\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx\) [107]

Optimal result

Integrand size = 38, antiderivative size = 266 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a^3 (3 A-17 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

1/10*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(17/2)+1/80*a^3*(3*A-17*B)*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e)
)^(13/2)-1/96*a^3*(3*A-17*B)*cos(f*x+e)^3/c/f/(c-c*sin(f*x+e))^(9/2)+1/128*a^3*(3*A-17*B)*cos(f*x+e)/c^3/f/(c-
c*sin(f*x+e))^(5/2)-1/512*a^3*(3*A-17*B)*cos(f*x+e)/c^4/f/(c-c*sin(f*x+e))^(3/2)-1/1024*a^3*(3*A-17*B)*arctanh
(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))/c^(11/2)/f*2^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3046, 2938, 2759, 2729, 2728, 212} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a^3 (3 A-17 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 c (3 A-17 B) \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}} \]

[In]

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

-1/512*(a^3*(3*A - 17*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*c^(11/2)
*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(10*f*(c - c*Sin[e + f*x])^(17/2)) + (a^3*(3*A - 17*B)*c*Cos[e + f*x]^5
)/(80*f*(c - c*Sin[e + f*x])^(13/2)) - (a^3*(3*A - 17*B)*Cos[e + f*x]^3)/(96*c*f*(c - c*Sin[e + f*x])^(9/2)) +
(a^3*(3*A - 17*B)*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (a^3*(3*A - 17*B)*Cos[e + f*x])/(512
*c^4*f*(c - c*Sin[e + f*x])^(3/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2938

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p
+ 1))), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {1}{20} \left (a^3 (3 A-17 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{15/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {1}{32} \left (a^3 (3 A-17 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx}{64 c^2} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{256 c^4} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{1024 c^5} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (a^3 (3 A-17 B)\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{512 c^5 f} \\ & = -\frac {a^3 (3 A-17 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 15.20 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.54 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (56370 A \cos \left (\frac {1}{2} (e+f x)\right )+38970 B \cos \left (\frac {1}{2} (e+f x)\right )-31140 A \cos \left (\frac {3}{2} (e+f x)\right )-38580 B \cos \left (\frac {3}{2} (e+f x)\right )-10404 A \cos \left (\frac {5}{2} (e+f x)\right )-12724 B \cos \left (\frac {5}{2} (e+f x)\right )+435 A \cos \left (\frac {7}{2} (e+f x)\right )+7775 B \cos \left (\frac {7}{2} (e+f x)\right )-45 A \cos \left (\frac {9}{2} (e+f x)\right )+255 B \cos \left (\frac {9}{2} (e+f x)\right )+(240+240 i) \sqrt [4]{-1} (3 A-17 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}+56370 A \sin \left (\frac {1}{2} (e+f x)\right )+38970 B \sin \left (\frac {1}{2} (e+f x)\right )+31140 A \sin \left (\frac {3}{2} (e+f x)\right )+38580 B \sin \left (\frac {3}{2} (e+f x)\right )-10404 A \sin \left (\frac {5}{2} (e+f x)\right )-12724 B \sin \left (\frac {5}{2} (e+f x)\right )-435 A \sin \left (\frac {7}{2} (e+f x)\right )-7775 B \sin \left (\frac {7}{2} (e+f x)\right )-45 A \sin \left (\frac {9}{2} (e+f x)\right )+255 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{122880 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}} \]

[In]

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(56370*A*Cos[(e + f*x)/2] + 38970*B*Cos[(e + f
*x)/2] - 31140*A*Cos[(3*(e + f*x))/2] - 38580*B*Cos[(3*(e + f*x))/2] - 10404*A*Cos[(5*(e + f*x))/2] - 12724*B*
Cos[(5*(e + f*x))/2] + 435*A*Cos[(7*(e + f*x))/2] + 7775*B*Cos[(7*(e + f*x))/2] - 45*A*Cos[(9*(e + f*x))/2] +
255*B*Cos[(9*(e + f*x))/2] + (240 + 240*I)*(-1)^(1/4)*(3*A - 17*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e +
f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10 + 56370*A*Sin[(e + f*x)/2] + 38970*B*Sin[(e + f*x)/2] + 31
140*A*Sin[(3*(e + f*x))/2] + 38580*B*Sin[(3*(e + f*x))/2] - 10404*A*Sin[(5*(e + f*x))/2] - 12724*B*Sin[(5*(e +
f*x))/2] - 435*A*Sin[(7*(e + f*x))/2] - 7775*B*Sin[(7*(e + f*x))/2] - 45*A*Sin[(9*(e + f*x))/2] + 255*B*Sin[(
9*(e + f*x))/2]))/(122880*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(235)=470\).

