7 a^2 b^4+b^6\right )}{(a-b)^7 (a+b)^7 (a+x)}+\frac{a+b}{2 (a-b)^7 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac{\log (1-\sin (c+d x))}{2 (a+b)^8 d}+\frac{\log (1+\sin (c+d x))}{2 (a-b)^8 d}-\frac{8 a b \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^8 d}+\frac{b}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac{a b}{3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac{b \left (3 a^2+b^2\right )}{5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac{a b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac{b \left (5 a^4+10 a^2 b^2+b^4\right )}{3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac{a b \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac{b \left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 2.54058, size = 365, normalized size = 0.95 \[ \frac{b \left (\frac{a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac{a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac{35 a^4 b^2+21 a^2 b^4+7 a^6+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}+\frac{10 a^2 b^2+5 a^4+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}+\frac{3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac{1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac{8 a \left (a^2+b^2\right ) \left (6 a^2 b^2+a^4+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac{a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^8}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(-Log[1 - Sin[c + d*x]]/(2*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]))))/d

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Maple [A]  time = 0.253, size = 699, normalized size = 1.8 \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)/(a+b*sin(d*x+c))^8,x)

[Out]

1/7/d*b/(a+b)/(a-b)/(a+b*sin(d*x+c))^7+1/3/d*a*b/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^6+3/5/d*b/(a+b)^3/(a-b)^3/(a
+b*sin(d*x+c))^5*a^2+1/5/d*b^3/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))^5+5/3/d*b/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^3*a
^4+10/3/d*b^3/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^3*a^2+1/3/d*b^5/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^3+7/d*b/(a+b)^
7/(a-b)^7/(a+b*sin(d*x+c))*a^6+35/d*b^3/(a+b)^7/(a-b)^7/(a+b*sin(d*x+c))*a^4+21/d*b^5/(a+b)^7/(a-b)^7/(a+b*sin
(d*x+c))*a^2+1/d*b^7/(a+b)^7/(a-b)^7/(a+b*sin(d*x+c))+1/d*b*a^3/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^4+1/d*b^3*a/(
a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^4+3/d*b*a^5/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^2+10/d*b^3*a^3/(a+b)^6/(a-b)^6/(a
+b*sin(d*x+c))^2+3/d*b^5*a/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^2-8/d*b*a^7/(a+b)^8/(a-b)^8*ln(a+b*sin(d*x+c))-56/
d*b^3*a^5/(a+b)^8/(a-b)^8*ln(a+b*sin(d*x+c))-56/d*b^5*a^3/(a+b)^8/(a-b)^8*ln(a+b*sin(d*x+c))-8/d*b^7*a/(a+b)^8
/(a-b)^8*ln(a+b*sin(d*x+c))-1/2/d/(a+b)^8*ln(sin(d*x+c)-1)+1/2*ln(1+sin(d*x+c))/(a-b)^8/d

