3.1559 \(\int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=550 \[ -\frac {\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {b^6 \left (-7 a^2 B+8 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^5}+\frac {\left (5 a^3 A-a^2 (25 A b-2 b B)+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^5}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A-8 a b B+7 A b^2\right )+\left (5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right ) \sin (c+d x)\right )}{24 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (5 a^4 A+2 a^3 b B-18 a^2 A b^2+46 a b^3 B-35 A b^4\right )+3 \left (5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right ) \sin (c+d x)\right )}{48 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {b \left (5 a^6 A+2 a^5 b B-23 a^4 A b^2-12 a^3 b^3 B+47 a^2 A b^4-54 a b^5 B+35 A b^6\right )}{16 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))} \]

[Out]

-1/32*(5*a^3*A+a^2*b*(25*A+2*B)+a*b^2*(47*A+10*B)+b^3*(35*A+16*B))*ln(1-sin(d*x+c))/(a+b)^5/d+1/32*(5*a^3*A-b^
3*(35*A-16*B)+a*b^2*(47*A-10*B)-a^2*(25*A*b-2*B*b))*ln(1+sin(d*x+c))/(a-b)^5/d+b^6*(8*A*a*b-7*B*a^2-B*b^2)*ln(
a+b*sin(d*x+c))/(a^2-b^2)^5/d-1/16*b*(5*A*a^6-23*A*a^4*b^2+47*A*a^2*b^4+35*A*b^6+2*B*a^5*b-12*B*a^3*b^3-54*B*a
*b^5)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))-1/6*sec(d*x+c)^6*(A*b-a*B-(A*a-B*b)*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+
c))+1/24*sec(d*x+c)^4*(b*(A*a^2+7*A*b^2-8*B*a*b)+(5*A*a^3-13*A*a*b^2+2*B*a^2*b+6*B*b^3)*sin(d*x+c))/(a^2-b^2)^
2/d/(a+b*sin(d*x+c))+1/48*sec(d*x+c)^2*(b*(5*A*a^4-18*A*a^2*b^2-35*A*b^4+2*B*a^3*b+46*B*a*b^3)+3*(5*A*a^5-18*A
*a^3*b^2+29*A*a*b^4+2*B*a^4*b-10*B*a^2*b^3-8*B*b^5)*sin(d*x+c))/(a^2-b^2)^3/d/(a+b*sin(d*x+c))

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Rubi [A]  time = 0.93, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2837, 823, 801} \[ -\frac {b \left (-23 a^4 A b^2+47 a^2 A b^4+5 a^6 A-12 a^3 b^3 B+2 a^5 b B-54 a b^5 B+35 A b^6\right )}{16 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {b^6 \left (-7 a^2 B+8 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {\left (a^2 b (25 A+2 B)+5 a^3 A+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^5}+\frac {\left (-a^2 (25 A b-2 b B)+5 a^3 A+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^5}-\frac {\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (\left (5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right ) \sin (c+d x)+b \left (a^2 A-8 a b B+7 A b^2\right )\right )}{24 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (3 \left (-18 a^3 A b^2+5 a^5 A-10 a^2 b^3 B+2 a^4 b B+29 a A b^4-8 b^5 B\right ) \sin (c+d x)+b \left (-18 a^2 A b^2+5 a^4 A+2 a^3 b B+46 a b^3 B-35 A b^4\right )\right )}{48 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

-((5*a^3*A + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B) + b^3*(35*A + 16*B))*Log[1 - Sin[c + d*x]])/(32*(a + b)^
5*d) + ((5*a^3*A - b^3*(35*A - 16*B) + a*b^2*(47*A - 10*B) - a^2*(25*A*b - 2*b*B))*Log[1 + Sin[c + d*x]])/(32*
(a - b)^5*d) + (b^6*(8*a*A*b - 7*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - (b*(5*a^6*A - 23*
a^4*A*b^2 + 47*a^2*A*b^4 + 35*A*b^6 + 2*a^5*b*B - 12*a^3*b^3*B - 54*a*b^5*B))/(16*(a^2 - b^2)^4*d*(a + b*Sin[c
+ d*x])) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(6*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (
Sec[c + d*x]^4*(b*(a^2*A + 7*A*b^2 - 8*a*b*B) + (5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B)*Sin[c + d*x]))/(2
4*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(5*a^4*A - 18*a^2*A*b^2 - 35*A*b^4 + 2*a^3*b*B +
46*a*b^3*B) + 3*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*Sin[c + d*x]))/(48*
(a^2 - b^2)^3*d*(a + b*Sin[c + d*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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2837

