\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx\) [307]

Optimal result

Integrand size = 37, antiderivative size = 284 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 a f}-\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}} \]

[Out]

-(A-B)*(c-d)^3*arctanh(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))*2^(1/2)/f/a^(1/2)-4/105*(7*A*d*(
21*c^2-12*c*d+7*d^2)+B*(36*c^3-63*c^2*d+144*c*d^2-37*d^3))*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-2/35*(7*A*d+6*B
*c-B*d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f*x+e))^(1/2)-2/7*B*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(
f*x+e))^(1/2)-2/105*d*(7*A*(9*c-d)*d+B*(24*c^2-15*c*d+31*d^2))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/a/f

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3062, 3047, 3102, 2830, 2728, 212} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 d \left (7 A d (9 c-d)+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 a f}-\frac {4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {2 (7 A d+6 B c-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}} \]

[In]

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

[Out]

-((Sqrt[2]*(A - B)*(c - d)^3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f))
- (4*(7*A*d*(21*c^2 - 12*c*d + 7*d^2) + B*(36*c^3 - 63*c^2*d + 144*c*d^2 - 37*d^3))*Cos[e + f*x])/(105*f*Sqrt[
a + a*Sin[e + f*x]]) - (2*d*(7*A*(9*c - d)*d + B*(24*c^2 - 15*c*d + 31*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f
*x]])/(105*a*f) - (2*(6*B*c + 7*A*d - B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(35*f*Sqrt[a + a*Sin[e + f*x]]
) - (2*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*f*Sqrt[a + a*Sin[e + f*x]])

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 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 {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {(c+d \sin (e+f x))^2 \left (\frac {1}{2} a (7 A c-B c+6 B d)+\frac {1}{2} a (6 B c+7 A d-B d) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{7 a} \\ & = -\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {4 \int \frac {(c+d \sin (e+f x)) \left (\frac {1}{4} a^2 \left (35 A c^2-11 B c^2-7 A c d+55 B c d+28 A d^2-4 B d^2\right )+\frac {1}{4} a^2 \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{35 a^2} \\ & = -\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} a^2 c \left (35 A c^2-11 B c^2-7 A c d+55 B c d+28 A d^2-4 B d^2\right )+\left (\frac {1}{4} a^2 d \left (35 A c^2-11 B c^2-7 A c d+55 B c d+28 A d^2-4 B d^2\right )+\frac {1}{4} a^2 c \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right )\right ) \sin (e+f x)+\frac {1}{4} a^2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{35 a^2} \\ & = -\frac {2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 a f}-\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {8 \int \frac {-\frac {1}{8} a^3 \left (B \left (33 c^3-189 c^2 d+27 c d^2-31 d^3\right )-7 A \left (15 c^3-3 c^2 d+21 c d^2-d^3\right )\right )+\frac {1}{4} a^3 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{105 a^3} \\ & = -\frac {4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 a f}-\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\left ((A-B) (c-d)^3\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = -\frac {4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 a f}-\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}-\frac {\left (2 (A-B) (c-d)^3\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 )}{f} \\ & = -\frac {\sqrt {2} (A-B) (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (36 c^3-63 c^2 d+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 d \left (7 A (9 c-d) d+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 a f}-\frac {2 (6 B c+7 A d-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.61 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((840+840 i) (-1)^{3/4} (A-B) (c-d)^3 \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )-105 \left (4 A d \left (6 c^2-3 c d+2 d^2\right )+B \left (8 c^3-12 c^2 d+24 c d^2-5 d^3\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-35 d \left (2 A (6 c-d) d+B \left (12 c^2-6 c d+5 d^2\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+21 d^2 (6 B c+2 A d-B d) \cos \left (\frac {5}{2} (e+f x)\right )+15 B d^3 \cos \left (\frac {7}{2} (e+f x)\right )+105 \left (4 A d \left (6 c^2-3 c d+2 d^2\right )+B \left (8 c^3-12 c^2 d+24 c d^2-5 d^3\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )-35 d \left (2 A (6 c-d) d+B \left (12 c^2-6 c d+5 d^2\right )\right ) \sin \left (\frac {3}{2} (e+f x)\right )+21 d^2 (-2 A d+B (-6 c+d)) \sin \left (\frac {5}{2} (e+f x)\right )+15 B d^3 \sin \left (\frac {7}{2} (e+f x)\right )\right )}{420 f \sqrt {a (1+\sin (e+f x))}} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((840 + 840*I)*(-1)^(3/4)*(A - B)*(c - d)^3*ArcTanh[(1/2 + I/2)*(-1)^(3
/4)*(-1 + Tan[(e + f*x)/4])] - 105*(4*A*d*(6*c^2 - 3*c*d + 2*d^2) + B*(8*c^3 - 12*c^2*d + 24*c*d^2 - 5*d^3))*C
os[(e + f*x)/2] - 35*d*(2*A*(6*c - d)*d + B*(12*c^2 - 6*c*d + 5*d^2))*Cos[(3*(e + f*x))/2] + 21*d^2*(6*B*c + 2
*A*d - B*d)*Cos[(5*(e + f*x))/2] + 15*B*d^3*Cos[(7*(e + f*x))/2] + 105*(4*A*d*(6*c^2 - 3*c*d + 2*d^2) + B*(8*c
^3 - 12*c^2*d + 24*c*d^2 - 5*d^3))*Sin[(e + f*x)/2] - 35*d*(2*A*(6*c - d)*d + B*(12*c^2 - 6*c*d + 5*d^2))*Sin[
(3*(e + f*x))/2] + 21*d^2*(-2*A*d + B*(-6*c + d))*Sin[(5*(e + f*x))/2] + 15*B*d^3*Sin[(7*(e + f*x))/2]))/(420*
f*Sqrt[a*(1 + Sin[e + f*x])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(559\) vs. \(2(259)=518\).

