\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx\) [314]

Optimal result

Integrand size = 37, antiderivative size = 283 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {(c-d)^2 (3 B (c-5 d)+A (c+11 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d \left (15 A c^2-99 B c^2-120 A c d+168 B c d+65 A d^2-93 B d^2\right ) \cos (e+f x)}{15 a f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (15 A c-51 B c-35 A d+39 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{30 a^2 f}+\frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}} \]

[Out]

-1/2*(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(f*x+e))^(3/2)-1/4*(c-d)^2*(3*B*(c-5*d)+A*(c+11*d))*arctanh
(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(3/2)/f*2^(1/2)+1/15*d*(15*A*c^2-120*A*c*d+65*A*d^2-
99*B*c^2+168*B*c*d-93*B*d^2)*cos(f*x+e)/a/f/(a+a*sin(f*x+e))^(1/2)+1/10*(5*A-9*B)*d*cos(f*x+e)*(c+d*sin(f*x+e)
)^2/a/f/(a+a*sin(f*x+e))^(1/2)+1/30*d^2*(15*A*c-35*A*d-51*B*c+39*B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/a^2/f

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3056, 3062, 3047, 3102, 2830, 2728, 212} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {(c-d)^2 (A (c+11 d)+3 B (c-5 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d^2 (15 A c-35 A d-51 B c+39 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{30 a^2 f}+\frac {d \left (15 A c^2-120 A c d+65 A d^2-99 B c^2+168 B c d-93 B d^2\right ) \cos (e+f x)}{15 a f \sqrt {a \sin (e+f x)+a}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac {d (5 A-9 B) \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a \sin (e+f x)+a}} \]

[In]

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

[Out]

-1/2*((c - d)^2*(3*B*(c - 5*d) + A*(c + 11*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]
])])/(Sqrt[2]*a^(3/2)*f) + (d*(15*A*c^2 - 99*B*c^2 - 120*A*c*d + 168*B*c*d + 65*A*d^2 - 93*B*d^2)*Cos[e + f*x]
)/(15*a*f*Sqrt[a + a*Sin[e + f*x]]) + (d^2*(15*A*c - 51*B*c - 35*A*d + 39*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e +
f*x]])/(30*a^2*f) + ((5*A - 9*B)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(10*a*f*Sqrt[a + a*Sin[e + f*x]]) - (
(A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(2*f*(a + a*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 2830

