3.4.51 \(\int \frac {\cot ^5(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\) [351]

Optimal. Leaf size=272 \[ -\frac {\left (8 a^2+20 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2} f}+\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}} \]

[Out]

-1/8*(8*a^2+20*a*b+35*b^2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(9/2)/f+arctanh((a+b*tan(f*x+e)^2)^(1/2
)/(a-b)^(1/2))/(a-b)^(5/2)/f+1/8*b*(4*a^3+3*a^2*b-50*a*b^2+35*b^3)/a^4/(a-b)^2/f/(a+b*tan(f*x+e)^2)^(1/2)+1/24
*b*(12*a^2+15*a*b-35*b^2)/a^3/(a-b)/f/(a+b*tan(f*x+e)^2)^(3/2)+1/8*(4*a+7*b)*cot(f*x+e)^2/a^2/f/(a+b*tan(f*x+e
)^2)^(3/2)-1/4*cot(f*x+e)^4/a/f/(a+b*tan(f*x+e)^2)^(3/2)

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Rubi [A]
time = 0.29, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3751, 457, 105, 156, 157, 162, 65, 214} \begin {gather*} \frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2+20 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{5/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((8*a^2 + 20*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(a^(9/2)*f) + ArcTanh[Sqrt[a + b*
Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) + (b*(12*a^2 + 15*a*b - 35*b^2))/(24*a^3*(a - b)*f*(a + b*Tan[e
+ f*x]^2)^(3/2)) + ((4*a + 7*b)*Cot[e + f*x]^2)/(8*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^4/(4*a*
f*(a + b*Tan[e + f*x]^2)^(3/2)) + (b*(4*a^3 + 3*a^2*b - 50*a*b^2 + 35*b^3))/(8*a^4*(a - b)^2*f*Sqrt[a + b*Tan[
e + f*x]^2])

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^5 \left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^3 (1+x) (a+b x)^{5/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (4 a+7 b)+\frac {7 b x}{2}}{x^2 (1+x) (a+b x)^{5/2}} \, dx,x,\tan ^2(e+f x)\right )}{4 a f}\\ &=\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (8 a^2+20 a b+35 b^2\right )+\frac {5}{4} b (4 a+7 b) x}{x (1+x) (a+b x)^{5/2}} \, dx,x,\tan ^2(e+f x)\right )}{4 a^2 f}\\ &=\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{8} (a-b) \left (8 a^2+20 a b+35 b^2\right )-\frac {3}{8} b \left (12 a^2+15 a b-35 b^2\right ) x}{x (1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{6 a^3 (a-b) f}\\ &=\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {\frac {3}{16} (a-b)^2 \left (8 a^2+20 a b+35 b^2\right )+\frac {3}{16} b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right ) x}{x (1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{3 a^4 (a-b)^2 f}\\ &=\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 (a-b)^2 f}+\frac {\left (8 a^2+20 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{16 a^4 f}\\ &=\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{(a-b)^2 b f}+\frac {\left (8 a^2+20 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{8 a^4 b f}\\ &=-\frac {\left (8 a^2+20 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2} f}+\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.31, size = 165, normalized size = 0.61 \begin {gather*} \frac {\cot ^2(e+f x) \left (8 a^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (3 a \cot ^2(e+f x) \left (-4 a-7 b+2 a \cot ^2(e+f x)\right )-\left (8 a^2+20 a b+35 b^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1+\frac {b \tan ^2(e+f x)}{a}\right )\right )\right )}{24 a^3 (-a+b) f \left (b+a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cot[e + f*x]^2*(8*a^3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(3*a*Cot[e +
f*x]^2*(-4*a - 7*b + 2*a*Cot[e + f*x]^2) - (8*a^2 + 20*a*b + 35*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*
Tan[e + f*x]^2)/a])))/(24*a^3*(-a + b)*f*(b + a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2])