Time = 3.56 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.97

method	result	size
parts	\(-\frac {A \,c^{3} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {c^{2} \left (3 d A +B c \right ) \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-2 \sqrt {a -a \sin \left (f x +e \right )}\right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {d^{2} \left (d A +3 B c \right ) \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (15 a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-6 \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}}+10 a \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}-30 a^{2} \sqrt {a -a \sin \left (f x +e \right )}\right )}{15 a^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}-\frac {d^{3} B \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (105 a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-30 \left (a -a \sin \left (f x +e \right )\right )^{\frac {7}{2}}+84 \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}} a -140 a^{2} \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}\right )}{105 a^{4} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {c d \left (d A +B c \right ) \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-3 a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+2 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}\right )}{a^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(560\)
default	\(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (105 A \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c^{3}-315 A \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c^{2} d +315 A \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c \,d^{2}-105 A \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) d^{3}-105 B \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c^{3}+315 B \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c^{2} d -315 B \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c \,d^{2}+105 B \,a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) d^{3}-30 d^{3} B \left (a -a \sin \left (f x +e \right )\right )^{\frac {7}{2}}+42 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}} a \,d^{3}+126 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}} a c \,d^{2}+84 d^{3} B a \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}}-210 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{2} c \,d^{2}-70 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{2} d^{3}-210 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{2} c^{2} d -210 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{2} c \,d^{2}-140 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{2} d^{3}+630 \sqrt {a -a \sin \left (f x +e \right )}\, A \,c^{2} d \,a^{3}+210 \sqrt {a -a \sin \left (f x +e \right )}\, A \,a^{3} d^{3}+210 \sqrt {a -a \sin \left (f x +e \right )}\, B \,c^{3} a^{3}+630 \sqrt {a -a \sin \left (f x +e \right )}\, B \,a^{3} c \,d^{2}\right )}{105 a^{4} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(610\)

