3.6.37 \(\int \frac {\cot ^5(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [537]

Optimal. Leaf size=208 \[ -\frac {\left (8 a^2+40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

-1/8*(8*a^2+40*a*b+35*b^2)*arctanh((a+b*sin(f*x+e)^2)^(1/2)/a^(1/2))/a^(9/2)/f+1/24*(8*a^2+40*a*b+35*b^2)/a^3/
f/(a+b*sin(f*x+e)^2)^(3/2)+1/8*(8*a+7*b)*csc(f*x+e)^2/a^2/f/(a+b*sin(f*x+e)^2)^(3/2)-1/4*csc(f*x+e)^4/a/f/(a+b
*sin(f*x+e)^2)^(3/2)+1/8*(8*a^2+40*a*b+35*b^2)/a^4/f/(a+b*sin(f*x+e)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3273, 91, 79, 53, 65, 214} \begin {gather*} \frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2+40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((8*a^2 + 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(a^(9/2)*f) + (8*a^2 + 40*a*b + 3
5*b^2)/(24*a^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((8*a + 7*b)*Csc[e + f*x]^2)/(8*a^2*f*(a + b*Sin[e + f*x]^2)^
(3/2)) - Csc[e + f*x]^4/(4*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (8*a^2 + 40*a*b + 35*b^2)/(8*a^4*f*Sqrt[a + b*S
in[e + f*x]^2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 214

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

Rule 3273

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m
+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^3 (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (-8 a-7 b)+2 a x}{x^2 (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2+40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2+40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^3 f}\\ &=\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^4 f}\\ &=\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{8 a^4 b f}\\ &=-\frac {\left (8 a^2+40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.57, size = 117, normalized size = 0.56 \begin {gather*} \frac {3 a \csc ^4(e+f x) \left (8 a+7 b-2 a \csc ^2(e+f x)\right )+\left (8 a^2+40 a b+35 b^2\right ) \csc ^2(e+f x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1+\frac {b \sin ^2(e+f x)}{a}\right )}{24 a^3 f \left (b+a \csc ^2(e+f x)\right ) \sqrt {a+b \sin ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*Csc[e + f*x]^4*(8*a + 7*b - 2*a*Csc[e + f*x]^2) + (8*a^2 + 40*a*b + 35*b^2)*Csc[e + f*x]^2*Hypergeometric
2F1[-3/2, 1, -1/2, 1 + (b*Sin[e + f*x]^2)/a])/(24*a^3*f*(b + a*Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(184)=368\).
time = 41.83, size = 988, normalized size = 4.75

method	result	size
default	\(\frac {-\frac {\left (a^{2}+4 a b +3 b^{2}\right ) \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{2 a^{4} \sqrt {-a b}\, \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} b \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{6 a^{3} \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {b \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{4} \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} \sqrt {-a b}\, \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )}-\frac {b \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{6 a^{3} \sqrt {-a b}\, \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}\, b^{2}}{12 a^{4} \sqrt {-a b}\, \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{a^{3} \sin \left (f x +e \right )^{2}}-\frac {5 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{a^{\frac {7}{2}}}+\frac {11 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 a^{4} \sin \left (f x +e \right )^{2}}-\frac {35 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 a^{\frac {9}{2}}}+\frac {\left (a^{2}+4 a b +3 b^{2}\right ) \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{2 a^{4} \sqrt {-a b}\, \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{a^{\frac {5}{2}}}-\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{4 a^{3} \sin \left (f x +e \right )^{4}}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} b \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{6 a^{3} \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {b \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{4} \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} \sqrt {-a b}\, \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {b \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{6 a^{3} \sqrt {-a b}\, \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}\, b^{2}}{12 a^{4} \sqrt {-a b}\, \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )}}{f}\)	\(988\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

