3.1.94 \(\int \cot ^4(e+f x) \sqrt {a+a \sin (e+f x)} \, dx\) [94]

Optimal. Leaf size=163 \[ \frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+\frac {11 a \cot (e+f x)}{8 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f} \]

[Out]

11/8*arctanh(cos(f*x+e)*a^(1/2)/(a+a*sin(f*x+e))^(1/2))*a^(1/2)/f-2*a*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)+11/8
*a*cot(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-1/12*a*cot(f*x+e)*csc(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-1/3*cot(f*x+e)*cs
c(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.25, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2797, 2725, 3123, 3059, 2851, 2852, 212} \begin {gather*} -\frac {2 a \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}+\frac {11 a \cot (e+f x)}{8 f \sqrt {a \sin (e+f x)+a}}+\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{8 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a
*Sin[e + f*x]]) + (11*a*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqr
t[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f)

Rule 212

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

Rule 2725

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

Rule 2797

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Int[(a + b*Sin[e + f*x
])^m, x] + Int[(a + b*Sin[e + f*x])^m*((1 - 2*Sin[e + f*x]^2)/Sin[e + f*x]^4), x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] &&  !LtQ[m, -1]

Rule 2851

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

Rule 2852

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

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 3123

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x]
)^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n
+ 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2,
0])

Rubi steps

\begin {align*} \int \cot ^4(e+f x) \sqrt {a+a \sin (e+f x)} \, dx &=\int \sqrt {a+a \sin (e+f x)} \, dx+\int \csc ^4(e+f x) \sqrt {a+a \sin (e+f x)} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {\int \csc ^3(e+f x) \left (\frac {a}{2}-\frac {9}{2} a \sin (e+f x)\right ) \sqrt {a+a \sin (e+f x)} \, dx}{3 a}\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {11}{8} \int \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+\frac {11 a \cot (e+f x)}{8 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {11}{16} \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+\frac {11 a \cot (e+f x)}{8 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {(11 a) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}\\ &=\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+\frac {11 a \cot (e+f x)}{8 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cot (e+f x) \csc (e+f x)}{12 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}\\ \end {align*}

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Mathematica [A]
time = 1.07, size = 309, normalized size = 1.90 \begin {gather*} \frac {\csc ^{10}\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sin (e+f x))} \left (252 \cos \left (\frac {1}{2} (e+f x)\right )-250 \cos \left (\frac {3}{2} (e+f x)\right )-114 \cos \left (\frac {5}{2} (e+f x)\right )+48 \cos \left (\frac {7}{2} (e+f x)\right )-252 \sin \left (\frac {1}{2} (e+f x)\right )+99 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-99 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-250 \sin \left (\frac {3}{2} (e+f x)\right )+114 \sin \left (\frac {5}{2} (e+f x)\right )-33 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))+33 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))+48 \sin \left (\frac {7}{2} (e+f x)\right )\right )}{24 f \left (1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(252*Cos[(e + f*x)/2] - 250*Cos[(3*(e + f*x))/2] - 114*Cos[(5*
(e + f*x))/2] + 48*Cos[(7*(e + f*x))/2] - 252*Sin[(e + f*x)/2] + 99*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2
]]*Sin[e + f*x] - 99*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 250*Sin[(3*(e + f*x))/2] + 11
4*Sin[(5*(e + f*x))/2] - 33*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 33*Log[1 - Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 48*Sin[(7*(e + f*x))/2]))/(24*f*(1 + Cot[(e + f*x)/2])*(Csc[(
e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)

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Maple [A]
time = 2.37, size = 170, normalized size = 1.04

method	result	size
default	\(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (48 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {7}{2}} \left (\sin ^{3}\left (f x +e \right )\right )-15 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {7}{2}}+56 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-33 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-33 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (f x +e \right )\right )\right )}{24 a^{\frac {7}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(48*(-a*(sin(f*x+e)-1))^(1/2)*a^(7/2)*sin(f*x+e)^3-15*(-a*(sin(
f*x+e)-1))^(1/2)*a^(7/2)+56*(-a*(sin(f*x+e)-1))^(3/2)*a^(5/2)-33*(-a*(sin(f*x+e)-1))^(5/2)*a^(3/2)-33*arctanh(
(-a*(sin(f*x+e)-1))^(1/2)/a^(1/2))*a^4*sin(f*x+e)^3)/a^(7/2)/sin(f*x+e)^3/cos(f*x+e)/(a+a*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(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*cot(f*x + e)^4, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (153) = 306\).
time = 0.39, size = 417, normalized size = 2.56 \begin {gather*} \frac {33 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, {\left (48 \, \cos \left (f x + e\right )^{4} - 33 \, \cos \left (f x + e\right )^{3} - 139 \, \cos \left (f x + e\right )^{2} + {\left (48 \, \cos \left (f x + e\right )^{3} + 81 \, \cos \left (f x + e\right )^{2} - 58 \, \cos \left (f x + e\right ) - 83\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 83\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} - {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2} - f \cos \left (f x + e\right ) - f\right )} \sin \left (f x + e\right ) + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/96*(33*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e
) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e
) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x
+ e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e
) - 1)) + 4*(48*cos(f*x + e)^4 - 33*cos(f*x + e)^3 - 139*cos(f*x + e)^2 + (48*cos(f*x + e)^3 + 81*cos(f*x + e)
^2 - 58*cos(f*x + e) - 83)*sin(f*x + e) + 25*cos(f*x + e) + 83)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^4 -
2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f*cos(f*x + e) - f)*sin(f*x + e) + f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{4}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*cot(e + f*x)**4, x)

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Giac [A]
time = 3.62, size = 223, normalized size = 1.37 \begin {gather*} \frac {\sqrt {2} {\left (33 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 192 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {4 \, {\left (132 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 112 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

1/96*sqrt(2)*(33*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi
+ 1/2*f*x + 1/2*e)))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 192*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi
+ 1/2*f*x + 1/2*e) + 4*(132*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 112*sgn(co
s(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 15*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1
/4*pi + 1/2*f*x + 1/2*e))/(2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^3)*sqrt(a)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^4\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4*(a + a*sin(e + f*x))^(1/2),x)

[Out]

int(cot(e + f*x)^4*(a + a*sin(e + f*x))^(1/2), x)

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