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Maxima [B]  time = 1.24815, size = 1566, normalized size = 4.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(1680*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*log(b*sin(d*x + c) + a)/(a^16 - 8*a^14*b^2 + 28*a^12*b^4
- 56*a^10*b^6 + 70*a^8*b^8 - 56*a^6*b^10 + 28*a^4*b^12 - 8*a^2*b^14 + b^16) - 2*(1443*a^12*b + 3704*a^10*b^3 +
1849*a^8*b^5 - 496*a^6*b^7 + 309*a^4*b^9 - 104*a^2*b^11 + 15*b^13 + 105*(7*a^6*b^7 + 35*a^4*b^9 + 21*a^2*b^11
+ b^13)*sin(d*x + c)^6 + 105*(45*a^7*b^6 + 217*a^5*b^8 + 119*a^3*b^10 + 3*a*b^12)*sin(d*x + c)^5 + 35*(365*a^
8*b^5 + 1680*a^6*b^7 + 826*a^4*b^9 + 8*a^2*b^11 + b^13)*sin(d*x + c)^4 + 35*(533*a^9*b^4 + 2304*a^7*b^6 + 994*
a^5*b^8 + 8*a^3*b^10 + a*b^12)*sin(d*x + c)^3 + 21*(743*a^10*b^3 + 2934*a^8*b^5 + 1099*a^6*b^7 + 29*a^4*b^9 -
6*a^2*b^11 + b^13)*sin(d*x + c)^2 + 7*(1023*a^11*b^2 + 3494*a^9*b^4 + 1219*a^7*b^6 + 29*a^5*b^8 - 6*a^3*b^10 +
a*b^12)*sin(d*x + c))/(a^21 - 7*a^19*b^2 + 21*a^17*b^4 - 35*a^15*b^6 + 35*a^13*b^8 - 21*a^11*b^10 + 7*a^9*b^1
2 - a^7*b^14 + (a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35*a^6*b^15 - 21*a^4*b^17 + 7*a^2*b^19 -
b^21)*sin(d*x + c)^7 + 7*(a^15*b^6 - 7*a^13*b^8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*a^5*b^16 + 7*a
^3*b^18 - a*b^20)*sin(d*x + c)^6 + 21*(a^16*b^5 - 7*a^14*b^7 + 21*a^12*b^9 - 35*a^10*b^11 + 35*a^8*b^13 - 21*a
^6*b^15 + 7*a^4*b^17 - a^2*b^19)*sin(d*x + c)^5 + 35*(a^17*b^4 - 7*a^15*b^6 + 21*a^13*b^8 - 35*a^11*b^10 + 35*
a^9*b^12 - 21*a^7*b^14 + 7*a^5*b^16 - a^3*b^18)*sin(d*x + c)^4 + 35*(a^18*b^3 - 7*a^16*b^5 + 21*a^14*b^7 - 35*
a^12*b^9 + 35*a^10*b^11 - 21*a^8*b^13 + 7*a^6*b^15 - a^4*b^17)*sin(d*x + c)^3 + 21*(a^19*b^2 - 7*a^17*b^4 + 21
*a^15*b^6 - 35*a^13*b^8 + 35*a^11*b^10 - 21*a^9*b^12 + 7*a^7*b^14 - a^5*b^16)*sin(d*x + c)^2 + 7*(a^20*b - 7*a
^18*b^3 + 21*a^16*b^5 - 35*a^14*b^7 + 35*a^12*b^9 - 21*a^10*b^11 + 7*a^8*b^13 - a^6*b^15)*sin(d*x + c)) - 105*
log(sin(d*x + c) + 1)/(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a*b^
7 + b^8) + 105*log(sin(d*x + c) - 1)/(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a
^2*b^6 + 8*a*b^7 + b^8))/d

