3.102 \(\int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=227 \[ -\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a \sin (e+f x)+a}}{24 f}+\frac{55 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{12 f} \]

[Out]

(55*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (9*a^3*Cos[e + f*x])/(40*f*Sqrt[
a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (17*a^2*Cot[e + f*x]*Sqrt[a + a
*Sin[e + f*x]])/(24*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (5*a*Cot[e + f*x]*Csc[e + f*x]*
(a + a*Sin[e + f*x])^(3/2))/(12*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(3*f)

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Rubi [A]  time = 0.627151, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 10, 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, 2981, 2773, 206} \[ -\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a \sin (e+f x)+a}}{24 f}+\frac{55 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{12 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]

[Out]

(55*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (9*a^3*Cos[e + f*x])/(40*f*Sqrt[
a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (17*a^2*Cot[e + f*x]*Sqrt[a + a
*Sin[e + f*x]])/(24*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (5*a*Cot[e + f*x]*Csc[e + f*x]*
(a + a*Sin[e + f*x])^(3/2))/(12*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(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 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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))^{5/2} \, dx &=\int (a+a \sin (e+f x))^{5/2} \, dx+\int \csc ^4(e+f x) (a+a \sin (e+f x))^{5/2} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac{\int \csc ^3(e+f x) \left (\frac{5 a}{2}-\frac{13}{2} a \sin (e+f x)\right ) (a+a \sin (e+f x))^{5/2} \, dx}{3 a}+\frac{1}{5} (8 a) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac{16 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac{\int \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2} \left (-\frac{17 a^2}{4}-\frac{57}{4} a^2 \sin (e+f x)\right ) \, dx}{6 a}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{64 a^3 \cos (e+f x)}{15 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a+a \sin (e+f x)}}{24 f}-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac{\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \left (-\frac{165 a^3}{8}-\frac{97}{8} a^3 \sin (e+f x)\right ) \, dx}{6 a}\\ &=-\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a+a \sin (e+f x)}}{24 f}-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}-\frac{1}{16} \left (55 a^2\right ) \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a+a \sin (e+f x)}}{24 f}-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac{\left (55 a^3\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{55 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 f}-\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a+a \sin (e+f x)}}{24 f}-\frac{2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}\\ \end{align*}

Mathematica [A]  time = 1.89914, size = 360, normalized size = 1.59 \[ -\frac{a^2 \csc ^{10}\left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sin (e+f x)+1)} \left (-108 \sin \left (\frac{1}{2} (e+f x)\right )+706 \sin \left (\frac{3}{2} (e+f x)\right )+450 \sin \left (\frac{5}{2} (e+f x)\right )-156 \sin \left (\frac{7}{2} (e+f x)\right )-100 \sin \left (\frac{9}{2} (e+f x)\right )+12 \sin \left (\frac{11}{2} (e+f x)\right )+108 \cos \left (\frac{1}{2} (e+f x)\right )+706 \cos \left (\frac{3}{2} (e+f x)\right )-450 \cos \left (\frac{5}{2} (e+f x)\right )-156 \cos \left (\frac{7}{2} (e+f x)\right )+100 \cos \left (\frac{9}{2} (e+f x)\right )+12 \cos \left (\frac{11}{2} (e+f x)\right )-2475 \sin (e+f x) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+2475 \sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+825 \sin (3 (e+f x)) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )-825 \sin (3 (e+f x)) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )\right )}{120 f \left (\cot \left (\frac{1}{2} (e+f x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (e+f x)\right )-\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]

[Out]

-(a^2*Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(108*Cos[(e + f*x)/2] + 706*Cos[(3*(e + f*x))/2] - 450*Co
s[(5*(e + f*x))/2] - 156*Cos[(7*(e + f*x))/2] + 100*Cos[(9*(e + f*x))/2] + 12*Cos[(11*(e + f*x))/2] - 108*Sin[
(e + f*x)/2] - 2475*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 2475*Log[1 - Cos[(e + f*x)/2]
+ Sin[(e + f*x)/2]]*Sin[e + f*x] + 706*Sin[(3*(e + f*x))/2] + 450*Sin[(5*(e + f*x))/2] + 825*Log[1 + Cos[(e +
f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 825*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x
)] - 156*Sin[(7*(e + f*x))/2] - 100*Sin[(9*(e + f*x))/2] + 12*Sin[(11*(e + f*x))/2]))/(120*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 = 0.694, size = 222, normalized size = 1. \begin{align*} -{\frac{1+\sin \left ( fx+e \right ) }{120\, \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 ( 48\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}\sqrt{a}-320\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}{a}^{3/2}+480\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{5/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}-825\,{\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}+135\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}\sqrt{a}-440\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{a}^{3/2}+345\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{5/2} \right ){\frac{1}{\sqrt{a}}}{\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))^(5/2),x)