[In]

int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-A*c^3*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/
a^(1/2))/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f+c^2*(3*A*d+B*c)*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(a^(1/2)
*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))-2*(a-a*sin(f*x+e))^(1/2))/a/cos(f*x+e)/(a+a*sin(f
*x+e))^(1/2)/f+1/15*d^2*(A*d+3*B*c)*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(15*a^(5/2)*2^(1/2)*arctanh(1/2*(
a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))-6*(a-a*sin(f*x+e))^(5/2)+10*a*(a-a*sin(f*x+e))^(3/2)-30*a^2*(a-a*sin(f*
x+e))^(1/2))/a^3/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f-1/105*d^3*B*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(105
*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))-30*(a-a*sin(f*x+e))^(7/2)+84*(a-a*sin(f*x
+e))^(5/2)*a-140*a^2*(a-a*sin(f*x+e))^(3/2))/a^4/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f+c*d*(A*d+B*c)*(1+sin(f*x+
e))*(-a*(sin(f*x+e)-1))^(1/2)*(-3*a^(3/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))+2*(a-a*s
in(f*x+e))^(3/2))/a^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. 629 vs. \(2 (259) = 518\).

Time = 0.29 (sec) , antiderivative size = 629, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {105 \, \sqrt {2} {\left ({\left (A - B\right )} a c^{3} - 3 \, {\left (A - B\right )} a c^{2} d + 3 \, {\left (A - B\right )} a c d^{2} - {\left (A - B\right )} a d^{3} + {\left ({\left (A - B\right )} a c^{3} - 3 \, {\left (A - B\right )} a c^{2} d + 3 \, {\left (A - B\right )} a c d^{2} - {\left (A - B\right )} a d^{3}\right )} \cos \left (f x + e\right ) + {\left ({\left (A - B\right )} a c^{3} - 3 \, {\left (A - B\right )} a c^{2} d + 3 \, {\left (A - B\right )} a c d^{2} - {\left (A - B\right )} a d^{3}\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\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 )}{\sqrt {a}} + 4 \, {\left (15 \, B d^{3} \cos \left (f x + e\right )^{4} - 105 \, B c^{3} - 105 \, {\left (3 \, A - 2 \, B\right )} c^{2} d + 21 \, {\left (10 \, A - 17 \, B\right )} c d^{2} - {\left (119 \, A - 92 \, B\right )} d^{3} + 3 \, {\left (21 \, B c d^{2} + {\left (7 \, A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (105 \, B c^{2} d + 21 \, {\left (5 \, A - 4 \, B\right )} c d^{2} - 4 \, {\left (7 \, A - 16 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, B c^{3} + 105 \, {\left (3 \, A - B\right )} c^{2} d - 21 \, {\left (5 \, A - 16 \, B\right )} c d^{2} + 2 \, {\left (56 \, A - 23 \, B\right )} d^{3}\right )} \cos \left (f x + e\right ) + {\left (15 \, B d^{3} \cos \left (f x + e\right )^{3} + 105 \, B c^{3} + 105 \, {\left (3 \, A - 2 \, B\right )} c^{2} d - 21 \, {\left (10 \, A - 17 \, B\right )} c d^{2} + {\left (119 \, A - 92 \, B\right )} d^{3} - 3 \, {\left (21 \, B c d^{2} + {\left (7 \, A - 6 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, B c^{2} d + 21 \, {\left (5 \, A - B\right )} c d^{2} - {\left (7 \, A - 46 \, 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}}{210 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/210*(105*sqrt(2)*((A - B)*a*c^3 - 3*(A - B)*a*c^2*d + 3*(A - B)*a*c*d^2 - (A - B)*a*d^3 + ((A - B)*a*c^3 - 3
*(A - B)*a*c^2*d + 3*(A - B)*a*c*d^2 - (A - B)*a*d^3)*cos(f*x + e) + ((A - B)*a*c^3 - 3*(A - B)*a*c^2*d + 3*(A
- B)*a*c*d^2 - (A - B)*a*d^3)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2
)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (
cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) + 4*(15*B*d^3*cos(f*x + e)^4 - 105*B*c^3 - 105*(3*
A - 2*B)*c^2*d + 21*(10*A - 17*B)*c*d^2 - (119*A - 92*B)*d^3 + 3*(21*B*c*d^2 + (7*A - B)*d^3)*cos(f*x + e)^3 -
(105*B*c^2*d + 21*(5*A - 4*B)*c*d^2 - 4*(7*A - 16*B)*d^3)*cos(f*x + e)^2 - (105*B*c^3 + 105*(3*A - B)*c^2*d -
21*(5*A - 16*B)*c*d^2 + 2*(56*A - 23*B)*d^3)*cos(f*x + e) + (15*B*d^3*cos(f*x + e)^3 + 105*B*c^3 + 105*(3*A -
2*B)*c^2*d - 21*(10*A - 17*B)*c*d^2 + (119*A - 92*B)*d^3 - 3*(21*B*c*d^2 + (7*A - 6*B)*d^3)*cos(f*x + e)^2 -
(105*B*c^2*d + 21*(5*A - B)*c*d^2 - (7*A - 46*B)*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a
*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)

