3.1551 \(\int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx\)

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

[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))*ln(1-sin(d*x+c))/(a+b)^4/d+1/32*(5*a^3*A-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-a*B-(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

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Rubi [A]  time = 0.68, antiderivative size = 383, 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} \[ \frac {b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {\left (a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac {\left (-a^2 b (20 A-B)+5 a^3 A+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^4}-\frac {\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right )}+\frac {\sec ^4(c+d x) \left (\left (5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right ) \sin (c+d x)+6 b^2 (A b-a B)\right )}{24 d \left (a^2-b^2\right )^2}-\frac {\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (-16 a^3 A b^2+5 a^5 A-4 a^2 b^3 B+a^4 b B+19 a A b^4-5 b^5 B\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]])/(32*(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)*Si
n[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 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)} \, dx &=\frac {b^7 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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*}

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Mathematica [A]  time = 2.49, size = 565, normalized size = 1.48 \[ \frac {\frac {768 b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}-\frac {48 \left (5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {48 \left (5 a^3 A+a^2 b (B-20 A)+a b^2 (29 A-4 B)+b^3 (5 B-16 A)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}+\frac {\sec ^6(c+d x) \left (198 a^5 A \sin (c+d x)+85 a^5 A \sin (3 (c+d x))+15 a^5 A \sin (5 (c+d x))+128 a^5 B-128 a^4 A b-114 a^4 b B \sin (c+d x)+17 a^4 b B \sin (3 (c+d x))+3 a^4 b B \sin (5 (c+d x))-480 a^3 A b^2 \sin (c+d x)-272 a^3 A b^2 \sin (3 (c+d x))-48 a^3 A b^2 \sin (5 (c+d x))-352 a^3 b^2 B+352 a^2 A b^3-96 b^2 \left (a^2-3 b^2\right ) (a B-A b) \cos (2 (c+d x))+264 a^2 b^3 B \sin (c+d x)-4 a^2 b^3 B \sin (3 (c+d x))-12 a^2 b^3 B \sin (5 (c+d x))-48 b^4 (A b-a B) \cos (4 (c+d x))+330 a A b^4 \sin (c+d x)+259 a A b^4 \sin (3 (c+d x))+57 a A b^4 \sin (5 (c+d x))+368 a b^4 B-368 A b^5-198 b^5 B \sin (c+d x)-85 b^5 B \sin (3 (c+d x))-15 b^5 B \sin (5 (c+d x))\right )}{\left (a^2-b^2\right )^3}}{768 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-48*(5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x
)/2]])/(a + b)^4 + (48*(5*a^3*A + a*b^2*(29*A - 4*B) + a^2*b*(-20*A + B) + b^3*(-16*A + 5*B))*Log[Cos[(c + d*x
)/2] + Sin[(c + d*x)/2]])/(a - b)^4 + (768*b^6*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 + (Sec[c + d
*x]^6*(-128*a^4*A*b + 352*a^2*A*b^3 - 368*A*b^5 + 128*a^5*B - 352*a^3*b^2*B + 368*a*b^4*B - 96*b^2*(a^2 - 3*b^
2)*(-(A*b) + a*B)*Cos[2*(c + d*x)] - 48*b^4*(A*b - a*B)*Cos[4*(c + d*x)] + 198*a^5*A*Sin[c + d*x] - 480*a^3*A*
b^2*Sin[c + d*x] + 330*a*A*b^4*Sin[c + d*x] - 114*a^4*b*B*Sin[c + d*x] + 264*a^2*b^3*B*Sin[c + d*x] - 198*b^5*
B*Sin[c + d*x] + 85*a^5*A*Sin[3*(c + d*x)] - 272*a^3*A*b^2*Sin[3*(c + d*x)] + 259*a*A*b^4*Sin[3*(c + d*x)] + 1
7*a^4*b*B*Sin[3*(c + d*x)] - 4*a^2*b^3*B*Sin[3*(c + d*x)] - 85*b^5*B*Sin[3*(c + d*x)] + 15*a^5*A*Sin[5*(c + d*
x)] - 48*a^3*A*b^2*Sin[5*(c + d*x)] + 57*a*A*b^4*Sin[5*(c + d*x)] + 3*a^4*b*B*Sin[5*(c + d*x)] - 12*a^2*b^3*B*
Sin[5*(c + d*x)] - 15*b^5*B*Sin[5*(c + d*x)]))/(a^2 - b^2)^3)/(768*d)

