\(\int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx\) [1551]

Optimal result

Integrand size = 31, antiderivative size = 383 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\left (5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^4 d}+\frac {\left (5 a^3 A-b^3 (16 A-5 B)+a b^2 (29 A-4 B)-a^2 b (20 A-B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^4 d}+\frac {b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}+\frac {\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d} \]

[Out]

-1/32*(5*A*a^3+a^2*b*(20*A+B)+a*b^2*(29*A+4*B)+b^3*(16*A+5*B))*ln(1-sin(d*x+c))/(a+b)^4/d+1/32*(5*A*a^3-b^3*(1
6*A-5*B)+a*b^2*(29*A-4*B)-a^2*b*(20*A-B))*ln(1+sin(d*x+c))/(a-b)^4/d+b^6*(A*b-B*a)*ln(a+b*sin(d*x+c))/(a^2-b^2
)^4/d-1/6*sec(d*x+c)^6*(A*b-B*a-(A*a-B*b)*sin(d*x+c))/(a^2-b^2)/d+1/24*sec(d*x+c)^4*(6*b^2*(A*b-B*a)+(5*A*a^3-
11*A*a*b^2+B*a^2*b+5*B*b^3)*sin(d*x+c))/(a^2-b^2)^2/d-1/16*sec(d*x+c)^2*(8*b^4*(A*b-B*a)-(5*A*a^5-16*A*a^3*b^2
+19*A*a*b^4+B*a^4*b-4*B*a^2*b^3-5*B*b^5)*sin(d*x+c))/(a^2-b^2)^3/d

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2916, 837, 815} \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right )}+\frac {b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {\left (5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac {\left (5 a^3 A-a^2 b (20 A-B)+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^4}+\frac {\sec ^4(c+d x) \left (\left (5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right ) \sin (c+d x)+6 b^2 (A b-a B)\right )}{24 d \left (a^2-b^2\right )^2}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A+a^4 b B-16 a^3 A b^2-4 a^2 b^3 B+19 a A b^4-5 b^5 B\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^3} \]

[In]

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

[Out]

-1/32*((5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*Log[1 - Sin[c + d*x]])/((a + b)^4*
d) + ((5*a^3*A - b^3*(16*A - 5*B) + a*b^2*(29*A - 4*B) - a^2*b*(20*A - B))*Log[1 + Sin[c + d*x]])/(32*(a - b)^
4*d) + (b^6*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*
Sin[c + d*x]))/(6*(a^2 - b^2)*d) + (Sec[c + d*x]^4*(6*b^2*(A*b - a*B) + (5*a^3*A - 11*a*A*b^2 + a^2*b*B + 5*b^
3*B)*Sin[c + d*x]))/(24*(a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(8*b^4*(A*b - a*B) - (5*a^5*A - 16*a^3*A*b^2 + 19*a
*A*b^4 + a^4*b*B - 4*a^2*b^3*B - 5*b^5*B)*Sin[c + d*x]))/(16*(a^2 - b^2)^3*d)

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 837

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] +
Dist[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 2916