(-1/2*(a^2+4*a*b+3*b^2)/a^4/(-a*b)^(1/2)/(sin(f*x+e)+(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/
a^2/b/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/6/a^3/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*
(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^4*b/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/
2)-1/12/a^2/(-a*b)^(1/2)/(sin(f*x+e)+(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/6/a^3/(-a*b)^(1/2)*
b/(sin(f*x+e)+(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^4/(-a*b)^(1/2)/(sin(f*x+e)+(-a*b)^(1/
2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)*b^2+1/a^3/sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)-5/a^(7/2)*b*ln((2*a+
2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))+11/8/a^4*b/sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)-35/8/a^(9/2)*
b^2*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))+1/2*(a^2+4*a*b+3*b^2)/a^4/(-a*b)^(1/2)/(sin(f*x+e)
-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/a^(5/2)*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin
(f*x+e))-1/4/a^3/sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2)-1/12/a^2/b/(sin(f*x+e)-(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)
^2+(a*b+b^2)/b)^(1/2)-1/6/a^3/(sin(f*x+e)-(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^4*b/(si
n(f*x+e)-(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)+1/12/a^2/(-a*b)^(1/2)/(sin(f*x+e)-(-a*b)^(1/2)/
b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)+1/6/a^3/(-a*b)^(1/2)*b/(sin(f*x+e)-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*
b+b^2)/b)^(1/2)+1/12/a^4/(-a*b)^(1/2)/(sin(f*x+e)-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)*b^2)/f

________________________________________________________________________________________

Maxima [A]
time = 0.38, size = 295, normalized size = 1.42 \begin {gather*} -\frac {\frac {24 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} + \frac {120 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {7}{2}}} + \frac {105 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {9}{2}}} - \frac {24}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} - \frac {8}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a} - \frac {120 \, b}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{3}} - \frac {40 \, b}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {105 \, b^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{4}} - \frac {35 \, b^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {24}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )^{2}} - \frac {21 \, b}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} \sin \left (f x + e\right )^{2}} + \frac {6}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )^{4}}}{24 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/24*(24*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/a^(5/2) + 120*b*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/a^
(7/2) + 105*b^2*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/a^(9/2) - 24/(sqrt(b*sin(f*x + e)^2 + a)*a^2) - 8/((b
*sin(f*x + e)^2 + a)^(3/2)*a) - 120*b/(sqrt(b*sin(f*x + e)^2 + a)*a^3) - 40*b/((b*sin(f*x + e)^2 + a)^(3/2)*a^
2) - 105*b^2/(sqrt(b*sin(f*x + e)^2 + a)*a^4) - 35*b^2/((b*sin(f*x + e)^2 + a)^(3/2)*a^3) - 24/((b*sin(f*x + e
)^2 + a)^(3/2)*a*sin(f*x + e)^2) - 21*b/((b*sin(f*x + e)^2 + a)^(3/2)*a^2*sin(f*x + e)^2) + 6/((b*sin(f*x + e)
^2 + a)^(3/2)*a*sin(f*x + e)^4))/f

