3.98 \(\int \cot ^4(e+f x) (a+a \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=197 \[ -\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a \sin (e+f x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f} \]

[Out]

(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a
+ a*Sin[e + f*x]]) + (29*a^2*Cot[e + f*x])/(24*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin
[e + f*x]])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f) - (Cot[e + f*x]*Csc[e + f*x]^
2*(a + a*Sin[e + f*x])^(3/2))/(3*f)

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Rubi [A]  time = 0.496362, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2718, 2647, 2646, 3044, 2975, 2980, 2773, 206} \[ -\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a \sin (e+f x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a
+ a*Sin[e + f*x]]) + (29*a^2*Cot[e + f*x])/(24*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin
[e + f*x]])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f) - (Cot[e + f*x]*Csc[e + f*x]^
2*(a + a*Sin[e + f*x])^(3/2))/(3*f)

Rule 2718

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 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

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 3044

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*Sin[
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
])

Rule 2975

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], 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] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| EqQ[c, 0])

Rule 2980

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 2773

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 206

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

Rubi steps

\begin{align*} \int \cot ^4(e+f x) (a+a \sin (e+f x))^{3/2} \, dx &=\int (a+a \sin (e+f x))^{3/2} \, dx+\int \csc ^4(e+f x) (a+a \sin (e+f x))^{3/2} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac{\int \csc ^3(e+f x) \left (\frac{3 a}{2}-\frac{11}{2} a \sin (e+f x)\right ) (a+a \sin (e+f x))^{3/2} \, dx}{3 a}+\frac{1}{3} (4 a) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a+a \sin (e+f x)}}{4 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac{\int \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)} \left (-\frac{29 a^2}{4}-\frac{41}{4} a^2 \sin (e+f x)\right ) \, dx}{6 a}\\ &=-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a+a \sin (e+f x)}}{4 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}-\frac{1}{16} (37 a) \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a+a \sin (e+f x)}}{4 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac{\left (37 a^2\right ) \operatorname{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{37 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 f}-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a+a \sin (e+f x)}}{4 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}\\ \end{align*}

Mathematica [A]  time = 1.35636, size = 334, normalized size = 1.7 \[ -\frac{a \csc ^{10}\left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sin (e+f x)+1)} \left (276 \sin \left (\frac{1}{2} (e+f x)\right )+326 \sin \left (\frac{3}{2} (e+f x)\right )-78 \sin \left (\frac{5}{2} (e+f x)\right )-72 \sin \left (\frac{7}{2} (e+f x)\right )-8 \sin \left (\frac{9}{2} (e+f x)\right )-276 \cos \left (\frac{1}{2} (e+f x)\right )+326 \cos \left (\frac{3}{2} (e+f x)\right )+78 \cos \left (\frac{5}{2} (e+f x)\right )-72 \cos \left (\frac{7}{2} (e+f x)\right )+8 \cos \left (\frac{9}{2} (e+f x)\right )-333 \sin (e+f x) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+333 \sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+111 \sin (3 (e+f x)) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )-111 \sin (3 (e+f x)) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )\right )}{24 f \left (\cot \left (\frac{1}{2} (e+f x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (e+f x)\right )-\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

-(a*Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(-276*Cos[(e + f*x)/2] + 326*Cos[(3*(e + f*x))/2] + 78*Cos[
(5*(e + f*x))/2] - 72*Cos[(7*(e + f*x))/2] + 8*Cos[(9*(e + f*x))/2] + 276*Sin[(e + f*x)/2] - 333*Log[1 + Cos[(
e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 333*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] +
326*Sin[(3*(e + f*x))/2] - 78*Sin[(5*(e + f*x))/2] + 111*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(
e + f*x)] - 111*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 72*Sin[(7*(e + f*x))/2] - 8*Si
n[(9*(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 = 0.651, size = 196, normalized size = 1. \begin{align*} -{\frac{1+\sin \left ( fx+e \right ) }{24\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 96\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{5/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}-16\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}{a}^{3/2}-111\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{3}{a}^{3}+15\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{5/2}+8\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{a}^{3/2}-15\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}\sqrt{a} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(a+a*sin(f*x+e))^(3/2),x)

[Out]