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/b)*x)^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*} \text {integral}& = \frac {b^7 \text {Subst}\left (\int \frac {A+\frac {B x}{b}}{(a+x) \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}-\frac {b^5 \text {Subst}\left (\int \frac {-5 a^2 A+6 A b^2-a b B-5 (a A-b B) x}{(a+x) \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}+\frac {\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}+\frac {b^3 \text {Subst}\left (\int \frac {3 \left (5 a^4 A-11 a^2 A b^2+8 A b^4+a^3 b B-3 a b^3 B\right )+3 \left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) x}{(a+x) \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}+\frac {\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}-\frac {b \text {Subst}\left (\int \frac {-3 \left (5 a^6 A-16 a^4 A b^2+19 a^2 A b^4-16 A b^6+a^5 b B-4 a^3 b^3 B+11 a b^5 B\right )-3 \left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) x}{(a+x) \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}+\frac {\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}-\frac {b \text {Subst}\left (\int \left (\frac {3 (a-b)^3 \left (-5 a^3 A-a^2 b (20 A+B)-a b^2 (29 A+4 B)-b^3 (16 A+5 B)\right )}{2 b (a+b) (b-x)}+\frac {48 b^5 (-A b+a B)}{(a-b) (a+b) (a+x)}+\frac {3 (a+b)^3 \left (-5 a^3 A+b^3 (16 A-5 B)-a b^2 (29 A-4 B)+a^2 b (20 A-B)\right )}{2 (a-b) 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 (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^4 d}+\frac {\left (5 a^3 A-b^3 (16 A-5 B)+a b^2 (29 A-4 B)-a^2 b (20 A-B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^4 d}+\frac {b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 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}+\frac {\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.28 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.52 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {b^7 \left (-\frac {\sec ^6(c+d x) \left (-A b^2+a b B-b (-a A+b B) \sin (c+d x)\right )}{6 b^8 \left (-a^2+b^2\right )}+\frac {-\frac {\sec ^4(c+d x) \left (-5 a b^2 (a A-b B)-b^2 \left (-5 a^2 A+6 A b^2-a b B\right )-b \left (-5 b^2 (a A-b B)-a \left (-5 a^2 A+6 A b^2-a b B\right )\right ) \sin (c+d x)\right )}{4 b^6 \left (-a^2+b^2\right )}+\frac {\frac {3 \left (\frac {(a-b)^3 \left (5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{2 b (a+b)}-\frac {(a+b)^3 \left (5 a^3 A-b^3 (16 A-5 B)+a b^2 (29 A-4 B)-a^2 b (20 A-B)\right ) \log (1+\sin (c+d x))}{2 (a-b) b}-\frac {16 b^5 (A b-a B) \log (a+b \sin (c+d x))}{(a-b) (a+b)}\right )}{2 b^2 \left (-a^2+b^2\right )}-\frac {\sec ^2(c+d x) \left (3 a b^2 \left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right )-3 b^2 \left (5 a^4 A-11 a^2 A b^2+8 A b^4+a^3 b B-3 a b^3 B\right )-b \left (3 b^2 \left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right )-3 a \left (5 a^4 A-11 a^2 A b^2+8 A b^4+a^3 b B-3 a b^3 B\right )\right ) \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right )}}{4 b^2 \left (-a^2+b^2\right )}}{6 b^2 \left (-a^2+b^2\right )}\right )}{d} \]

[In]

Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),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)) + (-1/4*(Sec
[c + d*x]^4*(-5*a*b^2*(a*A - b*B) - b^2*(-5*a^2*A + 6*A*b^2 - a*b*B) - b*(-5*b^2*(a*A - b*B) - a*(-5*a^2*A + 6
*A*b^2 - a*b*B))*Sin[c + d*x]))/(b^6*(-a^2 + b^2)) + ((3*(((a - b)^3*(5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A
+ 4*B) + b^3*(16*A + 5*B))*Log[1 - Sin[c + d*x]])/(2*b*(a + b)) - ((a + b)^3*(5*a^3*A - b^3*(16*A - 5*B) + a*
b^2*(29*A - 4*B) - a^2*b*(20*A - B))*Log[1 + Sin[c + d*x]])/(2*(a - b)*b) - (16*b^5*(A*b - a*B)*Log[a + b*Sin[
c + d*x]])/((a - b)*(a + b))))/(2*b^2*(-a^2 + b^2)) - (Sec[c + d*x]^2*(3*a*b^2*(5*a^3*A - 11*a*A*b^2 + a^2*b*B
+ 5*b^3*B) - 3*b^2*(5*a^4*A - 11*a^2*A*b^2 + 8*A*b^4 + a^3*b*B - 3*a*b^3*B) - b*(3*b^2*(5*a^3*A - 11*a*A*b^2
+ a^2*b*B + 5*b^3*B) - 3*a*(5*a^4*A - 11*a^2*A*b^2 + 8*A*b^4 + a^3*b*B - 3*a*b^3*B))*Sin[c + d*x]))/(2*b^4*(-a
^2 + b^2)))/(4*b^2*(-a^2 + b^2)))/(6*b^2*(-a^2 + b^2))))/d

