3.8.41 \(\int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx\) [741]

Optimal. Leaf size=361 \[ \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)+2/3*b*(6*a*b*c*d-3*a^2*d^2-b^2
*(4*c^2-d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/(c^2-d^2)/f+2/3*(9*a^2*b*c*d^2-3*a^3*d^3-9*a*b^2*d*(2*c^2-
d^2)+b^3*(8*c^3-5*c*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/
4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^3/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2
/3*b*(18*a*b*c*d-9*a^2*d^2-b^2*(8*c^2+d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Elli
pticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))
^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2871, 3102, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 f \left (c^2-d^2\right )}-\frac {2 b \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b*(6*a*b*c
*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)*f) - (2*(9*a^2*b
*c*d^2 - 3*a^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c +
d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(18*a*b*c*d - 9
*a^2*d^2 - b^2*(8*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])
/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

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

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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

Rule 2831

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

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/
(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

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*} \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (2 b (b c-a d)^2-a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac {1}{2} \left (a^2 b c d-b^3 c d-a^3 d^2-a b^2 \left (2 c^2-3 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{d \left (c^2-d^2\right )}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {4 \int \frac {-\frac {1}{4} d \left (3 a^3 c d+9 a b^2 c d-9 a^2 b d^2-b^3 \left (2 c^2+d^2\right )\right )+\frac {1}{4} \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^2 \left (c^2-d^2\right )}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^3}-\frac {\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d^3 \left (c^2-d^2\right )}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {\left (\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 d^3 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 2.15, size = 311, normalized size = 0.86 \begin {gather*} \frac {2 \left (\frac {\left (d^2 \left (-3 a^3 c d-9 a b^2 c d+9 a^2 b d^2+b^3 \left (2 c^2+d^2\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+\left (9 a^2 b c d^2-3 a^3 d^3+9 a b^2 d \left (-2 c^2+d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d) (c+d)}-\frac {d \cos (e+f x) \left (9 a b^2 c^2 d-9 a^2 b c d^2+3 a^3 d^3+b^3 \left (-4 c^3+c d^2\right )+b^3 d \left (-c^2+d^2\right ) \sin (e+f x)\right )}{-c^2+d^2}\right )}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((d^2*(-3*a^3*c*d - 9*a*b^2*c*d + 9*a^2*b*d^2 + b^3*(2*c^2 + d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/
(c + d)] + (9*a^2*b*c*d^2 - 3*a^3*d^3 + 9*a*b^2*d*(-2*c^2 + d^2) + b^3*(8*c^3 - 5*c*d^2))*((c + d)*EllipticE[(
-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e +
f*x])/(c + d)])/((c - d)*(c + d)) - (d*Cos[e + f*x]*(9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 3*a^3*d^3 + b^3*(-4*c^3
+ c*d^2) + b^3*d*(-c^2 + d^2)*Sin[e + f*x]))/(-c^2 + d^2)))/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1397\) vs. \(2(409)=818\).
time = 23.59, size = 1398, normalized size = 3.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b/d^3*(d^2*b^2*(-2/3/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3*(1
/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3/d*c*(1/d*c-1
)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+
e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(
((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+2*(3*a*b*d^2-b^2*c*d)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))
^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*
((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(
1/2),((c-d)/(c+d))^(1/2)))+6*a^2*d^2*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(
(-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2
),((c-d)/(c+d))^(1/2))-6*a*b*c*d*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-si
n(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))+2*b^2*c^2*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*
x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)
/(c+d))^(1/2)))+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^3*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)
