3.1.98 \(\int \cot ^4(e+f x) (a+a \sin (e+f x))^{3/2} \, dx\) [98]

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

[Out]

37/8*a^(3/2)*arctanh(cos(f*x+e)*a^(1/2)/(a+a*sin(f*x+e))^(1/2))/f-1/3*cot(f*x+e)*csc(f*x+e)^2*(a+a*sin(f*x+e))
^(3/2)/f-8/3*a^2*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)+29/24*a^2*cot(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-2/3*a*cos(f
*x+e)*(a+a*sin(f*x+e))^(1/2)/f-1/4*a*cot(f*x+e)*csc(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.34, antiderivative size = 197, normalized size of antiderivative = 1.00, 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 = {2797, 2726, 2725, 3123, 3054, 3059, 2852, 212} \begin {gather*} \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 {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 {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} \end {gather*}

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 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 2726

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] &&
EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 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 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 3054

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
*Sin[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 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) (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 ) \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 {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*}

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Mathematica [A]
time = 1.04, size = 334, normalized size = 1.70 \begin {gather*} -\frac {a \csc ^{10}\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sin (e+f x))} \left (-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 )+276 \sin \left (\frac {1}{2} (e+f x)\right )-333 \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)+333 \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)+326 \sin \left (\frac {3}{2} (e+f x)\right )-78 \sin \left (\frac {5}{2} (e+f x)\right )+111 \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))-111 \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))-72 \sin \left (\frac {7}{2} (e+f x)\right )-8 \sin \left (\frac {9}{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*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

-1/24*(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]]*Si
n[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*Sin[(9*(e + f*x))/2]))/(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 = 2.28, size = 196, normalized size = 0.99

method	result	size
default	\(\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (16 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) a^{\frac {3}{2}}-96 a^{\frac {5}{2}} \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin ^{3}\left (f x +e \right )\right )+111 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) a^{3}+15 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \sqrt {a}-8 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-15 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {5}{2}}\right )}{24 a^{\frac {3}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(196\)

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,method=_RETURNVERBOSE)

[Out]

1/24*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)/a^(3/2)*(16*(-a*(sin(f*x+e)-1))^(3/2)*sin(f*x+e)^3*a^(3/2)-96*a^
(5/2)*(-a*(sin(f*x+e)-1))^(1/2)*sin(f*x+e)^3+111*arctanh((-a*(sin(f*x+e)-1))^(1/2)/a^(1/2))*sin(f*x+e)^3*a^3+1
5*(-a*(sin(f*x+e)-1))^(5/2)*a^(1/2)-8*(-a*(sin(f*x+e)-1))^(3/2)*a^(3/2)-15*(-a*(sin(f*x+e)-1))^(1/2)*a^(5/2))/
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))^(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] Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (183) = 366\).
time = 0.37, size = 463, normalized size = 2.35 \begin {gather*} \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 {gather*}

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3003 deep

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Giac [A]
time = 9.89, size = 260, normalized size = 1.32 \begin {gather*} -\frac {\sqrt {2} {\left (128 \, a \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} - 111 \, \sqrt {2} a \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 ) - 384 \, a \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 (60 \, a \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} - 16 \, a \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 \, a \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))^(3/2),x, algorithm="giac")

[Out]

-1/96*sqrt(2)*(128*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 111*sqrt(2)*a*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)) - 384*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 4*(60
*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 16*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2
*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 15*a*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\,{\left (a+a\,\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(cot(e + f*x)^4*(a + a*sin(e + f*x))^(3/2),x)

[Out]

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

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