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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 107.50, size = 790286, normalized size = 2905.46

method	result	size
default	\(\text {Expression too large to display}\)	\(790286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (250) = 500\).
time = 3.50, size = 2501, normalized size = 9.19 \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*tan(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(24*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + e)^4)*sqrt(a - b)*log((b*tan(f*x +
e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*((8*a^5*b^2 - 4*a^4*b^3 -
a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85
*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f
*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(6*
a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x
+ e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b -
9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6
*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^
9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/48*(48*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*
tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + 3*((8*a^5*b^2 - 4*a^4*
b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4
+ 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*
tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) -
2*(6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*ta
n(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^
6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 +
3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 -
3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/24*(3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85
*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f
*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(-a)*arctan
(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 12*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x +
e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2
+ 1)) - (6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b
^6)*tan(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7
- 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*
b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (
a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/24*(3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^
5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)
*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(-a)*
arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 24*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(
f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (6*a^7 - 18*a^6*b + 18*a^5
*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 - 4*(6*a^6*b -
3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3
- 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f
*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b
^3)*f*tan(f*x + e)^4)]

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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 \tan ^{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*tan(f*x+e)**2)**(5/2),x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [B]
time = 14.01, size = 2500, normalized size = 9.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*(a + b*tan(e + f*x)^2)^2*(15*a^2*b - 250*a*b^2 + 12*a^3 + 175*b^3))/(24*(a^3*b - a^4)*(a - b)) - b^3/(3*a*
(a - b)) + (b*(a + b*tan(e + f*x)^2)^3*(3*a^2*b - 50*a*b^2 + 4*a^3 + 35*b^3))/(8*(a^3*b - a^4)*(a*b - a^2)) +
(b*(10*a*b^2 - 7*b^3)*(a + b*tan(e + f*x)^2))/(3*a*(a - b)*(a*b - a^2)))/(f*(a + b*tan(e + f*x)^2)^(7/2) + a^2
*f*(a + b*tan(e + f*x)^2)^(3/2) - 2*a*f*(a + b*tan(e + f*x)^2)^(5/2)) - (atan((a^16*f^3*(a + b*tan(e + f*x)^2)
^(1/2)*128i - a^11*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*
f^2 + 10*a^3*b^2*f^2)*128i + b^11*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^
2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*1225i + a^8*b^8*f^3*(a + b*tan(e + f*x)^2)^(1/2)*64i - a^9*b^7*f^3*(a + b