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {\frac {\left (A b -B a \right ) b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {A +B}{3 \left (16 a +16 b \right ) \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -3 A b -B a -2 B b}{32 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+14 A a b +11 A \,b^{2}+B \,a^{2}+4 B a b +5 B \,b^{2}}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 A \,a^{3}-20 A \,a^{2} b -29 A a \,b^{2}-16 A \,b^{3}-B \,a^{2} b -4 B a \,b^{2}-5 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{4}}-\frac {A -B}{3 \left (16 a -16 b \right ) \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -3 A b -B a +2 B b}{32 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-14 A a b +11 A \,b^{2}-B \,a^{2}+4 B a b -5 B \,b^{2}}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 A \,a^{3}-20 A \,a^{2} b +29 A a \,b^{2}-16 A \,b^{3}+B \,a^{2} b -4 B a \,b^{2}+5 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{4}}}{d}\)	\(393\)
default	\(\frac {\frac {\left (A b -B a \right ) b^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {A +B}{3 \left (16 a +16 b \right ) \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -3 A b -B a -2 B b}{32 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+14 A a b +11 A \,b^{2}+B \,a^{2}+4 B a b +5 B \,b^{2}}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 A \,a^{3}-20 A \,a^{2} b -29 A a \,b^{2}-16 A \,b^{3}-B \,a^{2} b -4 B a \,b^{2}-5 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{4}}-\frac {A -B}{3 \left (16 a -16 b \right ) \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -3 A b -B a +2 B b}{32 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-14 A a b +11 A \,b^{2}-B \,a^{2}+4 B a b -5 B \,b^{2}}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 A \,a^{3}-20 A \,a^{2} b +29 A a \,b^{2}-16 A \,b^{3}+B \,a^{2} b -4 B a \,b^{2}+5 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{4}}}{d}\)	\(393\)
parallelrisch	\(\frac {48 b^{6} \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A b -B a \right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-15 \left (\left (\frac {16 A}{5}+B \right ) b^{3}+\frac {29 a \left (A +\frac {4 B}{29}\right ) b^{2}}{5}+4 \left (A +\frac {B}{20}\right ) a^{2} b +A \,a^{3}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+15 \left (\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (-\frac {16 A}{5}+B \right ) b^{3}+\frac {29 \left (A -\frac {4 B}{29}\right ) a \,b^{2}}{5}-4 \left (A -\frac {B}{20}\right ) a^{2} b +A \,a^{3}\right ) \left (a +b \right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8 \left (15 \left (a^{2}-\frac {b^{2}}{2}\right ) \left (A b -B a \right ) \left (a^{2}-\frac {7 b^{2}}{5}\right ) \cos \left (2 d x +2 c \right )+6 \left (a^{4}-\frac {7}{2} a^{2} b^{2}+\frac {7}{2} b^{4}\right ) \left (A b -B a \right ) \cos \left (4 d x +4 c \right )+\left (a^{4}-\frac {7}{2} a^{2} b^{2}+\frac {11}{2} b^{4}\right ) \left (A b -B a \right ) \cos \left (6 d x +6 c \right )+\left (\frac {85}{4} A \,a^{5}-\frac {85}{4} B \,b^{5}-68 A \,a^{3} b^{2}+\frac {259}{4} A a \,b^{4}+\frac {17}{4} B \,a^{4} b -B \,a^{2} b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {15}{4} B \,b^{5}+\frac {15}{4} A \,a^{5}-12 A \,a^{3} b^{2}+\frac {57}{4} A a \,b^{4}+\frac {3}{4} B \,a^{4} b -3 B \,a^{2} b^{3}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {99}{2} B \,b^{5}+\frac {99}{2} A \,a^{5}-120 A \,a^{3} b^{2}+\frac {165}{2} A a \,b^{4}-\frac {57}{2} B \,a^{4} b +66 B \,a^{2} b^{3}\right ) \sin \left (d x +c \right )-22 \left (A b -B a \right ) \left (a^{4}-\frac {53}{22} a^{2} b^{2}+\frac {37}{22} b^{4}\right )\right ) \left (a -b \right )}{15}\right ) \left (a +b \right )}{48 \left (a -b \right )^{4} \left (a +b \right )^{4} d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(617\)
norman	\(\text {Expression too large to display}\)	\(1436\)
risch	\(\text {Expression too large to display}\)	\(3078\)