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac {b^7 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{b}}{(a+x)^2 \left (b^2-x^2\right )^4} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {b^5 \operatorname {Subst}\left (\int \frac {-5 a^2 A+7 A b^2-2 a b B-6 (a A-b B) x}{(a+x)^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{6 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {b^3 \operatorname {Subst}\left (\int \frac {15 a^4 A-34 a^2 A b^2+35 A b^4+6 a^3 b B-22 a b^3 B+4 \left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) x}{(a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}\\ &=-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (5 a^4 A-18 a^2 A b^2-35 A b^4+2 a^3 b B+46 a b^3 B\right )+3 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {b \operatorname {Subst}\left (\int \frac {-3 \left (5 a^6 A-13 a^4 A b^2+11 a^2 A b^4-35 A b^6+2 a^5 b B-8 a^3 b^3 B+38 a b^5 B\right )-6 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) x}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d}\\ &=-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (5 a^4 A-18 a^2 A b^2-35 A b^4+2 a^3 b B+46 a b^3 B\right )+3 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {b \operatorname {Subst}\left (\int \left (\frac {3 (a-b)^3 \left (-5 a^3 A-a^2 b (25 A+2 B)-a b^2 (47 A+10 B)-b^3 (35 A+16 B)\right )}{2 b (a+b)^2 (b-x)}-\frac {3 \left (5 a^6 A-23 a^4 A b^2+47 a^2 A b^4+35 A b^6+2 a^5 b B-12 a^3 b^3 B-54 a b^5 B\right )}{\left (a^2-b^2\right ) (a+x)^2}+\frac {48 b^5 \left (-8 a A b+7 a^2 B+b^2 B\right )}{\left (-a^2+b^2\right )^2 (a+x)}+\frac {3 (a+b)^3 \left (-5 a^3 A+b^3 (35 A-16 B)-a b^2 (47 A-10 B)+a^2 (25 A b-2 b B)\right )}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d}\\ &=-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^5 d}+\frac {\left (5 a^3 A-b^3 (35 A-16 B)+a b^2 (47 A-10 B)-a^2 (25 A b-2 b B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^5 d}+\frac {b^6 \left (8 a A b-7 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {b \left (5 a^6 A-23 a^4 A b^2+47 a^2 A b^4+35 A b^6+2 a^5 b B-12 a^3 b^3 B-54 a b^5 B\right )}{16 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (5 a^4 A-18 a^2 A b^2-35 A b^4+2 a^3 b B+46 a b^3 B\right )+3 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.21, size = 766, normalized size = 1.39 \[ \frac {b^7 \left (\frac {\frac {\frac {\left (6 a \left (5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right )-3 \left (5 a^6 A+2 a^5 b B-13 a^4 A b^2-8 a^3 b^3 B+11 a^2 A b^4+38 a b^5 B-35 A b^6\right )\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )-6 \left (5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right ) \left (-\frac {\log (a+b \sin (c+d x))}{a^2-b^2}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)}\right )}{2 b^2 \left (b^2-a^2\right )}-\frac {\sec ^2(c+d x) \left (4 a b^2 \left (5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right )-b \left (4 b^2 \left (5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right )-a \left (15 a^4 A+6 a^3 b B-34 a^2 A b^2-22 a b^3 B+35 A b^4\right )\right ) \sin (c+d x)-b^2 \left (15 a^4 A+6 a^3 b B-34 a^2 A b^2-22 a b^3 B+35 A b^4\right )\right )}{2 b^4 \left (b^2-a^2\right ) (a+b \sin (c+d x))}}{4 b^2 \left (b^2-a^2\right )}-\frac {\sec ^4(c+d x) \left (-b \left (-a \left (-5 a^2 A-2 a b B+7 A b^2\right )-6 b^2 (a A-b B)\right ) \sin (c+d x)-b^2 \left (-5 a^2 A-2 a b B+7 A b^2\right )-6 a b^2 (a A-b B)\right )}{4 b^6 \left (b^2-a^2\right ) (a+b \sin (c+d x))}}{6 b^2 \left (b^2-a^2\right )}-\frac {\sec ^6(c+d x) \left (-b (b B-a A) \sin (c+d x)+a b B-A b^2\right )}{6 b^8 \left (b^2-a^2\right ) (a+b \sin (c+d x))}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