Sympy [F]

\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{3}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral((A + B*sin(e + f*x))*(c + d*sin(e + f*x))**3/sqrt(a*(sin(e + f*x) + 1)), x)

Maxima [F]

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

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (259) = 518\).

Time = 0.41 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {105 \, \sqrt {2} {\left (A \sqrt {a} c^{3} - B \sqrt {a} c^{3} - 3 \, A \sqrt {a} c^{2} d + 3 \, B \sqrt {a} c^{2} d + 3 \, A \sqrt {a} c d^{2} - 3 \, B \sqrt {a} c d^{2} - A \sqrt {a} d^{3} + 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 \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {105 \, \sqrt {2} {\left (A \sqrt {a} c^{3} - B \sqrt {a} c^{3} - 3 \, A \sqrt {a} c^{2} d + 3 \, B \sqrt {a} c^{2} d + 3 \, A \sqrt {a} c d^{2} - 3 \, B \sqrt {a} c d^{2} - A \sqrt {a} d^{3} + 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 \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, \sqrt {2} {\left (120 \, B a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 252 \, B a^{\frac {13}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 84 \, A a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 168 \, B a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 210 \, B a^{\frac {13}{2}} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 210 \, A a^{\frac {13}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 210 \, B a^{\frac {13}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 70 \, A a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 140 \, B a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 105 \, B a^{\frac {13}{2}} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 315 \, A a^{\frac {13}{2}} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 315 \, B a^{\frac {13}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 105 \, A a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{7} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{210 \, f} \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

1/210*(105*sqrt(2)*(A*sqrt(a)*c^3 - B*sqrt(a)*c^3 - 3*A*sqrt(a)*c^2*d + 3*B*sqrt(a)*c^2*d + 3*A*sqrt(a)*c*d^2
- 3*B*sqrt(a)*c*d^2 - A*sqrt(a)*d^3 + B*sqrt(a)*d^3)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a*sgn(cos(-1/4*p
i + 1/2*f*x + 1/2*e))) - 105*sqrt(2)*(A*sqrt(a)*c^3 - B*sqrt(a)*c^3 - 3*A*sqrt(a)*c^2*d + 3*B*sqrt(a)*c^2*d +
3*A*sqrt(a)*c*d^2 - 3*B*sqrt(a)*c*d^2 - A*sqrt(a)*d^3 + B*sqrt(a)*d^3)*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1
)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 4*sqrt(2)*(120*B*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 2
52*B*a^(13/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 84*A*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 16
8*B*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 210*B*a^(13/2)*c^2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 21
0*A*a^(13/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 210*B*a^(13/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 +
70*A*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 140*B*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 105
*B*a^(13/2)*c^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 315*A*a^(13/2)*c^2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 315*B*a
^(13/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 105*A*a^(13/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/(a^7*sgn(cos
(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

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

[In]

int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^3)/(a + a*sin(e + f*x))^(1/2),x)

[Out]

int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^3)/(a + a*sin(e + f*x))^(1/2), x)