-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*
((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/
2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*
((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e)
)/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e
)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.20, size = 1083, normalized size = 3.00 \begin {gather*} \frac {{\left (\sqrt {2} {\left (16 \, b^{3} c^{4} d - 36 \, a b^{2} c^{3} d^{2} + 2 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )} c^{2} d^{3} + 3 \, {\left (a^{3} + 15 \, a b^{2}\right )} c d^{4} - 3 \, {\left (9 \, a^{2} b + b^{3}\right )} d^{5}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (16 \, b^{3} c^{5} - 36 \, a b^{2} c^{4} d + 2 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )} c^{3} d^{2} + 3 \, {\left (a^{3} + 15 \, a b^{2}\right )} c^{2} d^{3} - 3 \, {\left (9 \, a^{2} b + b^{3}\right )} c d^{4}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (16 \, b^{3} c^{4} d - 36 \, a b^{2} c^{3} d^{2} + 2 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )} c^{2} d^{3} + 3 \, {\left (a^{3} + 15 \, a b^{2}\right )} c d^{4} - 3 \, {\left (9 \, a^{2} b + b^{3}\right )} d^{5}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (16 \, b^{3} c^{5} - 36 \, a b^{2} c^{4} d + 2 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )} c^{3} d^{2} + 3 \, {\left (a^{3} + 15 \, a b^{2}\right )} c^{2} d^{3} - 3 \, {\left (9 \, a^{2} b + b^{3}\right )} c d^{4}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (\sqrt {2} {\left (8 i \, b^{3} c^{3} d^{2} - 18 i \, a b^{2} c^{2} d^{3} + i \, {\left (9 \, a^{2} b - 5 \, b^{3}\right )} c d^{4} - 3 i \, {\left (a^{3} - 3 \, a b^{2}\right )} d^{5}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (8 i \, b^{3} c^{4} d - 18 i \, a b^{2} c^{3} d^{2} + i \, {\left (9 \, a^{2} b - 5 \, b^{3}\right )} c^{2} d^{3} - 3 i \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (-8 i \, b^{3} c^{3} d^{2} + 18 i \, a b^{2} c^{2} d^{3} - i \, {\left (9 \, a^{2} b - 5 \, b^{3}\right )} c d^{4} + 3 i \, {\left (a^{3} - 3 \, a b^{2}\right )} d^{5}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-8 i \, b^{3} c^{4} d + 18 i \, a b^{2} c^{3} d^{2} - i \, {\left (9 \, a^{2} b - 5 \, b^{3}\right )} c^{2} d^{3} + 3 i \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 6 \, {\left ({\left (b^{3} c^{2} d^{3} - b^{3} d^{5}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} - 3 \, a^{3} d^{5} + {\left (9 \, a^{2} b - b^{3}\right )} c d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{9 \, {\left ({\left (c^{2} d^{5} - d^{7}\right )} f \sin \left (f x + e\right ) + {\left (c^{3} d^{4} - c d^{6}\right )} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*((sqrt(2)*(16*b^3*c^4*d - 36*a*b^2*c^3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15*a*b^2)*c*d^4 - 3*(9
*a^2*b + b^3)*d^5)*sin(f*x + e) + sqrt(2)*(16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 8*b^3)*c^3*d^2 + 3*(a^3
+ 15*a*b^2)*c^2*d^3 - 3*(9*a^2*b + b^3)*c*d^4))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*
(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(16*b^3*c^4*d - 3
6*a*b^2*c^3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15*a*b^2)*c*d^4 - 3*(9*a^2*b + b^3)*d^5)*sin(f*x + e)
+ sqrt(2)*(16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 8*b^3)*c^3*d^2 + 3*(a^3 + 15*a*b^2)*c^2*d^3 - 3*(9*a^2*
b + b^3)*c*d^4))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/
3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(8*I*b^3*c^3*d^2 - 18*I*a*b^2*c^2*d^3 + I*(9
*a^2*b - 5*b^3)*c*d^4 - 3*I*(a^3 - 3*a*b^2)*d^5)*sin(f*x + e) + sqrt(2)*(8*I*b^3*c^4*d - 18*I*a*b^2*c^3*d^2 +
I*(9*a^2*b - 5*b^3)*c^2*d^3 - 3*I*(a^3 - 3*a*b^2)*c*d^4))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2,
-8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3
, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(-8*I*b^3*c^3*d^2 + 18*I*a*b^2*c^2*d^3
- I*(9*a^2*b - 5*b^3)*c*d^4 + 3*I*(a^3 - 3*a*b^2)*d^5)*sin(f*x + e) + sqrt(2)*(-8*I*b^3*c^4*d + 18*I*a*b^2*c^3
*d^2 - I*(9*a^2*b - 5*b^3)*c^2*d^3 + 3*I*(a^3 - 3*a*b^2)*c*d^4))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^
2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*
c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 6*((b^3*c^2*d^3 - b^3*d^5)*cos(f*x + e)*
sin(f*x + e) + (4*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 - 3*a^3*d^5 + (9*a^2*b - b^3)*c*d^4)*cos(f*x + e))*sqrt(d*sin(
f*x + e) + c))/((c^2*d^5 - d^7)*f*sin(f*x + e) + (c^3*d^4 - c*d^6)*f)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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