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Fricas [B]  time = 26.269, size = 7699, normalized size = 20. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(2886*a^14*b + 35728*a^12*b^3 + 113862*a^10*b^5 + 11760*a^8*b^7 - 97230*a^6*b^9 - 62496*a^4*b^11 - 4158
*a^2*b^13 - 352*b^15 - 210*(7*a^8*b^7 + 28*a^6*b^9 - 14*a^4*b^11 - 20*a^2*b^13 - b^15)*cos(d*x + c)^6 + 70*(36
5*a^10*b^5 + 1378*a^8*b^7 - 602*a^6*b^9 - 944*a^4*b^11 - 187*a^2*b^13 - 10*b^15)*cos(d*x + c)^4 - 14*(2229*a^1
2*b^3 + 10223*a^10*b^5 + 7960*a^8*b^7 - 10490*a^6*b^9 - 8915*a^4*b^11 - 949*a^2*b^13 - 58*b^15)*cos(d*x + c)^2
- 1680*(a^14*b + 28*a^12*b^3 + 189*a^10*b^5 + 400*a^8*b^7 + 315*a^6*b^9 + 84*a^4*b^11 + 7*a^2*b^13 - 7*(a^8*b
^7 + 7*a^6*b^9 + 7*a^4*b^11 + a^2*b^13)*cos(d*x + c)^6 + 7*(5*a^10*b^5 + 38*a^8*b^7 + 56*a^6*b^9 + 26*a^4*b^11
+ 3*a^2*b^13)*cos(d*x + c)^4 - 7*(3*a^12*b^3 + 31*a^10*b^5 + 94*a^8*b^7 + 94*a^6*b^9 + 31*a^4*b^11 + 3*a^2*b^
13)*cos(d*x + c)^2 + (7*a^13*b^2 + 84*a^11*b^4 + 315*a^9*b^6 + 400*a^7*b^8 + 189*a^5*b^10 + 28*a^3*b^12 + a*b^
14 - (a^7*b^8 + 7*a^5*b^10 + 7*a^3*b^12 + a*b^14)*cos(d*x + c)^6 + 3*(7*a^9*b^6 + 50*a^7*b^8 + 56*a^5*b^10 + 1
4*a^3*b^12 + a*b^14)*cos(d*x + c)^4 - (35*a^11*b^4 + 287*a^9*b^6 + 542*a^7*b^8 + 350*a^5*b^10 + 63*a^3*b^12 +
3*a*b^14)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 105*(a^15 + 8*a^14*b + 49*a^13*b^2 + 224*a^1
2*b^3 + 693*a^11*b^4 + 1512*a^10*b^5 + 2485*a^9*b^6 + 3200*a^8*b^7 + 3235*a^7*b^8 + 2520*a^6*b^9 + 1491*a^5*b^
10 + 672*a^4*b^11 + 231*a^3*b^12 + 56*a^2*b^13 + 7*a*b^14 - 7*(a^9*b^6 + 8*a^8*b^7 + 28*a^7*b^8 + 56*a^6*b^9 +
70*a^5*b^10 + 56*a^4*b^11 + 28*a^3*b^12 + 8*a^2*b^13 + a*b^14)*cos(d*x + c)^6 + 7*(5*a^11*b^4 + 40*a^10*b^5 +
143*a^9*b^6 + 304*a^8*b^7 + 434*a^7*b^8 + 448*a^6*b^9 + 350*a^5*b^10 + 208*a^4*b^11 + 89*a^3*b^12 + 24*a^2*b^
13 + 3*a*b^14)*cos(d*x + c)^4 - 7*(3*a^13*b^2 + 24*a^12*b^3 + 94*a^11*b^4 + 248*a^10*b^5 + 493*a^9*b^6 + 752*a
^8*b^7 + 868*a^7*b^8 + 752*a^6*b^9 + 493*a^5*b^10 + 248*a^4*b^11 + 94*a^3*b^12 + 24*a^2*b^13 + 3*a*b^14)*cos(d
*x + c)^2 + (7*a^14*b + 56*a^13*b^2 + 231*a^12*b^3 + 672*a^11*b^4 + 1491*a^10*b^5 + 2520*a^9*b^6 + 3235*a^8*b^
7 + 3200*a^7*b^8 + 2485*a^6*b^9 + 1512*a^5*b^10 + 693*a^4*b^11 + 224*a^3*b^12 + 49*a^2*b^13 + 8*a*b^14 + b^15
- (a^8*b^7 + 8*a^7*b^8 + 28*a^6*b^9 + 56*a^5*b^10 + 70*a^4*b^11 + 56*a^3*b^12 + 28*a^2*b^13 + 8*a*b^14 + b^15)
*cos(d*x + c)^6 + 3*(7*a^10*b^5 + 56*a^9*b^6 + 197*a^8*b^7 + 400*a^7*b^8 + 518*a^6*b^9 + 448*a^5*b^10 + 266*a^
4*b^11 + 112*a^3*b^12 + 35*a^2*b^13 + 8*a*b^14 + b^15)*cos(d*x + c)^4 - (35*a^12*b^3 + 280*a^11*b^4 + 1022*a^1
0*b^5 + 2296*a^9*b^6 + 3629*a^8*b^7 + 4336*a^7*b^8 + 4004*a^6*b^9 + 2800*a^5*b^10 + 1421*a^4*b^11 + 504*a^3*b^
12 + 126*a^2*b^13 + 24*a*b^14 + 3*b^15)*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 105*(a^15 - 8*a^
14*b + 49*a^13*b^2 - 224*a^12*b^3 + 693*a^11*b^4 - 1512*a^10*b^5 + 2485*a^9*b^6 - 3200*a^8*b^7 + 3235*a^7*b^8