[In]

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

[Out]

1/d*((A*b-B*a)*b^6/(a+b)^4/(a-b)^4*ln(a+b*sin(d*x+c))-1/3*(A+B)/(16*a+16*b)/(sin(d*x+c)-1)^3-1/32*(-2*A*a-3*A*
b-B*a-2*B*b)/(a+b)^2/(sin(d*x+c)-1)^2-1/32*(5*A*a^2+14*A*a*b+11*A*b^2+B*a^2+4*B*a*b+5*B*b^2)/(a+b)^3/(sin(d*x+
c)-1)+1/32/(a+b)^4*(-5*A*a^3-20*A*a^2*b-29*A*a*b^2-16*A*b^3-B*a^2*b-4*B*a*b^2-5*B*b^3)*ln(sin(d*x+c)-1)-1/3*(A
-B)/(16*a-16*b)/(1+sin(d*x+c))^3-1/32*(2*A*a-3*A*b-B*a+2*B*b)/(a-b)^2/(1+sin(d*x+c))^2-1/32*(5*A*a^2-14*A*a*b+
11*A*b^2-B*a^2+4*B*a*b-5*B*b^2)/(a-b)^3/(1+sin(d*x+c))+1/32*(5*A*a^3-20*A*a^2*b+29*A*a*b^2-16*A*b^3+B*a^2*b-4*
B*a*b^2+5*B*b^3)/(a-b)^4*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 2.48 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {16 \, B a^{7} - 16 \, A a^{6} b - 48 \, B a^{5} b^{2} + 48 \, A a^{4} b^{3} + 48 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5} - 16 \, B a b^{6} + 16 \, A b^{7} - 96 \, {\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (b \sin \left (d x + c\right ) + a\right ) + 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} - {\left (35 \, A - 16 \, B\right )} a b^{6} - {\left (16 \, A - 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} - {\left (35 \, A + 16 \, B\right )} a b^{6} + {\left (16 \, A + 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, {\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} - 24 \, {\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, A a^{7} - 8 \, B a^{6} b - 24 \, A a^{5} b^{2} + 24 \, B a^{4} b^{3} + 24 \, A a^{3} b^{4} - 24 \, B a^{2} b^{5} - 8 \, A a b^{6} + 8 \, B b^{7} + 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} - B a^{2} b^{5} - 19 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} + 3 \, B a^{4} b^{3} + 27 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} - 11 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{96 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{6}} \]

[In]

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

[Out]