(b^7*(-1/6*(Sec[c + d*x]^6*(-(A*b^2) + a*b*B - b*(-(a*A) + b*B)*Sin[c + d*x]))/(b^8*(-a^2 + b^2)*(a + b*Sin[c
+ d*x])) + (-1/4*(Sec[c + d*x]^4*(-6*a*b^2*(a*A - b*B) - b^2*(-5*a^2*A + 7*A*b^2 - 2*a*b*B) - b*(-6*b^2*(a*A -
b*B) - a*(-5*a^2*A + 7*A*b^2 - 2*a*b*B))*Sin[c + d*x]))/(b^6*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-1/2*(Sec[
c + d*x]^2*(4*a*b^2*(5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B) - b^2*(15*a^4*A - 34*a^2*A*b^2 + 35*A*b^4 + 6
*a^3*b*B - 22*a*b^3*B) - b*(4*b^2*(5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B) - a*(15*a^4*A - 34*a^2*A*b^2 +
35*A*b^4 + 6*a^3*b*B - 22*a*b^3*B))*Sin[c + d*x]))/(b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-6*(5*a^5*A - 18
*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)) + Log[1
+ Sin[c + d*x]]/(2*(a - b)*b) - Log[a + b*Sin[c + d*x]]/(a^2 - b^2)) + (6*a*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b
^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B) - 3*(5*a^6*A - 13*a^4*A*b^2 + 11*a^2*A*b^4 - 35*A*b^6 + 2*a^5*b*B - 8
*a^3*b^3*B + 38*a*b^5*B))*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) -
(2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(2*b^2*(-a^2 + b^
2)))/(4*b^2*(-a^2 + b^2)))/(6*b^2*(-a^2 + b^2))))/d

