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

Optimal. Leaf size=284 \[ -\frac{2 d \left (7 A d (9 c-d)+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 a f}-\frac{4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (-63 c^2 d+36 c^3+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+6 B c-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (A-B) (c-d)^3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}} \]

[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]])

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Rubi [A]  time = 1.00144, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{2 d \left (7 A d (9 c-d)+B \left (24 c^2-15 c d+31 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 a f}-\frac{4 \left (7 A d \left (21 c^2-12 c d+7 d^2\right )+B \left (-63 c^2 d+36 c^3+144 c d^2-37 d^3\right )\right ) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+6 B c-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (A-B) (c-d)^3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[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 2983

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] && (I
ntegerQ[n] || EqQ[m + 1/2, 0])

Rule 2968

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 3023

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]

Rule 2751

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*Sin
[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 2649

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 206

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

Rubi steps

\begin{align*} \int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt{a+a \sin (e+f x)}} \, dx &=-\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 ) \operatorname{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 \tanh ^{-1}\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]  time = 0.87627, size = 375, normalized size = 1.32 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (105 \left (4 A d \left (6 c^2-3 c d+2 d^2\right )+B \left (-12 c^2 d+8 c^3+24 c d^2-5 d^3\right )\right ) \sin \left (\frac{1}{2} (e+f x)\right )-35 d \left (2 A d (6 c-d)+B \left (12 c^2-6 c d+5 d^2\right )\right ) \sin \left (\frac{3}{2} (e+f x)\right )-105 \left (4 A d \left (6 c^2-3 c d+2 d^2\right )+B \left (-12 c^2 d+8 c^3+24 c d^2-5 d^3\right )\right ) \cos \left (\frac{1}{2} (e+f x)\right )-35 d \left (2 A d (6 c-d)+B \left (12 c^2-6 c d+5 d^2\right )\right ) \cos \left (\frac{3}{2} (e+f x)\right )+21 d^2 (B (d-6 c)-2 A d) \sin \left (\frac{5}{2} (e+f x)\right )+21 d^2 (2 A d+6 B c-B d) \cos \left (\frac{5}{2} (e+f x)\right )+(840+840 i) (-1)^{3/4} (A-B) (c-d)^3 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (e+f x)\right )-1\right )\right )+15 B d^3 \sin \left (\frac{7}{2} (e+f x)\right )+15 B d^3 \cos \left (\frac{7}{2} (e+f x)\right )\right )}{420 f \sqrt{a (\sin (e+f x)+1)}} \]

Antiderivative was successfully verified.

[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])])