1/96*(16*B*a^7 - 16*A*a^6*b - 48*B*a^5*b^2 + 48*A*a^4*b^3 + 48*B*a^3*b^4 - 48*A*a^2*b^5 - 16*B*a*b^6 + 16*A*b^
7 - 96*(B*a*b^6 - A*b^7)*cos(d*x + c)^6*log(b*sin(d*x + c) + a) + 3*(5*A*a^7 + B*a^6*b - 21*A*a^5*b^2 - 5*B*a^
4*b^3 + 35*A*a^3*b^4 + 15*B*a^2*b^5 - (35*A - 16*B)*a*b^6 - (16*A - 5*B)*b^7)*cos(d*x + c)^6*log(sin(d*x + c)
+ 1) - 3*(5*A*a^7 + B*a^6*b - 21*A*a^5*b^2 - 5*B*a^4*b^3 + 35*A*a^3*b^4 + 15*B*a^2*b^5 - (35*A + 16*B)*a*b^6 +
(16*A + 5*B)*b^7)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 48*(B*a^3*b^4 - A*a^2*b^5 - B*a*b^6 + A*b^7)*cos(d*
x + c)^4 - 24*(B*a^5*b^2 - A*a^4*b^3 - 2*B*a^3*b^4 + 2*A*a^2*b^5 + B*a*b^6 - A*b^7)*cos(d*x + c)^2 + 2*(8*A*a^
7 - 8*B*a^6*b - 24*A*a^5*b^2 + 24*B*a^4*b^3 + 24*A*a^3*b^4 - 24*B*a^2*b^5 - 8*A*a*b^6 + 8*B*b^7 + 3*(5*A*a^7 +
B*a^6*b - 21*A*a^5*b^2 - 5*B*a^4*b^3 + 35*A*a^3*b^4 - B*a^2*b^5 - 19*A*a*b^6 + 5*B*b^7)*cos(d*x + c)^4 + 2*(5
*A*a^7 + B*a^6*b - 21*A*a^5*b^2 + 3*B*a^4*b^3 + 27*A*a^3*b^4 - 9*B*a^2*b^5 - 11*A*a*b^6 + 5*B*b^7)*cos(d*x + c
)^2)*sin(d*x + c))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d*cos(d*x + c)^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\frac {96 \, {\left (B a b^{6} - A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (5 \, A a^{3} - {\left (20 \, A - B\right )} a^{2} b + {\left (29 \, A - 4 \, B\right )} a b^{2} - {\left (16 \, A - 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (5 \, A a^{3} + {\left (20 \, A + B\right )} a^{2} b + {\left (29 \, A + 4 \, B\right )} a b^{2} + {\left (16 \, A + 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (8 \, B a^{5} - 8 \, A a^{4} b - 28 \, B a^{3} b^{2} + 28 \, A a^{2} b^{3} + 44 \, B a b^{4} - 44 \, A b^{5} + 3 \, {\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 4 \, B a^{2} b^{3} + 19 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{5} + 24 \, {\left (B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{4} - 8 \, {\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 2 \, B a^{2} b^{3} + 17 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{3} + 12 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} - 5 \, B a b^{4} + 5 \, A b^{5}\right )} \sin \left (d x + c\right )^{2} + 3 \, {\left (11 \, A a^{5} - B a^{4} b - 32 \, A a^{3} b^{2} + 4 \, B a^{2} b^{3} + 29 \, A a b^{4} - 11 \, B b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{6} - a^{6} + 3 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + b^{6} - 3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{2}}}{96 \, d} \]

[In]

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

[Out]