Time = 5.05 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {a^{3} \left (15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-75 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )-180 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+300 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+240 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \sin \left (f x +e \right )-90 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+840 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}+3072 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}-3360 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}+1440 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}+510 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+5480 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}-17408 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}+19040 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}-8160 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}-720 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}+4080 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{15360 c^{\frac {23}{2}} \left (\sin \left (f x +e \right )-1\right )^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(526\)
parts	\(\text {Expression too large to display}\)	\(1807\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE)

[Out]

1/15360*a^3*(15*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^4*sin(f*
x+e)-75*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^4-180*2^(1/2)*ar
ctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^2*sin(f*x+e)+300*2^(1/2)*arctanh(1
/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^2+240*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e)
)^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*sin(f*x+e)-90*A*(c+c*sin(f*x+e))^(9/2)*c^(3/2)+840*A*(c+c*sin(f*x+e))^
(7/2)*c^(5/2)+3072*A*(c+c*sin(f*x+e))^(5/2)*c^(7/2)-3360*A*(c+c*sin(f*x+e))^(3/2)*c^(9/2)+1440*A*(c+c*sin(f*x+
e))^(1/2)*c^(11/2)+510*B*(c+c*sin(f*x+e))^(9/2)*c^(3/2)+5480*B*(c+c*sin(f*x+e))^(7/2)*c^(5/2)-17408*B*(c+c*sin
(f*x+e))^(5/2)*c^(7/2)+19040*B*(c+c*sin(f*x+e))^(3/2)*c^(9/2)-8160*B*(c+c*sin(f*x+e))^(1/2)*c^(11/2)-720*A*2^(
1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6+4080*B*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)
*2^(1/2)/c^(1/2))*c^6)*(c*(1+sin(f*x+e)))^(1/2)/c^(23/2)/(sin(f*x+e)-1)^4/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (235) = 470\).

Time = 0.29 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.86 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {15 \, \sqrt {2} {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 18 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 20 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 48 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} + {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 6 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 12 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) + 32 \, {\left (3 \, A - 17 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (39 \, A + 803 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 4 \, {\left (609 \, A + 389 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (449 \, A + 869 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (143 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3} + {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 80 \, {\left (3 \, A + 47 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (223 \, A + 443 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (113 \, A + 213 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{30720 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

-1/30720*(15*sqrt(2)*((3*A - 17*B)*a^3*cos(f*x + e)^6 - 5*(3*A - 17*B)*a^3*cos(f*x + e)^5 - 18*(3*A - 17*B)*a^
3*cos(f*x + e)^4 + 20*(3*A - 17*B)*a^3*cos(f*x + e)^3 + 48*(3*A - 17*B)*a^3*cos(f*x + e)^2 - 16*(3*A - 17*B)*a
^3*cos(f*x + e) - 32*(3*A - 17*B)*a^3 + ((3*A - 17*B)*a^3*cos(f*x + e)^5 + 6*(3*A - 17*B)*a^3*cos(f*x + e)^4 -
12*(3*A - 17*B)*a^3*cos(f*x + e)^3 - 32*(3*A - 17*B)*a^3*cos(f*x + e)^2 + 16*(3*A - 17*B)*a^3*cos(f*x + e) +
32*(3*A - 17*B)*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c
)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x +
e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(15*(3*A - 17*B)*a^3*cos(f*x + e)^5 - 5*(39*A
+ 803*B)*a^3*cos(f*x + e)^4 + 4*(609*A + 389*B)*a^3*cos(f*x + e)^3 + 12*(449*A + 869*B)*a^3*cos(f*x + e)^2 -
24*(143*A + 43*B)*a^3*cos(f*x + e) - 6144*(A + B)*a^3 + (15*(3*A - 17*B)*a^3*cos(f*x + e)^4 + 80*(3*A + 47*B)*
a^3*cos(f*x + e)^3 + 12*(223*A + 443*B)*a^3*cos(f*x + e)^2 - 24*(113*A + 213*B)*a^3*cos(f*x + e) - 6144*(A + B
)*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(
f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos
(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x +
e) + 32*c^6*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3/(-c*sin(f*x + e) + c)^(11/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (235) = 470\).

Time = 0.56 (sec) , antiderivative size = 990, normalized size of antiderivative = 3.72 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, algorithm="giac")

[Out]

-1/245760*(120*sqrt(2)*(3*A*a^3*sqrt(c) - 17*B*a^3*sqrt(c))*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/
4*pi + 1/2*f*x + 1/2*e) + 1))/(c^6*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*(6*A*a^3*sqrt(c) + 6*B*a^3*s
qrt(c) + 15*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 75*B*a^3
*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 30*A*a^3*sqrt(c)*(cos(-1/
4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 290*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f
*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 120*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e)
- 1)^3/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 360*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(
-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 60*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2
*f*x + 1/2*e) + 1)^4 - 900*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e
) + 1)^4 + 822*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5 - 4
658*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5)*(cos(-1/4*pi
+ 1/2*f*x + 1/2*e) + 1)^5/(c^6*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + s
qrt(2)*(60*A*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 900*B*a^
3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 120*A*a^3*c^(49/2)*(cos
(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 360*B*a^3*c^(49/2)*(cos(-1/4*pi +
1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 30*A*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/
2*e) - 1)^3/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 290*B*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3
/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 15*A*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*p
i + 1/2*f*x + 1/2*e) + 1)^4 + 75*B*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x
+ 1/2*e) + 1)^4 + 6*A*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)
^5 + 6*B*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5)/(c^30*sgn
(sin(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(11/2),x)

[Out]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(11/2), x)