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

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3062

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*
(m + n + 1))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c
*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] &&
(IntegerQ[n] || EqQ[m + 1/2, 0])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {(c+d \sin (e+f x))^2 \left (\frac {1}{2} a (3 B (c-2 d)+A (c+6 d))-\frac {1}{2} a (5 A-9 B) d \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2} \\ & = \frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {(c+d \sin (e+f x)) \left (\frac {1}{4} a^2 \left (5 A \left (c^2+7 c d-4 d^2\right )+3 B \left (5 c^2-13 c d+12 d^2\right )\right )-\frac {1}{4} a^2 d (15 A c-51 B c-35 A d+39 B d) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{5 a^3} \\ & = \frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\frac {1}{4} a^2 c \left (5 A \left (c^2+7 c d-4 d^2\right )+3 B \left (5 c^2-13 c d+12 d^2\right )\right )+\left (-\frac {1}{4} a^2 c d (15 A c-51 B c-35 A d+39 B d)+\frac {1}{4} a^2 d \left (5 A \left (c^2+7 c d-4 d^2\right )+3 B \left (5 c^2-13 c d+12 d^2\right )\right )\right ) \sin (e+f x)-\frac {1}{4} a^2 d^2 (15 A c-51 B c-35 A d+39 B d) \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{5 a^3} \\ & = \frac {d^2 (15 A c-51 B c-35 A d+39 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{30 a^2 f}+\frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {2 \int \frac {\frac {1}{8} a^3 \left (3 B \left (15 c^3-39 c^2 d+53 c d^2-13 d^3\right )+5 A \left (3 c^3+21 c^2 d-15 c d^2+7 d^3\right )\right )-\frac {1}{4} a^3 d \left (15 A c^2-99 B c^2-120 A c d+168 B c d+65 A d^2-93 B d^2\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{15 a^4} \\ & = \frac {d \left (15 A c^2-99 B c^2-120 A c d+168 B c d+65 A d^2-93 B d^2\right ) \cos (e+f x)}{15 a f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (15 A c-51 B c-35 A d+39 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{30 a^2 f}+\frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\left ((c-d)^2 (3 B (c-5 d)+A (c+11 d))\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a} \\ & = \frac {d \left (15 A c^2-99 B c^2-120 A c d+168 B c d+65 A d^2-93 B d^2\right ) \cos (e+f x)}{15 a f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (15 A c-51 B c-35 A d+39 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{30 a^2 f}+\frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\left ((c-d)^2 (3 B (c-5 d)+A (c+11 d))\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a f} \\ & = -\frac {(c-d)^2 (3 B (c-5 d)+A (c+11 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d \left (15 A c^2-99 B c^2-120 A c d+168 B c d+65 A d^2-93 B d^2\right ) \cos (e+f x)}{15 a f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (15 A c-51 B c-35 A d+39 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{30 a^2 f}+\frac {(5 A-9 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.77 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.42 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-30 A c^3 \cos \left (\frac {1}{2} (e+f x)\right )+30 B c^3 \cos \left (\frac {1}{2} (e+f x)\right )+90 A c^2 d \cos \left (\frac {1}{2} (e+f x)\right )-270 B c^2 d \cos \left (\frac {1}{2} (e+f x)\right )-270 A c d^2 \cos \left (\frac {1}{2} (e+f x)\right )+330 B c d^2 \cos \left (\frac {1}{2} (e+f x)\right )+110 A d^3 \cos \left (\frac {1}{2} (e+f x)\right )-165 B d^3 \cos \left (\frac {1}{2} (e+f x)\right )-180 B c^2 d \cos \left (\frac {3}{2} (e+f x)\right )-180 A c d^2 \cos \left (\frac {3}{2} (e+f x)\right )+210 B c d^2 \cos \left (\frac {3}{2} (e+f x)\right )+70 A d^3 \cos \left (\frac {3}{2} (e+f x)\right )-123 B d^3 \cos \left (\frac {3}{2} (e+f x)\right )+30 B c d^2 \cos \left (\frac {5}{2} (e+f x)\right )+10 A d^3 \cos \left (\frac {5}{2} (e+f x)\right )-9 B d^3 \cos \left (\frac {5}{2} (e+f x)\right )+3 B d^3 \cos \left (\frac {7}{2} (e+f x)\right )+30 A c^3 \sin \left (\frac {1}{2} (e+f x)\right )-30 B c^3 \sin \left (\frac {1}{2} (e+f x)\right )-90 A c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+270 B c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+270 A c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-330 B c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-110 A d^3 \sin \left (\frac {1}{2} (e+f x)\right )+165 B d^3 \sin \left (\frac {1}{2} (e+f x)\right )+(30+30 i) (-1)^{3/4} (c-d)^2 (3 B (c-5 d)+A (c+11 d)) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \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 )^2-180 B c^2 d \sin \left (\frac {3}{2} (e+f x)\right )-180 A c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+210 B c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+70 A d^3 \sin \left (\frac {3}{2} (e+f x)\right )-123 B d^3 \sin \left (\frac {3}{2} (e+f x)\right )-30 B c d^2 \sin \left (\frac {5}{2} (e+f x)\right )-10 A d^3 \sin \left (\frac {5}{2} (e+f x)\right )+9 B d^3 \sin \left (\frac {5}{2} (e+f x)\right )+3 B d^3 \sin \left (\frac {7}{2} (e+f x)\right )\right )}{60 f (a (1+\sin (e+f x)))^{3/2}} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-30*A*c^3*Cos[(e + f*x)/2] + 30*B*c^3*Cos[(e + f*x)/2] + 90*A*c^2*d*Co
s[(e + f*x)/2] - 270*B*c^2*d*Cos[(e + f*x)/2] - 270*A*c*d^2*Cos[(e + f*x)/2] + 330*B*c*d^2*Cos[(e + f*x)/2] +
110*A*d^3*Cos[(e + f*x)/2] - 165*B*d^3*Cos[(e + f*x)/2] - 180*B*c^2*d*Cos[(3*(e + f*x))/2] - 180*A*c*d^2*Cos[(
3*(e + f*x))/2] + 210*B*c*d^2*Cos[(3*(e + f*x))/2] + 70*A*d^3*Cos[(3*(e + f*x))/2] - 123*B*d^3*Cos[(3*(e + f*x
))/2] + 30*B*c*d^2*Cos[(5*(e + f*x))/2] + 10*A*d^3*Cos[(5*(e + f*x))/2] - 9*B*d^3*Cos[(5*(e + f*x))/2] + 3*B*d
^3*Cos[(7*(e + f*x))/2] + 30*A*c^3*Sin[(e + f*x)/2] - 30*B*c^3*Sin[(e + f*x)/2] - 90*A*c^2*d*Sin[(e + f*x)/2]
+ 270*B*c^2*d*Sin[(e + f*x)/2] + 270*A*c*d^2*Sin[(e + f*x)/2] - 330*B*c*d^2*Sin[(e + f*x)/2] - 110*A*d^3*Sin[(
e + f*x)/2] + 165*B*d^3*Sin[(e + f*x)/2] + (30 + 30*I)*(-1)^(3/4)*(c - d)^2*(3*B*(c - 5*d) + A*(c + 11*d))*Arc
Tanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 180*B*c^2*d*Sin
[(3*(e + f*x))/2] - 180*A*c*d^2*Sin[(3*(e + f*x))/2] + 210*B*c*d^2*Sin[(3*(e + f*x))/2] + 70*A*d^3*Sin[(3*(e +
f*x))/2] - 123*B*d^3*Sin[(3*(e + f*x))/2] - 30*B*c*d^2*Sin[(5*(e + f*x))/2] - 10*A*d^3*Sin[(5*(e + f*x))/2] +
9*B*d^3*Sin[(5*(e + f*x))/2] + 3*B*d^3*Sin[(7*(e + f*x))/2]))/(60*f*(a*(1 + Sin[e + f*x]))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(816\) vs. \(2(256)=512\).