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fricas [A]  time = 4.32, size = 643, normalized size = 1.68 \[ \frac {16 \, B a^{7} - 16 \, A a^{6} b - 48 \, B a^{5} b^{2} + 48 \, A a^{4} b^{3} + 48 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5} - 16 \, B a b^{6} + 16 \, A b^{7} - 96 \, {\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (b \sin \left (d x + c\right ) + a\right ) + 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} - {\left (35 \, A - 16 \, B\right )} a b^{6} - {\left (16 \, A - 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} - {\left (35 \, A + 16 \, B\right )} a b^{6} + {\left (16 \, A + 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, {\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} - 24 \, {\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, A a^{7} - 8 \, B a^{6} b - 24 \, A a^{5} b^{2} + 24 \, B a^{4} b^{3} + 24 \, A a^{3} b^{4} - 24 \, B a^{2} b^{5} - 8 \, A a b^{6} + 8 \, B b^{7} + 3 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} - B a^{2} b^{5} - 19 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} + 3 \, B a^{4} b^{3} + 27 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} - 11 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{96 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{6}} \]

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)),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)

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giac [B]  time = 0.32, size = 907, normalized size = 2.37 \[ -\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} \]

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)),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

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maple [B]  time = 0.50, size = 990, normalized size = 2.58 \[ \text {result too large to display} \]

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)),x)

[Out]

-1/2/d/(a-b)^4*ln(1+sin(d*x+c))*A*b^3+5/32/d/(a-b)^4*ln(1+sin(d*x+c))*B*b^3-5/32/d/(a+b)^3/(sin(d*x+c)-1)*a^2*
A+1/32/d/(a-b)^2/(1+sin(d*x+c))^2*a*B-1/32/d/(a+b)^4*ln(sin(d*x+c)-1)*B*a^2*b-29/32/d/(a+b)^4*ln(sin(d*x+c)-1)
*A*a*b^2-5/32/d/(a+b)^4*ln(sin(d*x+c)-1)*B*b^3-5/32/d/(a-b)^3/(1+sin(d*x+c))*a^2*A-11/32/d/(a-b)^3/(1+sin(d*x+
c))*A*b^2+1/32/d/(a-b)^3/(1+sin(d*x+c))*B*a^2-1/8/d/(a+b)^3/(sin(d*x+c)-1)*B*a*b-1/3/d/(16*a+16*b)/(sin(d*x+c)
-1)^3*A-1/3/d/(16*a+16*b)/(sin(d*x+c)-1)^3*B-1/3/d/(16*a-16*b)/(1+sin(d*x+c))^3*A+1/3/d/(16*a-16*b)/(1+sin(d*x
+c))^3*B+3/32/d/(a+b)^2/(sin(d*x+c)-1)^2*A*b-5/32/d/(a+b)^3/(sin(d*x+c)-1)*B*b^2-7/16/d/(a+b)^3/(sin(d*x+c)-1)
*A*a*b+5/32/d/(a-b)^4*ln(1+sin(d*x+c))*a^3*A-1/d*b^6/(a+b)^4/(a-b)^4*ln(a+b*sin(d*x+c))*a*B-1/8/d/(a+b)^4*ln(s
in(d*x+c)-1)*B*a*b^2+1/32/d/(a-b)^4*ln(1+sin(d*x+c))*B*a^2*b-5/8/d/(a+b)^4*ln(sin(d*x+c)-1)*A*a^2*b+1/d*b^7/(a
+b)^4/(a-b)^4*ln(a+b*sin(d*x+c))*A-5/8/d/(a-b)^4*ln(1+sin(d*x+c))*A*a^2*b-1/16/d/(a-b)^2/(1+sin(d*x+c))^2*a*A+
3/32/d/(a-b)^2/(1+sin(d*x+c))^2*A*b+7/16/d/(a-b)^3/(1+sin(d*x+c))*A*a*b+29/32/d/(a-b)^4*ln(1+sin(d*x+c))*A*a*b
^2-1/8/d/(a-b)^3/(1+sin(d*x+c))*B*a*b-1/8/d/(a-b)^4*ln(1+sin(d*x+c))*B*a*b^2+5/32/d/(a-b)^3/(1+sin(d*x+c))*B*b
^2-1/32/d/(a+b)^3/(sin(d*x+c)-1)*B*a^2+1/16/d/(a+b)^2/(sin(d*x+c)-1)^2*B*b+1/16/d/(a+b)^2/(sin(d*x+c)-1)^2*a*A
-5/32/d/(a+b)^4*ln(sin(d*x+c)-1)*a^3*A+1/32/d/(a+b)^2/(sin(d*x+c)-1)^2*a*B-11/32/d/(a+b)^3/(sin(d*x+c)-1)*A*b^
2-1/16/d/(a-b)^2/(1+sin(d*x+c))^2*B*b-1/2/d/(a+b)^4*ln(sin(d*x+c)-1)*A*b^3