[Out]

-1/120*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(48*(-a*(-1+sin(f*x+e)))^(5/2)*sin(f*x+e)^3*a^(1/2)-320*(-a*(
-1+sin(f*x+e)))^(3/2)*sin(f*x+e)^3*a^(3/2)+480*(-a*(-1+sin(f*x+e)))^(1/2)*a^(5/2)*sin(f*x+e)^3-825*arctanh((-a
*(-1+sin(f*x+e)))^(1/2)/a^(1/2))*sin(f*x+e)^3*a^3+135*(-a*(-1+sin(f*x+e)))^(5/2)*a^(1/2)-440*(-a*(-1+sin(f*x+e
)))^(3/2)*a^(3/2)+345*(-a*(-1+sin(f*x+e)))^(1/2)*a^(5/2))/sin(f*x+e)^3/a^(1/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/
2)/f

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Maxima [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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 1.73164, size = 1249, normalized size = 5.5 \begin{align*} \frac{825 \,{\left (a^{2} \cos \left (f x + e\right )^{4} - 2 \, a^{2} \cos \left (f x + e\right )^{2} + a^{2} -{\left (a^{2} \cos \left (f x + e\right )^{3} + a^{2} \cos \left (f x + e\right )^{2} - a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sin \left (f x + e\right )\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 \, a^{2} \cos \left (f x + e\right )^{6} + 224 \, a^{2} \cos \left (f x + e\right )^{5} - 128 \, a^{2} \cos \left (f x + e\right )^{4} - 583 \, a^{2} \cos \left (f x + e\right )^{3} + 147 \, a^{2} \cos \left (f x + e\right )^{2} + 399 \, a^{2} \cos \left (f x + e\right ) - 27 \, a^{2} +{\left (48 \, a^{2} \cos \left (f x + e\right )^{5} - 176 \, a^{2} \cos \left (f x + e\right )^{4} - 304 \, a^{2} \cos \left (f x + e\right )^{3} + 279 \, a^{2} \cos \left (f x + e\right )^{2} + 426 \, a^{2} \cos \left (f x + e\right ) + 27 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{480 \,{\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))^(5/2),x, algorithm="fricas")

[Out]

1/480*(825*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2 - (a^2*cos(f*x + e)^3 + a^2*cos(f*x + e)^2 - a^2*c
os(f*x + e) - a^2)*sin(f*x + e))*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*a^2*cos(f*x + e)^6 + 224*a^2*cos(f*x + e)^5 - 128*a^2*cos(f*x + e)^4
- 583*a^2*cos(f*x + e)^3 + 147*a^2*cos(f*x + e)^2 + 399*a^2*cos(f*x + e) - 27*a^2 + (48*a^2*cos(f*x + e)^5 -
176*a^2*cos(f*x + e)^4 - 304*a^2*cos(f*x + e)^3 + 279*a^2*cos(f*x + e)^2 + 426*a^2*cos(f*x + e) + 27*a^2)*sin(
f*x + e))*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))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 3.1945, size = 1135, normalized size = 5. \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))^(5/2),x, algorithm="giac")

[Out]

-1/240*(1650*a^3*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) - 825*a^(5/2)*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) - (8250*sqrt(2)*a^3*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) -
4125*sqrt(2)*sqrt(-a)*a^(5/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 11550*a^3*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sq
rt(-a)) - 5775*sqrt(-a)*a^(5/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 728*sqrt(2)*sqrt(-a)*a^(5/2) + 1030*sqrt(-a)*
a^(5/2))*sgn(tan(1/2*f*x + 1/2*e) + 1)/(5*sqrt(2)*sqrt(-a) + 7*sqrt(-a)) + (346*a^5*sgn(tan(1/2*f*x + 1/2*e) +
1) + (405*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1) + (100*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1) - (545*a^5*sgn(tan(1/2*f
*x + 1/2*e) + 1) + (720*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1) + (641*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1) + 5*(20*a^5
*sgn(tan(1/2*f*x + 1/2*e) + 1) + (2*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1)*tan(1/2*f*x + 1/2*e) + 15*a^5*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))*tan(1/2*f*x + 1/2*e))*t
an(1/2*f*x + 1/2*e))*tan(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)^(5/2) - 10*(15
*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3*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^(7/2)*sgn(tan(1/2*f*x + 1/2*e) + 1) -
24*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2)*sgn(tan(1/2*f*x + 1/2*e) + 1)
- 15*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))*a^5*sgn(tan(1/2*f*x + 1/2*e) + 1) +
14*a^(11/2)*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