Time = 3.18 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.89

method	result	size
parts	\(\text {Expression too large to display}\)	\(817\)
default	\(\text {Expression too large to display}\)	\(1030\)

[In]

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

[Out]

-1/4*A*c^3/a^(7/2)*(2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*sin(f*x+e)+2*(a-a*sin(f*x+
e))^(1/2)*a^(3/2)+2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2)*(-a*(sin(f*x+e)-1))^(1/2)/c
os(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f+1/4*c^2*(3*A*d+B*c)/a^(5/2)*(-3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*
2^(1/2)/a^(1/2))*a*sin(f*x+e)-3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a+2*(a-a*sin(f*x+e
))^(1/2)*a^(1/2))*(-a*(sin(f*x+e)-1))^(1/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f-1/12*d^2*(A*d+3*B*c)*(-8*(a-a*
sin(f*x+e))^(3/2)*sin(f*x+e)*a^(1/2)-24*(a-a*sin(f*x+e))^(1/2)*sin(f*x+e)*a^(3/2)+33*2^(1/2)*arctanh(1/2*(a-a*
sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*sin(f*x+e)-8*(a-a*sin(f*x+e))^(3/2)*a^(1/2)-30*(a-a*sin(f*x+e))^(1/2)*a
^(3/2)+33*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2)*(-a*(sin(f*x+e)-1))^(1/2)/a^(7/2)/c
os(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f+1/20*d^3*B*(-8*(a-a*sin(f*x+e))^(5/2)*a^(1/2)*sin(f*x+e)-80*(a-a*sin(f*x+e)
)^(1/2)*a^(5/2)*sin(f*x+e)-8*(a-a*sin(f*x+e))^(5/2)*a^(1/2)-90*(a-a*sin(f*x+e))^(1/2)*a^(5/2)+75*2^(1/2)*arcta
nh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^3+75*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^
(1/2)/a^(1/2))*a^3)*(-a*(sin(f*x+e)-1))^(1/2)/a^(9/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f-3/4*c*d*(A*d+B*c)/a^
(5/2)*(-7*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a*sin(f*x+e)+8*(a-a*sin(f*x+e))^(1/2)*si
n(f*x+e)*a^(1/2)-7*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a+10*(a-a*sin(f*x+e))^(1/2)*a^(
1/2))*(-a*(sin(f*x+e)-1))^(1/2)/cos(f*x+e)/(a+a*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. 784 vs. \(2 (256) = 512\).