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (184) = 368\).
time = 0.56, size = 984, normalized size = 4.73 \begin {gather*} \left [\frac {3 \, {\left ({\left (8 \, a^{2} b^{2} + 40 \, a b^{3} + 35 \, b^{4}\right )} \cos \left (f x + e\right )^{8} - 2 \, {\left (8 \, a^{3} b + 56 \, a^{2} b^{2} + 115 \, a b^{3} + 70 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + {\left (8 \, a^{4} + 88 \, a^{3} b + 323 \, a^{2} b^{2} + 450 \, a b^{3} + 210 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{4} + 56 \, a^{3} b + 123 \, a^{2} b^{2} + 110 \, a b^{3} + 35 \, b^{4} - 2 \, {\left (8 \, a^{4} + 64 \, a^{3} b + 171 \, a^{2} b^{2} + 185 \, a b^{3} + 70 \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (3 \, {\left (8 \, a^{3} b + 40 \, a^{2} b^{2} + 35 \, a b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (32 \, a^{4} + 232 \, a^{3} b + 500 \, a^{2} b^{2} + 315 \, a b^{3}\right )} \cos \left (f x + e\right )^{4} - 50 \, a^{4} - 205 \, a^{3} b - 260 \, a^{2} b^{2} - 105 \, a b^{3} + {\left (88 \, a^{4} + 413 \, a^{3} b + 640 \, a^{2} b^{2} + 315 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{48 \, {\left (a^{5} b^{2} f \cos \left (f x + e\right )^{8} - 2 \, {\left (a^{6} b + 2 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{6} + {\left (a^{7} + 6 \, a^{6} b + 6 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f\right )}}, \frac {3 \, {\left ({\left (8 \, a^{2} b^{2} + 40 \, a b^{3} + 35 \, b^{4}\right )} \cos \left (f x + e\right )^{8} - 2 \, {\left (8 \, a^{3} b + 56 \, a^{2} b^{2} + 115 \, a b^{3} + 70 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + {\left (8 \, a^{4} + 88 \, a^{3} b + 323 \, a^{2} b^{2} + 450 \, a b^{3} + 210 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{4} + 56 \, a^{3} b + 123 \, a^{2} b^{2} + 110 \, a b^{3} + 35 \, b^{4} - 2 \, {\left (8 \, a^{4} + 64 \, a^{3} b + 171 \, a^{2} b^{2} + 185 \, a b^{3} + 70 \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left (3 \, {\left (8 \, a^{3} b + 40 \, a^{2} b^{2} + 35 \, a b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (32 \, a^{4} + 232 \, a^{3} b + 500 \, a^{2} b^{2} + 315 \, a b^{3}\right )} \cos \left (f x + e\right )^{4} - 50 \, a^{4} - 205 \, a^{3} b - 260 \, a^{2} b^{2} - 105 \, a b^{3} + {\left (88 \, a^{4} + 413 \, a^{3} b + 640 \, a^{2} b^{2} + 315 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{24 \, {\left (a^{5} b^{2} f \cos \left (f x + e\right )^{8} - 2 \, {\left (a^{6} b + 2 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{6} + {\left (a^{7} + 6 \, a^{6} b + 6 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56*a^2*b^2 + 115*a*b^3 + 70*b^4)*cos(f
*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b + 123*a^2
*b^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sqrt(a)*lo
g(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(3*(8*a
^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^6 - (32*a^4 + 232*a^3*b + 500*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^4 -
50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a*b^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^2)
*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^5*b^2*f*cos(f*x + e)^8 - 2*(a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 +
6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 - 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^2 + (a^7 + 2*a^6*b + a^5*
b^2)*f), 1/24*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56*a^2*b^2 + 115*a*b^3 + 70*b^
4)*cos(f*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b +
123*a^2*b^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sq
rt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (3*(8*a^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^
6 - (32*a^4 + 232*a^3*b + 500*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^4 - 50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a*b
^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^5*b^2*
f*cos(f*x + e)^8 - 2*(a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 + 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 - 2*(
a^7 + 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^2 + (a^7 + 2*a^6*b + a^5*b^2)*f)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**5/(a+b*sin(f*x+e)**2)**(5/2),x)

[Out]

Integral(cot(e + f*x)**5/(a + b*sin(e + f*x)**2)**(5/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (191) = 382\).
time = 1.25, size = 1411, normalized size = 6.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

-1/192*(((((3*((a^17*b^2 + 2*a^16*b^3 + a^15*b^4)*tan(1/2*f*x + 1/2*e)^2/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4) -
(9*a^17*b^2 + 32*a^16*b^3 + 37*a^15*b^4 + 14*a^14*b^5)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e
)^2 - 2*(197*a^17*b^2 + 1106*a^16*b^3 + 2181*a^15*b^4 + 1832*a^14*b^5 + 560*a^13*b^6)/(a^18*b^2 + 2*a^17*b^3 +
a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - 6*(165*a^17*b^2 + 1072*a^16*b^3 + 2761*a^15*b^4 + 3526*a^14*b^5 + 2232*a^
13*b^6 + 560*a^12*b^7)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - 3*(307*a^17*b^2 + 1958*a^1
6*b^3 + 4835*a^15*b^4 + 5792*a^14*b^5 + 3376*a^13*b^6 + 768*a^12*b^7)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(
1/2*f*x + 1/2*e)^2 - (295*a^17*b^2 + 1552*a^16*b^3 + 2859*a^15*b^4 + 2242*a^14*b^5 + 640*a^13*b^6)/(a^18*b^2 +
2*a^17*b^3 + a^16*b^4))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 +
a)^(3/2) - 24*(8*a^2 + 40*a*b + 35*b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)
^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a^4) - 12*(8*a^(5/2) +
40*a^(3/2)*b + 35*sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*
tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^5 + 12*(6*(sqrt(a)*tan
(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2
+ a))^3*a^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2
+ 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 22*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4
+ 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sq
rt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 8*(sqr
t(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1
/2*e)^2 + a))^2*a^(3/2)*b - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*
x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 28*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x
+ 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b - 26*(sqrt(a)*tan(1/2*f*x + 1
/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2
- 7*a^(7/2) - 12*a^(5/2)*b)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*
x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2*a^4))/f

________________________________________________________________________________________

Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^5/(a + b*sin(e + f*x)^2)^(5/2),x)

[Out]

\text{Hanged}

________________________________________________________________________________________