-1/96*(96*(B*a*b^6 - A*b^7)*log(b*sin(d*x + c) + a)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) - 3*(5*A*a
^3 - (20*A - B)*a^2*b + (29*A - 4*B)*a*b^2 - (16*A - 5*B)*b^3)*log(sin(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^
2 - 4*a*b^3 + b^4) + 3*(5*A*a^3 + (20*A + B)*a^2*b + (29*A + 4*B)*a*b^2 + (16*A + 5*B)*b^3)*log(sin(d*x + c) -
1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(8*B*a^5 - 8*A*a^4*b - 28*B*a^3*b^2 + 28*A*a^2*b^3 + 44*B*
a*b^4 - 44*A*b^5 + 3*(5*A*a^5 + B*a^4*b - 16*A*a^3*b^2 - 4*B*a^2*b^3 + 19*A*a*b^4 - 5*B*b^5)*sin(d*x + c)^5 +
24*(B*a*b^4 - A*b^5)*sin(d*x + c)^4 - 8*(5*A*a^5 + B*a^4*b - 16*A*a^3*b^2 - 2*B*a^2*b^3 + 17*A*a*b^4 - 5*B*b^5
)*sin(d*x + c)^3 + 12*(B*a^3*b^2 - A*a^2*b^3 - 5*B*a*b^4 + 5*A*b^5)*sin(d*x + c)^2 + 3*(11*A*a^5 - B*a^4*b - 3
2*A*a^3*b^2 + 4*B*a^2*b^3 + 29*A*a*b^4 - 11*B*b^5)*sin(d*x + c))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sin(d*x
+ c)^6 - a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sin(d*x + c)^4 + 3*(a^6 - 3
*a^4*b^2 + 3*a^2*b^4 - b^6)*sin(d*x + c)^2))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 907 vs. \(2 (373) = 746\).

Time = 0.41 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.37 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\frac {96 \, {\left (B a b^{7} - A b^{8}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} + \frac {3 \, {\left (5 \, A a^{3} + 20 \, A a^{2} b + B a^{2} b + 29 \, A a b^{2} + 4 \, B a b^{2} + 16 \, A b^{3} + 5 \, B b^{3}\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {3 \, {\left (5 \, A a^{3} - 20 \, A a^{2} b + B a^{2} b + 29 \, A a b^{2} - 4 \, B a b^{2} - 16 \, A b^{3} + 5 \, B b^{3}\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (44 \, B a b^{6} \sin \left (d x + c\right )^{6} - 44 \, A b^{7} \sin \left (d x + c\right )^{6} + 15 \, A a^{7} \sin \left (d x + c\right )^{5} + 3 \, B a^{6} b \sin \left (d x + c\right )^{5} - 63 \, A a^{5} b^{2} \sin \left (d x + c\right )^{5} - 15 \, B a^{4} b^{3} \sin \left (d x + c\right )^{5} + 105 \, A a^{3} b^{4} \sin \left (d x + c\right )^{5} - 3 \, B a^{2} b^{5} \sin \left (d x + c\right )^{5} - 57 \, A a b^{6} \sin \left (d x + c\right )^{5} + 15 \, B b^{7} \sin \left (d x + c\right )^{5} + 24 \, B a^{3} b^{4} \sin \left (d x + c\right )^{4} - 24 \, A a^{2} b^{5} \sin \left (d x + c\right )^{4} - 156 \, B a b^{6} \sin \left (d x + c\right )^{4} + 156 \, A b^{7} \sin \left (d x + c\right )^{4} - 40 \, A a^{7} \sin \left (d x + c\right )^{3} - 8 \, B a^{6} b \sin \left (d x + c\right )^{3} + 168 \, A a^{5} b^{2} \sin \left (d x + c\right )^{3} + 24 \, B a^{4} b^{3} \sin \left (d x + c\right )^{3} - 264 \, A a^{3} b^{4} \sin \left (d x + c\right )^{3} + 24 \, B a^{2} b^{5} \sin \left (d x + c\right )^{3} + 136 \, A a b^{6} \sin \left (d x + c\right )^{3} - 40 \, B b^{7} \sin \left (d x + c\right )^{3} + 12 \, B a^{5} b^{2} \sin \left (d x + c\right )^{2} - 12 \, A a^{4} b^{3} \sin \left (d x + c\right )^{2} - 72 \, B a^{3} b^{4} \sin \left (d x + c\right )^{2} + 72 \, A a^{2} b^{5} \sin \left (d x + c\right )^{2} + 192 \, B a b^{6} \sin \left (d x + c\right )^{2} - 192 \, A b^{7} \sin \left (d x + c\right )^{2} + 33 \, A a^{7} \sin \left (d x + c\right ) - 3 \, B a^{6} b \sin \left (d x + c\right ) - 129 \, A a^{5} b^{2} \sin \left (d x + c\right ) + 15 \, B a^{4} b^{3} \sin \left (d x + c\right ) + 183 \, A a^{3} b^{4} \sin \left (d x + c\right ) - 45 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 87 \, A a b^{6} \sin \left (d x + c\right ) + 33 \, B b^{7} \sin \left (d x + c\right ) + 8 \, B a^{7} - 8 \, A a^{6} b - 36 \, B a^{5} b^{2} + 36 \, A a^{4} b^{3} + 72 \, B a^{3} b^{4} - 72 \, A a^{2} b^{5} - 88 \, B a b^{6} + 88 \, A b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{96 \, d} \]