Time = 0.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.77 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {15 \, \sqrt {2} {\left (2 \, {\left (A + 3 \, B\right )} c^{3} + 6 \, {\left (3 \, A - 7 \, B\right )} c^{2} d - 6 \, {\left (7 \, A - 11 \, B\right )} c d^{2} + 2 \, {\left (11 \, A - 15 \, B\right )} d^{3} - {\left ({\left (A + 3 \, B\right )} c^{3} + 3 \, {\left (3 \, A - 7 \, B\right )} c^{2} d - 3 \, {\left (7 \, A - 11 \, B\right )} c d^{2} + {\left (11 \, A - 15 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left ({\left (A + 3 \, B\right )} c^{3} + 3 \, {\left (3 \, A - 7 \, B\right )} c^{2} d - 3 \, {\left (7 \, A - 11 \, B\right )} c d^{2} + {\left (11 \, A - 15 \, B\right )} d^{3}\right )} \cos \left (f x + e\right ) + {\left (2 \, {\left (A + 3 \, B\right )} c^{3} + 6 \, {\left (3 \, A - 7 \, B\right )} c^{2} d - 6 \, {\left (7 \, A - 11 \, B\right )} c d^{2} + 2 \, {\left (11 \, A - 15 \, B\right )} d^{3} + {\left ({\left (A + 3 \, B\right )} c^{3} + 3 \, {\left (3 \, A - 7 \, B\right )} c^{2} d - 3 \, {\left (7 \, A - 11 \, B\right )} c d^{2} + {\left (11 \, A - 15 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\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 (12 \, B d^{3} \cos \left (f x + e\right )^{4} - 15 \, {\left (A - B\right )} c^{3} + 45 \, {\left (A - B\right )} c^{2} d - 45 \, {\left (A - B\right )} c d^{2} + 15 \, {\left (A - B\right )} d^{3} + 4 \, {\left (15 \, B c d^{2} + {\left (5 \, A - 3 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 4 \, {\left (45 \, B c^{2} d + 15 \, {\left (3 \, A - 4 \, B\right )} c d^{2} - 4 \, {\left (5 \, A - 9 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 15 \, {\left ({\left (A - B\right )} c^{3} - 3 \, {\left (A - 5 \, B\right )} c^{2} d + 15 \, {\left (A - B\right )} c d^{2} - {\left (5 \, A - 9 \, B\right )} d^{3}\right )} \cos \left (f x + e\right ) + {\left (12 \, B d^{3} \cos \left (f x + e\right )^{3} + 15 \, {\left (A - B\right )} c^{3} - 45 \, {\left (A - B\right )} c^{2} d + 45 \, {\left (A - B\right )} c d^{2} - 15 \, {\left (A - B\right )} d^{3} - 4 \, {\left (15 \, B c d^{2} + {\left (5 \, A - 6 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 60 \, {\left (3 \, B c^{2} d + 3 \, {\left (A - B\right )} c d^{2} - {\left (A - 2 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{120 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