-1/24*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(96*(-a*(-1+sin(f*x+e)))^(1/2)*a^(5/2)*sin(f*x+e)^3-16*(-a*(-1
+sin(f*x+e)))^(3/2)*sin(f*x+e)^3*a^(3/2)-111*arctanh((-a*(-1+sin(f*x+e)))^(1/2)/a^(1/2))*sin(f*x+e)^3*a^3+15*(
-a*(-1+sin(f*x+e)))^(1/2)*a^(5/2)+8*(-a*(-1+sin(f*x+e)))^(3/2)*a^(3/2)-15*(-a*(-1+sin(f*x+e)))^(5/2)*a^(1/2))/
a^(3/2)/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., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*cot(f*x + e)^4, x)

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Fricas [B]  time = 1.72276, size = 1123, normalized size = 5.7 \begin{align*} \frac{111 \,{\left (a \cos \left (f x + e\right )^{4} - 2 \, a \cos \left (f x + e\right )^{2} -{\left (a \cos \left (f x + e\right )^{3} + a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) + a\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 (16 \, a \cos \left (f x + e\right )^{5} - 64 \, a \cos \left (f x + e\right )^{4} - 17 \, a \cos \left (f x + e\right )^{3} + 165 \, a \cos \left (f x + e\right )^{2} + 9 \, a \cos \left (f x + e\right ) -{\left (16 \, a \cos \left (f x + e\right )^{4} + 80 \, a \cos \left (f x + e\right )^{3} + 63 \, a \cos \left (f x + e\right )^{2} - 102 \, a \cos \left (f x + e\right ) - 93 \, a\right )} \sin \left (f x + e\right ) - 93 \, a\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/96*(111*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 - (a*cos(f*x + e)^3 + a*cos(f*x + e)^2 - a*cos(f*x + e) - a)*
sin(f*x + e) + a)*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*(16*a*cos(f*x + e)^5 - 64*a*cos(f*x + e)^4 - 17*a*cos(f*x + e)^3 + 165*a*cos(f*x + e)^2
+ 9*a*cos(f*x + e) - (16*a*cos(f*x + e)^4 + 80*a*cos(f*x + e)^3 + 63*a*cos(f*x + e)^2 - 102*a*cos(f*x + e) -
93*a)*sin(f*x + e) - 93*a)*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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(a+a*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 3.10235, size = 1058, normalized size = 5.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

-1/48*(222*a^2*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))*sgn(tan(1
/2*f*x + 1/2*e) + 1)/sqrt(-a) - 111*a^(3/2)*log(abs(-sqrt(a)*tan(1/2*f*x + 1/2*e) + sqrt(a*tan(1/2*f*x + 1/2*e
)^2 + a)))*sgn(tan(1/2*f*x + 1/2*e) + 1) - (1110*sqrt(2)*a^2*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 55
5*sqrt(2)*sqrt(-a)*a^(3/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 1554*a^2*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-
a)) - 777*sqrt(-a)*a^(3/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 1192*sqrt(2)*sqrt(-a)*a^(3/2) + 1706*sqrt(-a)*a^(3
/2))*sgn(tan(1/2*f*x + 1/2*e) + 1)/(5*sqrt(2)*sqrt(-a) + 7*sqrt(-a)) + (182*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1)
- (105*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - (138*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - (178*a^3*sgn(tan(1/2*f*x +
1/2*e) + 1) - (18*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - (2*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1)*tan(1/2*f*x + 1/2*
e) + 9*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1))*tan(1/2*f*x + 1/2*e))*tan(1/2*f*x + 1/2*e))*tan(1/2*f*x + 1/2*e))*ta
n(1/2*f*x + 1/2*e))*tan(1/2*f*x + 1/2*e))/(a*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) - 2*(9*(sqrt(a)*tan(1/2*f*x + 1
/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*sgn(tan(1/2*f*x + 1/2*e) + 1) - 18*(sqrt(a)*tan(1/2*f*x + 1/
2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*sgn(tan(1/2*f*x + 1/2*e) + 1) + 48*(sqrt(a)*tan(1/2*f*x +
1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*sgn(tan(1/2*f*x + 1/2*e) + 1) - 9*(sqrt(a)*tan(1/2*f*x
+ 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*sgn(tan(1/2*f*x + 1/2*e) + 1) - 22*a^(9/2)*sgn(tan(1/2*f*x
+ 1/2*e) + 1))/((sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^3)/f