*tan(e + f*x)^2)^(1/2)*576i + a^10*b^6*f^3*(a + b*tan(e + f*x)^2)^(1/2)*2240i - a^11*b^5*f^3*(a + b*tan(e + f*
x)^2)^(1/2)*4928i + a^12*b^4*f^3*(a + b*tan(e + f*x)^2)^(1/2)*6720i - a^13*b^3*f^3*(a + b*tan(e + f*x)^2)^(1/2
)*5824i + a^14*b^2*f^3*(a + b*tan(e + f*x)^2)^(1/2)*3136i - a^15*b*f^3*(a + b*tan(e + f*x)^2)^(1/2)*960i + a^2
*b^9*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b
^2*f^2)*16885i - a^3*b^8*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^
2*b^3*f^2 + 10*a^3*b^2*f^2)*19875i + a^4*b^7*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 -
5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*11859i - a^5*b^6*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5
*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*4053i + a^6*b^5*f*(a + b*tan(e + f*x)^2)^(
1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*2919i - a^7*b^4*f*(a +
b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*2625
i + a^8*b^3*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 1
0*a^3*b^2*f^2)*904i - a^9*b^2*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 -
10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*256i - a*b^10*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2
- 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*7175i + a^10*b*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5
*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*320i)/((a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 -
5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)^(3/2)*(1225*b^9 - 4725*a*b^8 + 6210*a^2*b^7 - 2730*a^3*b^6 + 18
9*a^4*b^5 - 945*a^5*b^4 + 840*a^6*b^3)))*1i)/(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 +
10*a^3*b^2*f^2)^(1/2) - (atanh((156800*a^2*b^17*f^2*(a + b*tan(e + f*x)^2)^(1/2)*(320*a^12*b + 64*a^13 + 1225
*a^9*b^4 + 1400*a^10*b^3 + 960*a^11*b^2)^(1/2))/(5488000*a^7*b^19*f^2 - 50960000*a^8*b^18*f^2 + 207491200*a^9*
b^17*f^2 - 483286400*a^10*b^16*f^2 + 704892160*a^11*b^15*f^2 - 668407040*a^12*b^14*f^2 + 435855616*a^13*b^13*f
^2 - 248036096*a^14*b^12*f^2 + 174993280*a^15*b^11*f^2 - 118823040*a^16*b^10*f^2 + 51799680*a^17*b^9*f^2 - 152
32896*a^18*b^8*f^2 + 7343616*a^19*b^7*f^2 - 3978240*a^20*b^6*f^2 + 860160*a^21*b^5*f^2) - (1545600*a^3*b^16*f^
2*(a + b*tan(e + f*x)^2)^(1/2)*(320*a^12*b + 64*a^13 + 1225*a^9*b^4 + 1400*a^10*b^3 + 960*a^11*b^2)^(1/2))/(54
88000*a^7*b^19*f^2 - 50960000*a^8*b^18*f^2 + 207491200*a^9*b^17*f^2 - 483286400*a^10*b^16*f^2 + 704892160*a^11
*b^15*f^2 - 668407040*a^12*b^14*f^2 + 435855616*a^13*b^13*f^2 - 248036096*a^14*b^12*f^2 + 174993280*a^15*b^11*
f^2 - 118823040*a^16*b^10*f^2 + 51799680*a^17*b^9*f^2 - 15232896*a^18*b^8*f^2 + 7343616*a^19*b^7*f^2 - 3978240
*a^20*b^6*f^2 + 860160*a^21*b^5*f^2) + (6775680*a^4*b^15*f^2*(a + b*tan(e + f*x)^2)^(1/2)*(320*a^12*b + 64*a^1
3 + 1225*a^9*b^4 + 1400*a^10*b^3 + 960*a^11*b^2)^(1/2))/(5488000*a^7*b^19*f^2 - 50960000*a^8*b^18*f^2 + 207491
200*a^9*b^17*f^2 - 483286400*a^10*b^16*f^2 + 704892160*a^11*b^15*f^2 - 668407040*a^12*b^14*f^2 + 435855616*a^1
3*b^13*f^2 - 248036096*a^14*b^12*f^2 + 174993280*a^15*b^11*f^2 - 118823040*a^16*b^10*f^2 + 51799680*a^17*b^9*f
^2 - 15232896*a^18*b^8*f^2 + 7343616*a^19*b^7*f^2 - 3978240*a^20*b^6*f^2 + 860160*a^21*b^5*f^2) - (17326720*a^
5*b^14*f^2*(a + b*tan(e + f*x)^2)^(1/2)*(320*a^12*b + 64*a^13 + 1225*a^9*b^4 + 1400*a^10*b^3 + 960*a^11*b^2)^(
1/2))/(5488000*a^7*b^19*f^2 - 50960000*a^8*b^18*f^2 + 207491200*a^9*b^17*f^2 - 483286400*a^10*b^16*f^2 + 70489
2160*a^11*b^15*f^2 - 668407040*a^12*b^14*f^2 + 435855616*a^13*b^13*f^2 - 248036096*a^14*b^12*f^2 + 174993280*a
^15*b^11*f^2 - 118823040*a^16*b^10*f^2 + 51799680*a^17*b^9*f^2 - 15232896*a^18*b^8*f^2 + 7343616*a^19*b^7*f^2
- 3978240*a^20*b^6*f^2 + 860160*a^21*b^5*f^2) + (28492032*a^6*b^13*f^2*(a + b*tan(e + f*x)^2)^(1/2)*(320*a^12*
b + 64*a^13 + 1225*a^9*b^4 + 1400*a^10*b^3 + 960*a^11*b^2)^(1/2))/(5488000*a^7*b^19*f^2 - 50960000*a^8*b^18*f^
2 + 207491200*a^9*b^17*f^2 - 483286400*a^10*b^16*f^2 + 704892160*a^11*b^15*f^2 - 668407040*a^12*b^14*f^2 + 435
855616*a^13*b^13*f^2 - 248036096*a^14*b^12*f^2 ...

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