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fricas [B]  time = 8.22, size = 1244, normalized size = 2.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/96*(16*B*a^9 - 16*A*a^8*b - 64*B*a^7*b^2 + 64*A*a^6*b^3 + 96*B*a^5*b^4 - 96*A*a^4*b^5 - 64*B*a^3*b^6 + 64*A*
a^2*b^7 + 16*B*a*b^8 - 16*A*b^9 - 6*(5*A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 14*B*a^5*b^4 + 70*A*a^4*b^5 - 42
*B*a^3*b^6 - 12*A*a^2*b^7 + 54*B*a*b^8 - 35*A*b^9)*cos(d*x + c)^6 + 2*(5*A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3
+ 42*B*a^5*b^4 + 6*A*a^4*b^5 - 90*B*a^3*b^6 + 52*A*a^2*b^7 + 46*B*a*b^8 - 35*A*b^9)*cos(d*x + c)^4 + 4*(A*a^8*
b - 8*B*a^7*b^2 + 4*A*a^6*b^3 + 24*B*a^5*b^4 - 18*A*a^4*b^5 - 24*B*a^3*b^6 + 20*A*a^2*b^7 + 8*B*a*b^8 - 7*A*b^
9)*cos(d*x + c)^2 - 96*((7*B*a^2*b^7 - 8*A*a*b^8 + B*b^9)*cos(d*x + c)^6*sin(d*x + c) + (7*B*a^3*b^6 - 8*A*a^2
*b^7 + B*a*b^8)*cos(d*x + c)^6)*log(b*sin(d*x + c) + a) + 3*((5*A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 14*B*a^
5*b^4 + 70*A*a^4*b^5 + 70*B*a^3*b^6 - 28*(5*A - 4*B)*a^2*b^7 - 2*(64*A - 35*B)*a*b^8 - (35*A - 16*B)*b^9)*cos(
d*x + c)^6*sin(d*x + c) + (5*A*a^9 + 2*B*a^8*b - 28*A*a^7*b^2 - 14*B*a^6*b^3 + 70*A*a^5*b^4 + 70*B*a^4*b^5 - 2
8*(5*A - 4*B)*a^3*b^6 - 2*(64*A - 35*B)*a^2*b^7 - (35*A - 16*B)*a*b^8)*cos(d*x + c)^6)*log(sin(d*x + c) + 1) -
3*((5*A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 14*B*a^5*b^4 + 70*A*a^4*b^5 + 70*B*a^3*b^6 - 28*(5*A + 4*B)*a^2*
b^7 + 2*(64*A + 35*B)*a*b^8 - (35*A + 16*B)*b^9)*cos(d*x + c)^6*sin(d*x + c) + (5*A*a^9 + 2*B*a^8*b - 28*A*a^7
*b^2 - 14*B*a^6*b^3 + 70*A*a^5*b^4 + 70*B*a^4*b^5 - 28*(5*A + 4*B)*a^3*b^6 + 2*(64*A + 35*B)*a^2*b^7 - (35*A +
16*B)*a*b^8)*cos(d*x + c)^6)*log(-sin(d*x + c) + 1) + 2*(8*A*a^9 - 8*B*a^8*b - 32*A*a^7*b^2 + 32*B*a^6*b^3 +
48*A*a^5*b^4 - 48*B*a^4*b^5 - 32*A*a^3*b^6 + 32*B*a^2*b^7 + 8*A*a*b^8 - 8*B*b^9 + 3*(5*A*a^9 + 2*B*a^8*b - 28*
A*a^7*b^2 - 14*B*a^6*b^3 + 70*A*a^5*b^4 + 14*B*a^4*b^5 - 76*A*a^3*b^6 + 6*B*a^2*b^7 + 29*A*a*b^8 - 8*B*b^9)*co
s(d*x + c)^4 + 2*(5*A*a^9 + 2*B*a^8*b - 28*A*a^7*b^2 + 54*A*a^5*b^4 - 12*B*a^4*b^5 - 44*A*a^3*b^6 + 16*B*a^2*b
^7 + 13*A*a*b^8 - 6*B*b^9)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b - 5*a^8*b^3 + 10*a^6*b^5 - 10*a^4*b^7 + 5*a^
2*b^9 - b^11)*d*cos(d*x + c)^6*sin(d*x + c) + (a^11 - 5*a^9*b^2 + 10*a^7*b^4 - 10*a^5*b^6 + 5*a^3*b^8 - a*b^10
)*d*cos(d*x + c)^6)