[In]

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

[Out]

-1/96*(96*(B*a*b^7 - A*b^8)*log(abs(b*sin(d*x + c) + a))/(a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9) + 3
*(5*A*a^3 + 20*A*a^2*b + B*a^2*b + 29*A*a*b^2 + 4*B*a*b^2 + 16*A*b^3 + 5*B*b^3)*log(abs(-sin(d*x + c) + 1))/(a
^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 3*(5*A*a^3 - 20*A*a^2*b + B*a^2*b + 29*A*a*b^2 - 4*B*a*b^2 - 16*A*
b^3 + 5*B*b^3)*log(abs(-sin(d*x + c) - 1))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 2*(44*B*a*b^6*sin(d*x
+ c)^6 - 44*A*b^7*sin(d*x + c)^6 + 15*A*a^7*sin(d*x + c)^5 + 3*B*a^6*b*sin(d*x + c)^5 - 63*A*a^5*b^2*sin(d*x
+ c)^5 - 15*B*a^4*b^3*sin(d*x + c)^5 + 105*A*a^3*b^4*sin(d*x + c)^5 - 3*B*a^2*b^5*sin(d*x + c)^5 - 57*A*a*b^6*
sin(d*x + c)^5 + 15*B*b^7*sin(d*x + c)^5 + 24*B*a^3*b^4*sin(d*x + c)^4 - 24*A*a^2*b^5*sin(d*x + c)^4 - 156*B*a
*b^6*sin(d*x + c)^4 + 156*A*b^7*sin(d*x + c)^4 - 40*A*a^7*sin(d*x + c)^3 - 8*B*a^6*b*sin(d*x + c)^3 + 168*A*a^
5*b^2*sin(d*x + c)^3 + 24*B*a^4*b^3*sin(d*x + c)^3 - 264*A*a^3*b^4*sin(d*x + c)^3 + 24*B*a^2*b^5*sin(d*x + c)^
3 + 136*A*a*b^6*sin(d*x + c)^3 - 40*B*b^7*sin(d*x + c)^3 + 12*B*a^5*b^2*sin(d*x + c)^2 - 12*A*a^4*b^3*sin(d*x
+ c)^2 - 72*B*a^3*b^4*sin(d*x + c)^2 + 72*A*a^2*b^5*sin(d*x + c)^2 + 192*B*a*b^6*sin(d*x + c)^2 - 192*A*b^7*si
n(d*x + c)^2 + 33*A*a^7*sin(d*x + c) - 3*B*a^6*b*sin(d*x + c) - 129*A*a^5*b^2*sin(d*x + c) + 15*B*a^4*b^3*sin(
d*x + c) + 183*A*a^3*b^4*sin(d*x + c) - 45*B*a^2*b^5*sin(d*x + c) - 87*A*a*b^6*sin(d*x + c) + 33*B*b^7*sin(d*x
+ c) + 8*B*a^7 - 8*A*a^6*b - 36*B*a^5*b^2 + 36*A*a^4*b^3 + 72*B*a^3*b^4 - 72*A*a^2*b^5 - 88*B*a*b^6 + 88*A*b^
7)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(sin(d*x + c)^2 - 1)^3))/d