- 2520*a^6*b^9 + 1491*a^5*b^10 - 672*a^4*b^11 + 231*a^3*b^12 - 56*a^2*b^13 + 7*a*b^14 - 7*(a^9*b^6 - 8*a^8*b^7
+ 28*a^7*b^8 - 56*a^6*b^9 + 70*a^5*b^10 - 56*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14)*cos(d*x + c)^6 + 7
*(5*a^11*b^4 - 40*a^10*b^5 + 143*a^9*b^6 - 304*a^8*b^7 + 434*a^7*b^8 - 448*a^6*b^9 + 350*a^5*b^10 - 208*a^4*b^
11 + 89*a^3*b^12 - 24*a^2*b^13 + 3*a*b^14)*cos(d*x + c)^4 - 7*(3*a^13*b^2 - 24*a^12*b^3 + 94*a^11*b^4 - 248*a^
10*b^5 + 493*a^9*b^6 - 752*a^8*b^7 + 868*a^7*b^8 - 752*a^6*b^9 + 493*a^5*b^10 - 248*a^4*b^11 + 94*a^3*b^12 - 2
4*a^2*b^13 + 3*a*b^14)*cos(d*x + c)^2 + (7*a^14*b - 56*a^13*b^2 + 231*a^12*b^3 - 672*a^11*b^4 + 1491*a^10*b^5
- 2520*a^9*b^6 + 3235*a^8*b^7 - 3200*a^7*b^8 + 2485*a^6*b^9 - 1512*a^5*b^10 + 693*a^4*b^11 - 224*a^3*b^12 + 49
*a^2*b^13 - 8*a*b^14 + b^15 - (a^8*b^7 - 8*a^7*b^8 + 28*a^6*b^9 - 56*a^5*b^10 + 70*a^4*b^11 - 56*a^3*b^12 + 28
*a^2*b^13 - 8*a*b^14 + b^15)*cos(d*x + c)^6 + 3*(7*a^10*b^5 - 56*a^9*b^6 + 197*a^8*b^7 - 400*a^7*b^8 + 518*a^6
*b^9 - 448*a^5*b^10 + 266*a^4*b^11 - 112*a^3*b^12 + 35*a^2*b^13 - 8*a*b^14 + b^15)*cos(d*x + c)^4 - (35*a^12*b
^3 - 280*a^11*b^4 + 1022*a^10*b^5 - 2296*a^9*b^6 + 3629*a^8*b^7 - 4336*a^7*b^8 + 4004*a^6*b^9 - 2800*a^5*b^10
+ 1421*a^4*b^11 - 504*a^3*b^12 + 126*a^2*b^13 - 24*a*b^14 + 3*b^15)*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x
+ c) + 1) + 14*(1023*a^13*b^2 + 5136*a^11*b^4 + 7255*a^9*b^6 - 5160*a^7*b^8 - 6435*a^5*b^10 - 1768*a^3*b^12 -
51*a*b^14 + 15*(45*a^9*b^6 + 172*a^7*b^8 - 98*a^5*b^10 - 116*a^3*b^12 - 3*a*b^14)*cos(d*x + c)^4 - 5*(533*a^1
1*b^4 + 2041*a^9*b^6 - 278*a^7*b^8 - 1574*a^5*b^10 - 703*a^3*b^12 - 19*a*b^14)*cos(d*x + c)^2)*sin(d*x + c))/(
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)*d*cos(d*x + c)^6 - 7*(5*a^19*b^4 - 37*a^17*b^6 + 116*a^15*b^8 - 196*a^13*b^10 + 182*a^11*b^12 - 70*a
^9*b^14 - 28*a^7*b^16 + 44*a^5*b^18 - 19*a^3*b^20 + 3*a*b^22)*d*cos(d*x + c)^4 + 7*(3*a^21*b^2 - 14*a^19*b^4 +
7*a^17*b^6 + 88*a^15*b^8 - 266*a^13*b^10 + 364*a^11*b^12 - 266*a^9*b^14 + 88*a^7*b^16 + 7*a^5*b^18 - 14*a^3*b
^20 + 3*a*b^22)*d*cos(d*x + c)^2 - (a^23 + 13*a^21*b^2 - 105*a^19*b^4 + 259*a^17*b^6 - 182*a^15*b^8 - 350*a^13
*b^10 + 910*a^11*b^12 - 890*a^9*b^14 + 421*a^7*b^16 - 63*a^5*b^18 - 21*a^3*b^20 + 7*a*b^22)*d + ((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^19 - 8*a^2*b^21 + b^23)*d*cos(d
*x + c)^6 - 3*(7*a^18*b^5 - 55*a^16*b^7 + 188*a^14*b^9 - 364*a^12*b^11 + 434*a^10*b^13 - 322*a^8*b^15 + 140*a^
6*b^17 - 28*a^4*b^19 - a^2*b^21 + b^23)*d*cos(d*x + c)^4 + (35*a^20*b^3 - 238*a^18*b^5 + 647*a^16*b^7 - 808*a^
14*b^9 + 182*a^12*b^11 + 812*a^10*b^13 - 1162*a^8*b^15 + 728*a^6*b^17 - 217*a^4*b^19 + 18*a^2*b^21 + 3*b^23)*d
*cos(d*x + c)^2 - (7*a^22*b - 21*a^20*b^3 - 63*a^18*b^5 + 421*a^16*b^7 - 890*a^14*b^9 + 910*a^12*b^11 - 350*a^
10*b^13 - 182*a^8*b^15 + 259*a^6*b^17 - 105*a^4*b^19 + 13*a^2*b^21 + b^23)*d)*sin(d*x + c))