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maxima [A]  time = 0.45, size = 632, normalized size = 1.65 \[ -\frac {\frac {96 \, {\left (B a b^{6} - A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (5 \, A a^{3} - {\left (20 \, A - B\right )} a^{2} b + {\left (29 \, A - 4 \, B\right )} a b^{2} - {\left (16 \, A - 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (5 \, A a^{3} + {\left (20 \, A + B\right )} a^{2} b + {\left (29 \, A + 4 \, B\right )} a b^{2} + {\left (16 \, A + 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (8 \, B a^{5} - 8 \, A a^{4} b - 28 \, B a^{3} b^{2} + 28 \, A a^{2} b^{3} + 44 \, B a b^{4} - 44 \, A b^{5} + 3 \, {\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 4 \, B a^{2} b^{3} + 19 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{5} + 24 \, {\left (B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{4} - 8 \, {\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 2 \, B a^{2} b^{3} + 17 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{3} + 12 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} - 5 \, B a b^{4} + 5 \, A b^{5}\right )} \sin \left (d x + c\right )^{2} + 3 \, {\left (11 \, A a^{5} - B a^{4} b - 32 \, A a^{3} b^{2} + 4 \, B a^{2} b^{3} + 29 \, A a b^{4} - 11 \, B b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{6} - a^{6} + 3 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + b^{6} - 3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{2}}}{96 \, d} \]

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)),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

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mupad [B]  time = 13.44, size = 729, normalized size = 1.90 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (5\,A\,a^3+\left (B-20\,A\right )\,a^2\,b+\left (29\,A-4\,B\right )\,a\,b^2+\left (5\,B-16\,A\right )\,b^3\right )}{d\,\left (32\,a^4-128\,a^3\,b+192\,a^2\,b^2-128\,a\,b^3+32\,b^4\right )}-\frac {\frac {-2\,B\,a^5+2\,A\,a^4\,b+7\,B\,a^3\,b^2-7\,A\,a^2\,b^3-11\,B\,a\,b^4+11\,A\,b^5}{12\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (A\,b^5-B\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {\sin \left (c+d\,x\right )\,\left (11\,A\,a^5-B\,a^4\,b-32\,A\,a^3\,b^2+4\,B\,a^2\,b^3+29\,A\,a\,b^4-11\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (B\,a^3\,b^2-A\,a^2\,b^3-5\,B\,a\,b^4+5\,A\,b^5\right )}{4\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (5\,A\,a^5+B\,a^4\,b-16\,A\,a^3\,b^2-2\,B\,a^2\,b^3+17\,A\,a\,b^4-5\,B\,b^5\right )}{6\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (5\,A\,a^5+B\,a^4\,b-16\,A\,a^3\,b^2-4\,B\,a^2\,b^3+19\,A\,a\,b^4-5\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^6+3\,{\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (5\,A\,a^3+\left (20\,A+B\right )\,a^2\,b+\left (29\,A+4\,B\right )\,a\,b^2+\left (16\,A+5\,B\right )\,b^3\right )}{d\,\left (32\,a^4+128\,a^3\,b+192\,a^2\,b^2+128\,a\,b^3+32\,b^4\right )}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (A\,b^7-B\,a\,b^6\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]

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))),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))

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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)),x)

[Out]

Timed out

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