Mupad [B] (verification not implemented)

Time = 12.96 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.90 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (5\,A\,a^3+\left (B-20\,A\right )\,a^2\,b+\left (29\,A-4\,B\right )\,a\,b^2+\left (5\,B-16\,A\right )\,b^3\right )}{d\,\left (32\,a^4-128\,a^3\,b+192\,a^2\,b^2-128\,a\,b^3+32\,b^4\right )}-\frac {\frac {-2\,B\,a^5+2\,A\,a^4\,b+7\,B\,a^3\,b^2-7\,A\,a^2\,b^3-11\,B\,a\,b^4+11\,A\,b^5}{12\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (A\,b^5-B\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {\sin \left (c+d\,x\right )\,\left (11\,A\,a^5-B\,a^4\,b-32\,A\,a^3\,b^2+4\,B\,a^2\,b^3+29\,A\,a\,b^4-11\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (B\,a^3\,b^2-A\,a^2\,b^3-5\,B\,a\,b^4+5\,A\,b^5\right )}{4\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (5\,A\,a^5+B\,a^4\,b-16\,A\,a^3\,b^2-2\,B\,a^2\,b^3+17\,A\,a\,b^4-5\,B\,b^5\right )}{6\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (5\,A\,a^5+B\,a^4\,b-16\,A\,a^3\,b^2-4\,B\,a^2\,b^3+19\,A\,a\,b^4-5\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^6+3\,{\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (5\,A\,a^3+\left (20\,A+B\right )\,a^2\,b+\left (29\,A+4\,B\right )\,a\,b^2+\left (16\,A+5\,B\right )\,b^3\right )}{d\,\left (32\,a^4+128\,a^3\,b+192\,a^2\,b^2+128\,a\,b^3+32\,b^4\right )}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (A\,b^7-B\,a\,b^6\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]

[In]

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

[Out]

(log(sin(c + d*x) + 1)*(5*A*a^3 - b^3*(16*A - 5*B) - a^2*b*(20*A - B) + a*b^2*(29*A - 4*B)))/(d*(32*a^4 - 128*
a^3*b - 128*a*b^3 + 32*b^4 + 192*a^2*b^2)) - ((11*A*b^5 - 2*B*a^5 - 7*A*a^2*b^3 + 7*B*a^3*b^2 + 2*A*a^4*b - 11
*B*a*b^4)/(12*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^4*(A*b^5 - B*a*b^4))/(2*(a^6 - b^6 + 3*a^2*
b^4 - 3*a^4*b^2)) - (sin(c + d*x)*(11*A*a^5 - 11*B*b^5 - 32*A*a^3*b^2 + 4*B*a^2*b^3 + 29*A*a*b^4 - B*a^4*b))/(
16*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)^2*(5*A*b^5 - A*a^2*b^3 + B*a^3*b^2 - 5*B*a*b^4))/(4*(a
^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^3*(5*A*a^5 - 5*B*b^5 - 16*A*a^3*b^2 - 2*B*a^2*b^3 + 17*A*a*
b^4 + B*a^4*b))/(6*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)^5*(5*A*a^5 - 5*B*b^5 - 16*A*a^3*b^2 -
4*B*a^2*b^3 + 19*A*a*b^4 + B*a^4*b))/(16*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)))/(d*(cos(c + d*x)^2 - 2*sin(c +
d*x)^2 + 3*sin(c + d*x)^4 - sin(c + d*x)^6)) - (log(sin(c + d*x) - 1)*(5*A*a^3 + b^3*(16*A + 5*B) + a*b^2*(29*
A + 4*B) + a^2*b*(20*A + B)))/(d*(128*a*b^3 + 128*a^3*b + 32*a^4 + 32*b^4 + 192*a^2*b^2)) + (log(a + b*sin(c +
d*x))*(A*b^7 - B*a*b^6))/(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2))