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giac [B]  time = 0.43, size = 1185, normalized size = 2.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/96*(96*(7*B*a^2*b^7 - 8*A*a*b^8 + B*b^9)*log(abs(b*sin(d*x + c) + a))/(a^10*b - 5*a^8*b^3 + 10*a^6*b^5 - 10
*a^4*b^7 + 5*a^2*b^9 - b^11) + 3*(5*A*a^3 + 25*A*a^2*b + 2*B*a^2*b + 47*A*a*b^2 + 10*B*a*b^2 + 35*A*b^3 + 16*B
*b^3)*log(abs(-sin(d*x + c) + 1))/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) - 3*(5*A*a^3 - 25*
A*a^2*b + 2*B*a^2*b + 47*A*a*b^2 - 10*B*a*b^2 - 35*A*b^3 + 16*B*b^3)*log(abs(-sin(d*x + c) - 1))/(a^5 - 5*a^4*
b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5) - 96*(7*B*a^2*b^7*sin(d*x + c) - 8*A*a*b^8*sin(d*x + c) + B*b^9*s
in(d*x + c) + 8*B*a^3*b^6 - 9*A*a^2*b^7 + A*b^9)/((a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^
10)*(b*sin(d*x + c) + a)) + 2*(308*B*a^2*b^6*sin(d*x + c)^6 - 352*A*a*b^7*sin(d*x + c)^6 + 44*B*b^8*sin(d*x +
c)^6 + 15*A*a^8*sin(d*x + c)^5 + 6*B*a^7*b*sin(d*x + c)^5 - 84*A*a^6*b^2*sin(d*x + c)^5 - 42*B*a^5*b^3*sin(d*x
+ c)^5 + 210*A*a^4*b^4*sin(d*x + c)^5 - 78*B*a^3*b^5*sin(d*x + c)^5 - 84*A*a^2*b^6*sin(d*x + c)^5 + 114*B*a*b
^7*sin(d*x + c)^5 - 57*A*b^8*sin(d*x + c)^5 + 120*B*a^4*b^4*sin(d*x + c)^4 - 144*A*a^3*b^5*sin(d*x + c)^4 - 10
20*B*a^2*b^6*sin(d*x + c)^4 + 1200*A*a*b^7*sin(d*x + c)^4 - 156*B*b^8*sin(d*x + c)^4 - 40*A*a^8*sin(d*x + c)^3
- 16*B*a^7*b*sin(d*x + c)^3 + 224*A*a^6*b^2*sin(d*x + c)^3 + 48*B*a^5*b^3*sin(d*x + c)^3 - 480*A*a^4*b^4*sin(
d*x + c)^3 + 240*B*a^3*b^5*sin(d*x + c)^3 + 160*A*a^2*b^6*sin(d*x + c)^3 - 272*B*a*b^7*sin(d*x + c)^3 + 136*A*
b^8*sin(d*x + c)^3 + 36*B*a^6*b^2*sin(d*x + c)^2 - 48*A*a^5*b^3*sin(d*x + c)^2 - 300*B*a^4*b^4*sin(d*x + c)^2
+ 384*A*a^3*b^5*sin(d*x + c)^2 + 1128*B*a^2*b^6*sin(d*x + c)^2 - 1392*A*a*b^7*sin(d*x + c)^2 + 192*B*b^8*sin(d
*x + c)^2 + 33*A*a^8*sin(d*x + c) - 6*B*a^7*b*sin(d*x + c) - 156*A*a^6*b^2*sin(d*x + c) + 42*B*a^5*b^3*sin(d*x
+ c) + 270*A*a^4*b^4*sin(d*x + c) - 210*B*a^3*b^5*sin(d*x + c) - 60*A*a^2*b^6*sin(d*x + c) + 174*B*a*b^7*sin(
d*x + c) - 87*A*b^8*sin(d*x + c) + 8*B*a^8 - 16*A*a^7*b - 52*B*a^6*b^2 + 96*A*a^5*b^3 + 180*B*a^4*b^4 - 288*A*
a^3*b^5 - 400*B*a^2*b^6 + 560*A*a*b^7 - 88*B*b^8)/((a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b
^10)*(sin(d*x + c)^2 - 1)^3))/d

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maple [B]  time = 0.86, size = 1080, normalized size = 1.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x)

[Out]

