3.5.90 \(\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx\) [490]

Optimal. Leaf size=298 \[ \frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {4 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

4/35*a^2*(c-7*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f-2/7*a^2*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d/f+4/35*a^2*
(c^2-7*c*d-10*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f+4/35*a^2*(c+3*d)*(c^2-10*c*d-7*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^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-4/35*a^2*(c^2-7*c*d-10*d^2)*(c^2-d^2)*(sin(1/2*e+1/4*P
i+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(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^2/f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2842, 2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\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 )}{35 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\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 )}{35 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(35*d*f) + (4*a^2*(c - 7*d)*Cos[e + f*x]*
(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) - (4*a^2*(c + 3
*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(35*d^2*f*Sq
rt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c^2 - 7*c*d - 10*d^2)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(35*d^2*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 2832

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2842

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {2 \int \left (6 a^2 d-a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2} \, dx}{7 d}\\ &=\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{2} a^2 d (9 c+7 d)-\frac {3}{2} a^2 \left (c^2-7 c d-10 d^2\right ) \sin (e+f x)\right ) \, dx}{35 d}\\ &=\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {8 \int \frac {\frac {3}{2} a^2 d \left (13 c^2+14 c d+5 d^2\right )-\frac {3}{4} a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d}\\ &=\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\left (2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{35 d^2}+\frac {\left (2 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{35 d^2}\\ &=\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\left (2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{35 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\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}{35 d^2 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {4 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\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)}}{35 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\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}}}{35 d^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.40, size = 262, normalized size = 0.88 \begin {gather*} \frac {a^2 \left (8 \left (c^4-6 c^3 d-44 c^2 d^2-58 c d^3-21 d^4\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-8 \left (c^4-7 c^3 d-11 c^2 d^2+7 c d^3+10 d^4\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d \cos (e+f x) \left (-4 c^3-112 c^2 d-106 c d^2-28 d^3+2 d^2 (13 c+14 d) \cos (2 (e+f x))-d \left (36 c^2+168 c d+95 d^2\right ) \sin (e+f x)+5 d^3 \sin (3 (e+f x))\right )\right )}{70 d^2 f \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(8*(c^4 - 6*c^3*d - 44*c^2*d^2 - 58*c*d^3 - 21*d^4)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[
(c + d*Sin[e + f*x])/(c + d)] - 8*(c^4 - 7*c^3*d - 11*c^2*d^2 + 7*c*d^3 + 10*d^4)*EllipticF[(-2*e + Pi - 2*f*x
)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(-4*c^3 - 112*c^2*d - 106*c*d^2 - 28*d
^3 + 2*d^2*(13*c + 14*d)*Cos[2*(e + f*x)] - d*(36*c^2 + 168*c*d + 95*d^2)*Sin[e + f*x] + 5*d^3*Sin[3*(e + f*x)
])))/(70*d^2*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. \(1315\) vs. \(2(340)=680\).
time = 4.87, size = 1316, normalized size = 4.42

method	result	size
default	\(\text {Expression too large to display}\)	\(1316\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*a^2*(-13*c*d^4*sin(f*x+e)^4-9*c^2*d^3*sin(f*x+e)^3-42*c*d^4*sin(f*x+e)^3-c^3*d^2*sin(f*x+e)^2-28*c^2*d^3
*sin(f*x+e)^2-7*c*d^4*sin(f*x+e)^2+9*c^2*d^3*sin(f*x+e)+42*c*d^4*sin(f*x+e)+28*((c+d*sin(f*x+e))/(c-d))^(1/2)*
(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d
)/(c+d))^(1/2))*c^2*d^3-74*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(
c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^4-5*d^5*sin(f*x+e)^5-14*d^5*sin(
f*x+e)^4-15*d^5*sin(f*x+e)^3+14*d^5*sin(f*x+e)^2+20*d^5*sin(f*x+e)+28*c^2*d^3+20*c*d^4+c^3*d^2-42*((c+d*sin(f*
x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))
/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^5+62*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(
1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^5-2*((c+d*sin(f*x+e
))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c
-d))^(1/2),((c-d)/(c+d))^(1/2))*c^5-64*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+s
in(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^3+68*((c+d*sin(f*x
+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/
(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^4+14*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*
(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d+2*((c+d*sin(f*
x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))
/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-68*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d
*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2+76*((c+d*si
n(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x
+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2)/d^3/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+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^2*(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 = 639, normalized size = 2.14 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {2} {\left (a^{2} c^{4} - 7 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 21 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\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 ) + 2 \, \sqrt {2} {\left (a^{2} c^{4} - 7 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 21 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\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 \, \sqrt {2} {\left (-i \, a^{2} c^{3} d + 7 i \, a^{2} c^{2} d^{2} + 37 i \, a^{2} c d^{3} + 21 i \, a^{2} d^{4}\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 \, \sqrt {2} {\left (i \, a^{2} c^{3} d - 7 i \, a^{2} c^{2} d^{2} - 37 i \, a^{2} c d^{3} - 21 i \, a^{2} d^{4}\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 (5 \, a^{2} d^{4} \cos \left (f x + e\right )^{3} - 2 \, {\left (4 \, a^{2} c d^{3} + 7 \, a^{2} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a^{2} c^{2} d^{2} + 28 \, a^{2} c d^{3} + 25 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{105 \, d^{3} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(2*sqrt(2)*(a^2*c^4 - 7*a^2*c^3*d + 2*a^2*c^2*d^2 + 21*a^2*c*d^3 + 15*a^2*d^4)*sqrt(I*d)*weierstrassPInv
erse(-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) + 2*sqrt(2)*(a^2*c^4 - 7*a^2*c^3*d + 2*a^2*c^2*d^2 + 21*a^2*c*d^3 + 15*a^2*d^4)*sqrt(-I*d)*weierstras
sPInverse(-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)*(-I*a^2*c^3*d + 7*I*a^2*c^2*d^2 + 37*I*a^2*c*d^3 + 21*I*a^2*d^4)*sqrt(I*d)*weierstr
assZeta(-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)*(I*a^2
*c^3*d - 7*I*a^2*c^2*d^2 - 37*I*a^2*c*d^3 - 21*I*a^2*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*(5*a^2*d^4*cos(f*x + e)^3 - 2*(4*a^2*c*d^3 +
7*a^2*d^4)*cos(f*x + e)*sin(f*x + e) - (a^2*c^2*d^2 + 28*a^2*c*d^3 + 25*a^2*d^4)*cos(f*x + e))*sqrt(d*sin(f*x
+ e) + c))/(d^3*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int c \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 2 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 2 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(c*sqrt(c + d*sin(e + f*x)), x) + Integral(2*c*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integ
ral(c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + In
tegral(2*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3,
x))

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

[Out]

integrate((a*sin(f*x + e) + a)^2*(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 {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\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 + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2),x)

[Out]

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

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