1/120*(15*sqrt(2)*(2*(A + 3*B)*c^3 + 6*(3*A - 7*B)*c^2*d - 6*(7*A - 11*B)*c*d^2 + 2*(11*A - 15*B)*d^3 - ((A +
3*B)*c^3 + 3*(3*A - 7*B)*c^2*d - 3*(7*A - 11*B)*c*d^2 + (11*A - 15*B)*d^3)*cos(f*x + e)^2 + ((A + 3*B)*c^3 + 3
*(3*A - 7*B)*c^2*d - 3*(7*A - 11*B)*c*d^2 + (11*A - 15*B)*d^3)*cos(f*x + e) + (2*(A + 3*B)*c^3 + 6*(3*A - 7*B)
*c^2*d - 6*(7*A - 11*B)*c*d^2 + 2*(11*A - 15*B)*d^3 + ((A + 3*B)*c^3 + 3*(3*A - 7*B)*c^2*d - 3*(7*A - 11*B)*c*
d^2 + (11*A - 15*B)*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x
+ e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e)
+ 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(12*B*d^3*cos(f*x + e)^4 - 1
5*(A - B)*c^3 + 45*(A - B)*c^2*d - 45*(A - B)*c*d^2 + 15*(A - B)*d^3 + 4*(15*B*c*d^2 + (5*A - 3*B)*d^3)*cos(f*
x + e)^3 - 4*(45*B*c^2*d + 15*(3*A - 4*B)*c*d^2 - 4*(5*A - 9*B)*d^3)*cos(f*x + e)^2 - 15*((A - B)*c^3 - 3*(A -
5*B)*c^2*d + 15*(A - B)*c*d^2 - (5*A - 9*B)*d^3)*cos(f*x + e) + (12*B*d^3*cos(f*x + e)^3 + 15*(A - B)*c^3 - 4
5*(A - B)*c^2*d + 45*(A - B)*c*d^2 - 15*(A - B)*d^3 - 4*(15*B*c*d^2 + (5*A - 6*B)*d^3)*cos(f*x + e)^2 - 60*(3*
B*c^2*d + 3*(A - B)*c*d^2 - (A - 2*B)*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*
x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (256) = 512\).

Time = 0.40 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\frac {15 \, \sqrt {2} {\left (A \sqrt {a} c^{3} + 3 \, B \sqrt {a} c^{3} + 9 \, A \sqrt {a} c^{2} d - 21 \, B \sqrt {a} c^{2} d - 21 \, A \sqrt {a} c d^{2} + 33 \, B \sqrt {a} c d^{2} + 11 \, A \sqrt {a} d^{3} - 15 \, B \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {15 \, \sqrt {2} {\left (A \sqrt {a} c^{3} + 3 \, B \sqrt {a} c^{3} + 9 \, A \sqrt {a} c^{2} d - 21 \, B \sqrt {a} c^{2} d - 21 \, A \sqrt {a} c d^{2} + 33 \, B \sqrt {a} c d^{2} + 11 \, A \sqrt {a} d^{3} - 15 \, B \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {30 \, \sqrt {2} {\left (A \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, A \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, B \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {16 \, \sqrt {2} {\left (12 \, B a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 30 \, B a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, A a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, B a^{\frac {17}{2}} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 45 \, A a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, B a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, A a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, B a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{120 \, f} \]

[In]

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

[Out]

1/120*(15*sqrt(2)*(A*sqrt(a)*c^3 + 3*B*sqrt(a)*c^3 + 9*A*sqrt(a)*c^2*d - 21*B*sqrt(a)*c^2*d - 21*A*sqrt(a)*c*d
^2 + 33*B*sqrt(a)*c*d^2 + 11*A*sqrt(a)*d^3 - 15*B*sqrt(a)*d^3)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a^2*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 15*sqrt(2)*(A*sqrt(a)*c^3 + 3*B*sqrt(a)*c^3 + 9*A*sqrt(a)*c^2*d - 21*B*sq
rt(a)*c^2*d - 21*A*sqrt(a)*c*d^2 + 33*B*sqrt(a)*c*d^2 + 11*A*sqrt(a)*d^3 - 15*B*sqrt(a)*d^3)*log(-sin(-1/4*pi
+ 1/2*f*x + 1/2*e) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 30*sqrt(2)*(A*sqrt(a)*c^3*sin(-1/4*pi + 1/
2*f*x + 1/2*e) - B*sqrt(a)*c^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3*A*sqrt(a)*c^2*d*sin(-1/4*pi + 1/2*f*x + 1/2*
e) + 3*B*sqrt(a)*c^2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 3*A*sqrt(a)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3*B
*sqrt(a)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - A*sqrt(a)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) + B*sqrt(a)*d^3*s
in(-1/4*pi + 1/2*f*x + 1/2*e))/((sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))
) + 16*sqrt(2)*(12*B*a^(17/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 30*B*a^(17/2)*c*d^2*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)^3 - 10*A*a^(17/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 45*B*a^(17/2)*c^2*d*sin(-1/4*pi + 1/2*f*x +
1/2*e) + 45*A*a^(17/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 45*B*a^(17/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2
*e) - 15*A*a^(17/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 30*B*a^(17/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/(a^
10*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

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

[In]

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

[Out]

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