-19/32/d/(a-b)^4/(1+sin(d*x+c))*A*b^2+1/16/d/(a+b)^3/(sin(d*x+c)-1)^2*a*A+1/8/d/(a+b)^3/(sin(d*x+c)-1)^2*A*b+1
/32/d/(a+b)^3/(sin(d*x+c)-1)^2*a*B+3/32/d/(a+b)^3/(sin(d*x+c)-1)^2*B*b-5/32/d/(a+b)^4/(sin(d*x+c)-1)*a^2*A-19/
32/d/(a+b)^4/(sin(d*x+c)-1)*A*b^2-1/32/d/(a+b)^4/(sin(d*x+c)-1)*B*a^2-11/32/d/(a+b)^4/(sin(d*x+c)-1)*B*b^2+1/3
2/d/(a-b)^4/(1+sin(d*x+c))*B*a^2+11/32/d/(a-b)^4/(1+sin(d*x+c))*B*b^2+5/32/d/(a-b)^5*ln(1+sin(d*x+c))*a^3*A-35
/32/d/(a-b)^5*ln(1+sin(d*x+c))*b^3*A+1/2/d/(a-b)^5*ln(1+sin(d*x+c))*B*b^3-5/32/d/(a+b)^5*ln(sin(d*x+c)-1)*a^3*
A-35/32/d/(a+b)^5*ln(sin(d*x+c)-1)*b^3*A+9/16/d/(a-b)^4/(1+sin(d*x+c))*A*a*b+8/d*b^7/(a+b)^5/(a-b)^5*ln(a+b*si
n(d*x+c))*A*a-7/d*b^6/(a+b)^5/(a-b)^5*ln(a+b*sin(d*x+c))*B*a^2+1/d*b^6/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))*a*B-1/
48/d/(a+b)^2/(sin(d*x+c)-1)^3*A-1/48/d/(a+b)^2/(sin(d*x+c)-1)^3*B-1/48/d/(a-b)^2/(1+sin(d*x+c))^3*A+1/48/d/(a-
b)^2/(1+sin(d*x+c))^3*B+1/8/d/(a-b)^3/(1+sin(d*x+c))^2*A*b+1/32/d/(a-b)^3/(1+sin(d*x+c))^2*a*B-3/32/d/(a-b)^3/
(1+sin(d*x+c))^2*B*b-5/32/d/(a-b)^4/(1+sin(d*x+c))*a^2*A-1/2/d/(a+b)^5*ln(sin(d*x+c)-1)*B*b^3-1/16/d/(a-b)^3/(
1+sin(d*x+c))^2*a*A-3/16/d/(a-b)^4/(1+sin(d*x+c))*B*a*b-25/32/d/(a-b)^5*ln(1+sin(d*x+c))*A*a^2*b-3/16/d/(a+b)^
4/(sin(d*x+c)-1)*B*a*b-25/32/d/(a+b)^5*ln(sin(d*x+c)-1)*A*a^2*b-47/32/d/(a+b)^5*ln(sin(d*x+c)-1)*A*a*b^2-1/16/
d/(a+b)^5*ln(sin(d*x+c)-1)*B*a^2*b-9/16/d/(a+b)^4/(sin(d*x+c)-1)*A*a*b+47/32/d/(a-b)^5*ln(1+sin(d*x+c))*A*a*b^
2+1/16/d/(a-b)^5*ln(1+sin(d*x+c))*B*a^2*b-5/16/d/(a-b)^5*ln(1+sin(d*x+c))*B*a*b^2-5/16/d/(a+b)^5*ln(sin(d*x+c)
-1)*B*a*b^2-1/d*b^8/(a+b)^5/(a-b)^5*ln(a+b*sin(d*x+c))*B-1/d*b^7/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))*A

