[next] [tail] [up]

3.2.1 \(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx\) [101]

Optimal. Leaf size=182 \[ \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(3/2)+14/3*a^2*(g*cos(f*x+e))^(5/2)/c/f/g
/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-14*a^2*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*Ellipt
icE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)/c/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+
e))^(1/2)

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Rubi [A]
time = 0.54, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2930, 2921, 2721, 2719} \begin {gather*} \frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (14*a^2*(g*Cos[e + f*
x])^(5/2))/(3*c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*C
os[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2921

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

Rule 2929

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

Rule 2930

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

Rubi steps

\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {(7 a) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {\left (7 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {\left (7 a^2 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {\left (7 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.18, size = 207, normalized size = 1.14 \begin {gather*} -\frac {2 (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-21 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {\cos (e+f x)} \left (\cos \left (\frac {1}{2} (e+f x)\right ) (12+\cos (e+f x))-(-12+\cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) (a (1+\sin (e+f x)))^{3/2}}{3 c f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(-21*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*
x)/2] - Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2]*(12 + Cos[e + f*x]) - (-12 + Cos[e + f*x])*Si
n[(e + f*x)/2]))*(a*(1 + Sin[e + f*x]))^(3/2))/(3*c*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
^3*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])

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Maple [C] Result contains complex when optimal does not.
time = 0.21, size = 2892, normalized size = 15.89

method result size
default \(\text {Expression too large to display}\) \(2892\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/f*(-1+cos(f*x+e))*(21*I*cos(f*x+e)*sin(f*x+e)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*
EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x
+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^4*(-cos(f*x+e)/(1+cos(f*
x+e))^2)^(3/2)-3*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*
x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^4*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-cos(f*x+e)^3*s
in(f*x+e)-21*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^2*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(
1/2)*(1/(1+cos(f*x+e)))^(1/2)-21*I*cos(f*x+e)*sin(f*x+e)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^
(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-9*cos(f*x+e)^2*sin(f*x+e)+21*I*EllipticF(I*(-1+cos(f*x+e))/sin
(f*x+e),I)*cos(f*x+e)^2*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)+12*ln(-2*(2*cos(
f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)
-1)/sin(f*x+e)^2)*cos(f*x+e)^3*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-12*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos
(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)
^3*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+18*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+
e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x
+e))^2)^(3/2)-18*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*
x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+12*ln(-2*(2*co
s(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/
2)-1)/sin(f*x+e)^2)*cos(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-3*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+co
s(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x
+e)/(1+cos(f*x+e))^2)^(3/2)*sin(f*x+e)-12*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^
2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^
2)^(3/2)+3*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(
1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sin(f*x+e)+42*I*cos(f*x+e)^2*(cos
(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+cos(f*x+e)))^(1/2)-42*I*cos(f*x+
e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+21*I
*cos(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+cos(f*x+e)))^(1/
2)-21*I*cos(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(
f*x+e),I)+21*I*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+
cos(f*x+e)))^(1/2)-21*I*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(-
1+cos(f*x+e))/sin(f*x+e),I)+33*cos(f*x+e)^2-8*cos(f*x+e)^3+3*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))
^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+co
s(f*x+e))^2)^(3/2)-3*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-co
s(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-9*cos(f*x+e)*sin(f*x+e)
*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*
x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+9*cos(f*x+e)*sin(f*x+e)*ln(-(2*cos(f*x+e)
^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/si
n(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-3*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-
cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^3*sin(f*x+e)*(-co
s(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+3*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos
(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^3*sin(f*x+e)*(-cos(f*x+e)/(1+cos(f*
x+e))^2)^(3/2)-9*cos(f*x+e)^2*sin(f*x+e)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)
^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+9
*cos(f*x+e)^2*sin(f*x+e)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*
(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+cos(f*x+e)^4)*(g*co
s(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(3/2)/(cos(f*x+e)^3-cos(f*x+e)^2*sin(f*x+e)-3*cos(f*x+e)^2-2*cos(f*x+e)*sin
(f*x+e)-2*cos(f*x+e)+4*sin(f*x+e)+4)/(-c*(sin(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((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(3/2)/(-c*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 = 0.13, size = 185, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (a g \sin \left (f x + e\right ) - 13 \, a g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 21 \, {\left (-i \, \sqrt {2} a g \sin \left (f x + e\right ) + i \, \sqrt {2} a g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 21 \, {\left (i \, \sqrt {2} a g \sin \left (f x + e\right ) - i \, \sqrt {2} a g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{3 \, {\left (c^{2} f \sin \left (f x + e\right ) - c^{2} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*(a*g*sin(f*x + e) - 13*a*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) - 2
1*(-I*sqrt(2)*a*g*sin(f*x + e) + I*sqrt(2)*a*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0,
cos(f*x + e) + I*sin(f*x + e))) - 21*(I*sqrt(2)*a*g*sin(f*x + e) - I*sqrt(2)*a*g)*sqrt(a*c*g)*weierstrassZeta(
-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(c^2*f*sin(f*x + e) - c^2*f)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(3/2)/(c-c*sin(f*x+e))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8010 deep

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Giac [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((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\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(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(3/2),x)

[Out]

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

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