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

[Out]

Timed out

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Giac [B]  time = 1.5458, size = 1364, normalized size = 3.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(1680*(a^7*b^2 + 7*a^5*b^4 + 7*a^3*b^6 + a*b^8)*log(abs(b*sin(d*x + c) + a))/(a^16*b - 8*a^14*b^3 + 28*
a^12*b^5 - 56*a^10*b^7 + 70*a^8*b^9 - 56*a^6*b^11 + 28*a^4*b^13 - 8*a^2*b^15 + b^17) - 105*log(abs(sin(d*x + c
) + 1))/(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a*b^7 + b^8) + 105
*log(abs(sin(d*x + c) - 1))/(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a^2*b^6 +
8*a*b^7 + b^8) - 2*(2178*a^7*b^8*sin(d*x + c)^7 + 15246*a^5*b^10*sin(d*x + c)^7 + 15246*a^3*b^12*sin(d*x + c)^
7 + 2178*a*b^14*sin(d*x + c)^7 + 15981*a^8*b^7*sin(d*x + c)^6 + 109662*a^6*b^9*sin(d*x + c)^6 + 105252*a^4*b^1
1*sin(d*x + c)^6 + 13146*a^2*b^13*sin(d*x + c)^6 - 105*b^15*sin(d*x + c)^6 + 50463*a^9*b^6*sin(d*x + c)^5 + 33
8226*a^7*b^8*sin(d*x + c)^5 + 309876*a^5*b^10*sin(d*x + c)^5 + 33558*a^3*b^12*sin(d*x + c)^5 - 315*a*b^14*sin(
d*x + c)^5 + 89005*a^10*b^5*sin(d*x + c)^4 + 579635*a^8*b^7*sin(d*x + c)^4 + 503720*a^6*b^9*sin(d*x + c)^4 + 4
7600*a^4*b^11*sin(d*x + c)^4 - 245*a^2*b^13*sin(d*x + c)^4 - 35*b^15*sin(d*x + c)^4 + 94885*a^11*b^4*sin(d*x +
c)^3 + 595595*a^9*b^6*sin(d*x + c)^3 + 487760*a^7*b^8*sin(d*x + c)^3 + 41720*a^5*b^10*sin(d*x + c)^3 - 245*a^
3*b^12*sin(d*x + c)^3 - 35*a*b^14*sin(d*x + c)^3 + 61341*a^12*b^3*sin(d*x + c)^2 + 366177*a^10*b^5*sin(d*x + c
)^2 + 281631*a^8*b^7*sin(d*x + c)^2 + 23268*a^6*b^9*sin(d*x + c)^2 - 735*a^4*b^11*sin(d*x + c)^2 + 147*a^2*b^1
3*sin(d*x + c)^2 - 21*b^15*sin(d*x + c)^2 + 22407*a^13*b^2*sin(d*x + c) + 124019*a^11*b^4*sin(d*x + c) + 90797
*a^9*b^6*sin(d*x + c) + 6916*a^7*b^8*sin(d*x + c) - 245*a^5*b^10*sin(d*x + c) + 49*a^3*b^12*sin(d*x + c) - 7*a
*b^14*sin(d*x + c) + 3621*a^14*b + 17507*a^12*b^3 + 13391*a^10*b^5 - 167*a^8*b^7 + 805*a^6*b^9 - 413*a^4*b^11
+ 119*a^2*b^13 - 15*b^15)/((a^16 - 8*a^14*b^2 + 28*a^12*b^4 - 56*a^10*b^6 + 70*a^8*b^8 - 56*a^6*b^10 + 28*a^4*
b^12 - 8*a^2*b^14 + b^16)*(b*sin(d*x + c) + a)^7))/d