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maxima [B]  time = 0.40, size = 1083, normalized size = 1.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/96*(96*(7*B*a^2*b^6 - 8*A*a*b^7 + B*b^8)*log(b*sin(d*x + c) + a)/(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^
6 + 5*a^2*b^8 - b^10) - 3*(5*A*a^3 - (25*A - 2*B)*a^2*b + (47*A - 10*B)*a*b^2 - (35*A - 16*B)*b^3)*log(sin(d*x
+ c) + 1)/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5) + 3*(5*A*a^3 + (25*A + 2*B)*a^2*b + (47*A
+ 10*B)*a*b^2 + (35*A + 16*B)*b^3)*log(sin(d*x + c) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 +
b^5) - 2*(8*B*a^7 - 16*A*a^6*b - 44*B*a^5*b^2 + 80*A*a^4*b^3 + 136*B*a^3*b^4 - 208*A*a^2*b^5 + 92*B*a*b^6 - 4
8*A*b^7 + 3*(5*A*a^6*b + 2*B*a^5*b^2 - 23*A*a^4*b^3 - 12*B*a^3*b^4 + 47*A*a^2*b^5 - 54*B*a*b^6 + 35*A*b^7)*sin
(d*x + c)^6 + 3*(5*A*a^7 + 2*B*a^6*b - 23*A*a^5*b^2 - 12*B*a^4*b^3 + 47*A*a^3*b^4 + 2*B*a^2*b^5 - 29*A*a*b^6 +
8*B*b^7)*sin(d*x + c)^5 - 8*(5*A*a^6*b + 2*B*a^5*b^2 - 23*A*a^4*b^3 - 19*B*a^3*b^4 + 55*A*a^2*b^5 - 55*B*a*b^
6 + 35*A*b^7)*sin(d*x + c)^4 - 4*(10*A*a^7 + 4*B*a^6*b - 46*A*a^5*b^2 - 17*B*a^4*b^3 + 86*A*a^3*b^4 - 2*B*a^2*
b^5 - 50*A*a*b^6 + 15*B*b^7)*sin(d*x + c)^3 + 3*(11*A*a^6*b + 10*B*a^5*b^2 - 57*A*a^4*b^3 - 76*B*a^3*b^4 + 161
*A*a^2*b^5 - 126*B*a*b^6 + 77*A*b^7)*sin(d*x + c)^2 + (33*A*a^7 + 2*B*a^6*b - 139*A*a^5*b^2 - 8*B*a^4*b^3 + 22
7*A*a^3*b^4 - 38*B*a^2*b^5 - 121*A*a*b^6 + 44*B*b^7)*sin(d*x + c))/(a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 +
a*b^8 - (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*sin(d*x + c)^7 - (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^
3*b^6 + a*b^8)*sin(d*x + c)^6 + 3*(a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*sin(d*x + c)^5 + 3*(a^9 -
4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*sin(d*x + c)^4 - 3*(a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9
)*sin(d*x + c)^3 - 3*(a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*sin(d*x + c)^2 + (a^8*b - 4*a^6*b^3 + 6
*a^4*b^5 - 4*a^2*b^7 + b^9)*sin(d*x + c)))/d

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mupad [B]  time = 14.38, size = 1024, normalized size = 1.86 \[ \frac {\frac {\sin \left (c+d\,x\right )\,\left (33\,A\,a^5+2\,B\,a^4\,b-106\,A\,a^3\,b^2-6\,B\,a^2\,b^3+121\,A\,a\,b^4-44\,B\,b^5\right )}{48\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^5\,\left (5\,A\,a^5+2\,B\,a^4\,b-18\,A\,a^3\,b^2-10\,B\,a^2\,b^3+29\,A\,a\,b^4-8\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (10\,A\,a^5+4\,B\,a^4\,b-36\,A\,a^3\,b^2-13\,B\,a^2\,b^3+50\,A\,a\,b^4-15\,B\,b^5\right )}{12\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {-2\,B\,a^7+4\,A\,a^6\,b+11\,B\,a^5\,b^2-20\,A\,a^4\,b^3-34\,B\,a^3\,b^4+52\,A\,a^2\,b^5-23\,B\,a\,b^6+12\,A\,b^7}{12\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^6\,\left (5\,A\,a^6\,b+2\,B\,a^5\,b^2-23\,A\,a^4\,b^3-12\,B\,a^3\,b^4+47\,A\,a^2\,b^5-54\,B\,a\,b^6+35\,A\,b^7\right )}{16\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (5\,A\,a^6\,b+2\,B\,a^5\,b^2-23\,A\,a^4\,b^3-19\,B\,a^3\,b^4+55\,A\,a^2\,b^5-55\,B\,a\,b^6+35\,A\,b^7\right )}{6\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (11\,A\,a^6\,b+10\,B\,a^5\,b^2-57\,A\,a^4\,b^3-76\,B\,a^3\,b^4+161\,A\,a^2\,b^5-126\,B\,a\,b^6+77\,A\,b^7\right )}{16\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left (-b\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,b\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,b\,{\sin \left (c+d\,x\right )}^3-3\,a\,{\sin \left (c+d\,x\right )}^2+b\,\sin \left (c+d\,x\right )+a\right )}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (7\,B\,a^2\,b^6-8\,A\,a\,b^7+B\,b^8\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (5\,A\,a^3+\left (25\,A+2\,B\right )\,a^2\,b+\left (47\,A+10\,B\right )\,a\,b^2+\left (35\,A+16\,B\right )\,b^3\right )}{d\,\left (32\,a^5+160\,a^4\,b+320\,a^3\,b^2+320\,a^2\,b^3+160\,a\,b^4+32\,b^5\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (5\,A\,a^3+\left (2\,B-25\,A\right )\,a^2\,b+\left (47\,A-10\,B\right )\,a\,b^2+\left (16\,B-35\,A\right )\,b^3\right )}{d\,\left (32\,a^5-160\,a^4\,b+320\,a^3\,b^2-320\,a^2\,b^3+160\,a\,b^4-32\,b^5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sin(c + d*x))/(cos(c + d*x)^7*(a + b*sin(c + d*x))^2),x)