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Maple [B]  time = 1.624, size = 610, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(105*A*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(
1/2)/a^(1/2))*c^3-315*A*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*c^2*d+315*A*a^(7/2
)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*c*d^2-105*A*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin
(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*d^3-105*B*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*
c^3+315*B*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*c^2*d-315*B*a^(7/2)*2^(1/2)*arct
anh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*c*d^2+105*B*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)
*2^(1/2)/a^(1/2))*d^3-30*B*d^3*(a-a*sin(f*x+e))^(7/2)+42*A*(a-a*sin(f*x+e))^(5/2)*a*d^3+126*B*(a-a*sin(f*x+e))
^(5/2)*a*c*d^2+84*B*(a-a*sin(f*x+e))^(5/2)*a*d^3-210*A*(a-a*sin(f*x+e))^(3/2)*a^2*c*d^2-70*A*(a-a*sin(f*x+e))^
(3/2)*a^2*d^3-210*B*(a-a*sin(f*x+e))^(3/2)*a^2*c^2*d-210*B*(a-a*sin(f*x+e))^(3/2)*a^2*c*d^2-140*B*(a-a*sin(f*x
+e))^(3/2)*a^2*d^3+630*A*c^2*d*a^3*(a-a*sin(f*x+e))^(1/2)+210*A*a^3*d^3*(a-a*sin(f*x+e))^(1/2)+210*B*c^3*a^3*(
a-a*sin(f*x+e))^(1/2)+630*B*a^3*c*d^2*(a-a*sin(f*x+e))^(1/2))/a^4/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.97671, size = 1543, normalized size = 5.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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Giac [B]  time = 2.12646, size = 2520, normalized size = 8.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/105*(210*sqrt(2)*(A*c^3 - B*c^3 - 3*A*c^2*d + 3*B*c^2*d + 3*A*c*d^2 - 3*B*c*d^2 - A*d^3 + B*d^3)*arctan(-1/2
*sqrt(2)*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a) + sqrt(a))/sqrt(-a))/(sqrt(-a)*sgn
(tan(1/2*f*x + 1/2*e) + 1)) + ((((((((105*B*a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1) + 315*A*a^3*c^2*d*sgn(tan(1/
2*f*x + 1/2*e) + 1) - 105*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 105*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e)
+ 1) + 273*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 91*A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - 43*B*a^3*
d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))*tan(1/2*f*x + 1/2*e)/a^12 - 105*(B*a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1) +
3*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 3*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 3*A*a^3*c*d^2*sgn(
tan(1/2*f*x + 1/2*e) + 1) + 3*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) +
1) - B*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)*tan(1/2*f*x + 1/2*e) + 7*(45*B*a^3*c^3*sgn(tan(1/2*f*x +
1/2*e) + 1) + 135*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 75*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 7
5*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 159*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 53*A*a^3*d^3*sgn
(tan(1/2*f*x + 1/2*e) + 1) - 29*B*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)*tan(1/2*f*x + 1/2*e) - 35*(9*B*
a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1) + 27*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 21*B*a^3*c^2*d*sgn(tan(
1/2*f*x + 1/2*e) + 1) - 21*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 33*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e)
+ 1) + 11*A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - 11*B*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)*tan(1/2
*f*x + 1/2*e) + 35*(9*B*a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1) + 27*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) -
21*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 21*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 33*B*a^3*c*d^2*
sgn(tan(1/2*f*x + 1/2*e) + 1) + 11*A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - 11*B*a^3*d^3*sgn(tan(1/2*f*x + 1/
2*e) + 1))/a^12)*tan(1/2*f*x + 1/2*e) - 7*(45*B*a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1) + 135*A*a^3*c^2*d*sgn(ta
n(1/2*f*x + 1/2*e) + 1) - 75*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 75*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*
e) + 1) + 159*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 53*A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - 29*B*a^
3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)*tan(1/2*f*x + 1/2*e) + 105*(B*a^3*c^3*sgn(tan(1/2*f*x + 1/2*e) + 1)
+ 3*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 3*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 3*A*a^3*c*d^2*s
gn(tan(1/2*f*x + 1/2*e) + 1) + 3*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + A*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e
) + 1) - B*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)*tan(1/2*f*x + 1/2*e) - (105*B*a^3*c^3*sgn(tan(1/2*f*x
+ 1/2*e) + 1) + 315*A*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1) - 105*B*a^3*c^2*d*sgn(tan(1/2*f*x + 1/2*e) + 1)
- 105*A*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 273*B*a^3*c*d^2*sgn(tan(1/2*f*x + 1/2*e) + 1) + 91*A*a^3*d^3
*sgn(tan(1/2*f*x + 1/2*e) + 1) - 43*B*a^3*d^3*sgn(tan(1/2*f*x + 1/2*e) + 1))/a^12)/(a*tan(1/2*f*x + 1/2*e)^2 +
a)^(7/2) - (210*sqrt(2)*A*a^13*c^3*arctan(sqrt(a)/sqrt(-a)) - 210*sqrt(2)*B*a^13*c^3*arctan(sqrt(a)/sqrt(-a))
- 630*sqrt(2)*A*a^13*c^2*d*arctan(sqrt(a)/sqrt(-a)) + 630*sqrt(2)*B*a^13*c^2*d*arctan(sqrt(a)/sqrt(-a)) + 630
*sqrt(2)*A*a^13*c*d^2*arctan(sqrt(a)/sqrt(-a)) - 630*sqrt(2)*B*a^13*c*d^2*arctan(sqrt(a)/sqrt(-a)) - 210*sqrt(
2)*A*a^13*d^3*arctan(sqrt(a)/sqrt(-a)) + 210*sqrt(2)*B*a^13*d^3*arctan(sqrt(a)/sqrt(-a)) - 105*sqrt(2)*B*sqrt(
-a)*sqrt(a)*c^3 - 315*sqrt(2)*A*sqrt(-a)*sqrt(a)*c^2*d + 210*sqrt(2)*B*sqrt(-a)*sqrt(a)*c^2*d + 210*sqrt(2)*A*
sqrt(-a)*sqrt(a)*c*d^2 - 357*sqrt(2)*B*sqrt(-a)*sqrt(a)*c*d^2 - 119*sqrt(2)*A*sqrt(-a)*sqrt(a)*d^3 + 92*sqrt(2
)*B*sqrt(-a)*sqrt(a)*d^3)*sgn(tan(1/2*f*x + 1/2*e) + 1)/(sqrt(-a)*a^13))/f