3.2.21 \(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx\) [121]

Optimal. Leaf size=298 \[ \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {462 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/5*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(5/2)-12*a^2*(g*cos(f*x+e))^(5/2)*(a+a*
sin(f*x+e))^(3/2)/c/f/g/(c-c*sin(f*x+e))^(3/2)-154/5*a^4*(g*cos(f*x+e))^(5/2)/c^2/f/g/(a+a*sin(f*x+e))^(1/2)/(
c-c*sin(f*x+e))^(1/2)+462/5*a^4*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(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^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-66/5*a^3*(
g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c^2/f/g/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.93, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {462 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 c^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}-\frac {12 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{c f g (c-c \sin (e+f x))^{3/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (12*a^2*(g*Cos[e
+ f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(c*f*g*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*(g*Cos[e + f*x])^(5/2)
)/(5*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (462*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e +
f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^3*(g*Cos
[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c^2*f*g*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))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {(3 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}+\frac {\left (33 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (231 a^3\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}{5 c^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (231 a^4\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}{5 c^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (231 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5 c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (231 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {12 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{c f g (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {462 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 c^2 f g \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 3.22, size = 267, normalized size = 0.90 \begin {gather*} -\frac {a^3 (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 \sqrt {a (1+\sin (e+f x))} \left (-1848 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 )^3+\sqrt {\cos (e+f x)} \left (487 \cos \left (\frac {1}{2} (e+f x)\right )+633 \cos \left (\frac {3}{2} (e+f x)\right )-17 \cos \left (\frac {5}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )+487 \sin \left (\frac {1}{2} (e+f x)\right )-633 \sin \left (\frac {3}{2} (e+f x)\right )-17 \sin \left (\frac {5}{2} (e+f x)\right )-\sin \left (\frac {7}{2} (e+f x)\right )\right )\right )}{20 c^2 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 ) (-1+\sin (e+f x))^2 \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])^(7/2))/(c - c*Sin[e + f*x])^(5/2),x]

[Out]

-1/20*(a^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sqrt[a*(1 + Sin[e + f*x])]*(-1848*El
lipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(487*Cos[(e + f*x)/2] + 6
33*Cos[(3*(e + f*x))/2] - 17*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2] + 487*Sin[(e + f*x)/2] - 633*Sin[(3*(
e + f*x))/2] - 17*Sin[(5*(e + f*x))/2] - Sin[(7*(e + f*x))/2])))/(c^2*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])

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Maple [C] Result contains complex when optimal does not.
time = 2.59, size = 3600, normalized size = 12.08

method	result	size
default	\(\text {Expression too large to display}\)	\(3600\)

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))^(7/2)/(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/f*(-1+cos(f*x+e))*(-sin(f*x+e)*cos(f*x+e)^5+231*I*cos(f*x+e)^4*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+co
s(f*x+e)))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-231*I*cos(f*x+e)^4*(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)-693*I*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x
+e)))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+cos(f*x+e)))^(1/2)+693*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)-462*I*cos(f*x+e)*(cos
(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+cos(f*x+e)))^(1/2)+462*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)-231*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)*sin(f*x+e)+693*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)*sin(f*x+e)-693*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)*sin(f*x+e)+462*I*cos(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*E
llipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-462*I*cos(f*x+e)*(cos(f*x+e)/(1+c
os(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)*sin(f*x+e)-80*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)^4*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+80*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(1+c
os(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)+40*cos(f*x+e)^4*sin(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*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)-40*cos(f*x+e)^4*sin(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*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)+69*cos(f*x+e)^3*sin(f*x+e)+231*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)*sin(f*x+e)-9*cos(f*x+e)^4*sin(f*x+e)-478*cos(f*x+e)^2*sin(f*x+e
)+cos(f*x+e)^6+80*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)-80*ln(-(2*c
os(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)+320*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)-320*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+co
s(f*x+e))^2)^(3/2)+280*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)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)-80*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)*sin(f*x+e)-280*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)+80*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)+446*cos(f*x+e)^2-85*cos(f*x+e)^3+40*cos(f*x+e)^5*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*ln(-(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)-40*cos(f*x+e)^5*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*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)-8*cos(
f*x+e)^5+80*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)-80*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)-280*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)+280*cos(f*x+e)*sin(f*x+e...

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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))^(7/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.14, size = 280, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (8 \, a^{3} g \cos \left (f x + e\right )^{2} - 146 \, a^{3} g + {\left (a^{3} g \cos \left (f x + e\right )^{2} + 162 \, a^{3} g\right )} \sin \left (f x + e\right )\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} + 231 \, {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} a^{3} g \sin \left (f x + e\right ) - 2 i \, \sqrt {2} a^{3} 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 ) + 231 \, {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} a^{3} g \sin \left (f x + e\right ) + 2 i \, \sqrt {2} a^{3} 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 )}{5 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} + 2 \, c^{3} f \sin \left (f x + e\right ) - 2 \, c^{3} 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))^(7/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/5*(2*(8*a^3*g*cos(f*x + e)^2 - 146*a^3*g + (a^3*g*cos(f*x + e)^2 + 162*a^3*g)*sin(f*x + e))*sqrt(g*cos(f*x
+ e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) + 231*(I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 2*I*sqrt(2)*a
^3*g*sin(f*x + e) - 2*I*sqrt(2)*a^3*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x +
e) + I*sin(f*x + e))) + 231*(-I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 2*I*sqrt(2)*a^3*g*sin(f*x + e) + 2*I*sqrt(2)*a
^3*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(c^3*f*co
s(f*x + e)^2 + 2*c^3*f*sin(f*x + e) - 2*c^3*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((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(5/2),x)

[Out]

Timed out

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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))^(7/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \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 )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,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))^(7/2))/(c - c*sin(e + f*x))^(5/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(5/2), x)

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