[Out]

((sin(c + d*x)*(33*A*a^5 - 44*B*b^5 - 106*A*a^3*b^2 - 6*B*a^2*b^3 + 121*A*a*b^4 + 2*B*a^4*b))/(48*(a^6 - b^6 +
3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^5*(5*A*a^5 - 8*B*b^5 - 18*A*a^3*b^2 - 10*B*a^2*b^3 + 29*A*a*b^4 + 2*B
*a^4*b))/(16*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)^3*(10*A*a^5 - 15*B*b^5 - 36*A*a^3*b^2 - 13*B
*a^2*b^3 + 50*A*a*b^4 + 4*B*a^4*b))/(12*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (12*A*b^7 - 2*B*a^7 + 52*A*a^2*
b^5 - 20*A*a^4*b^3 - 34*B*a^3*b^4 + 11*B*a^5*b^2 + 4*A*a^6*b - 23*B*a*b^6)/(12*(a^2 - b^2)*(a^6 - b^6 + 3*a^2*
b^4 - 3*a^4*b^2)) + (sin(c + d*x)^6*(35*A*b^7 + 47*A*a^2*b^5 - 23*A*a^4*b^3 - 12*B*a^3*b^4 + 2*B*a^5*b^2 + 5*A
*a^6*b - 54*B*a*b^6))/(16*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (sin(c + d*x)^4*(35*A*b^7 + 55*A*
a^2*b^5 - 23*A*a^4*b^3 - 19*B*a^3*b^4 + 2*B*a^5*b^2 + 5*A*a^6*b - 55*B*a*b^6))/(6*(a^2 - b^2)*(a^6 - b^6 + 3*a
^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^2*(77*A*b^7 + 161*A*a^2*b^5 - 57*A*a^4*b^3 - 76*B*a^3*b^4 + 10*B*a^5*b^2
+ 11*A*a^6*b - 126*B*a*b^6))/(16*(a^2 - b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)))/(d*(a + b*sin(c + d*x) - 3*
a*sin(c + d*x)^2 + 3*a*sin(c + d*x)^4 - a*sin(c + d*x)^6 - 3*b*sin(c + d*x)^3 + 3*b*sin(c + d*x)^5 - b*sin(c +
d*x)^7)) - (log(a + b*sin(c + d*x))*(B*b^8 + 7*B*a^2*b^6 - 8*A*a*b^7))/(d*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b
^6 + 10*a^6*b^4 - 5*a^8*b^2)) - (log(sin(c + d*x) - 1)*(5*A*a^3 + b^3*(35*A + 16*B) + a^2*b*(25*A + 2*B) + a*b
^2*(47*A + 10*B)))/(d*(160*a*b^4 + 160*a^4*b + 32*a^5 + 32*b^5 + 320*a^2*b^3 + 320*a^3*b^2)) + (log(sin(c + d*
x) + 1)*(5*A*a^3 - b^3*(35*A - 16*B) - a^2*b*(25*A - 2*B) + a*b^2*(47*A - 10*B)))/(d*(160*a*b^4 - 160*a^4*b +
32*a^5 - 32*b^5 - 320*a^2*b^3 + 320*a^3*b^2))

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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)**7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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