3.4.38 \(\int \frac {\cot ^5(e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [338]

Optimal. Leaf size=215 \[ -\frac {\left (8 a^2+12 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, 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+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (8 a^2+12 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 f (a-b) \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)^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

-1/8*((8*a^2 + 12*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(a^(7/2)*f) + ArcTanh[Sqrt[a + b*
Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) + (b*(4*a^2 + 3*a*b - 15*b^2))/(8*a^3*(a - b)*f*Sqrt[a + b*Tan[
e + f*x]^2]) + ((4*a + 5*b)*Cot[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^4/(4*a*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 )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^5 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^3 (1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (4 a+5 b)+\frac {5 b x}{2}}{x^2 (1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{4 a f}\\ &=\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (8 a^2+12 a b+15 b^2\right )+\frac {3}{4} b (4 a+5 b) x}{x (1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{4 a^2 f}\\ &=\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{8} (a-b) \left (8 a^2+12 a b+15 b^2\right )-\frac {1}{8} b \left (4 a^2+3 a b-15 b^2\right ) x}{x (1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 a^3 (a-b) f}\\ &=\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a 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) f}+\frac {\left (8 a^2+12 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{16 a^3 f}\\ &=\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a 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) b f}+\frac {\left (8 a^2+12 a b+15 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^3 b f}\\ &=-\frac {\left (8 a^2+12 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a 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 = 0.80, size = 142, normalized size = 0.66 \begin {gather*} \frac {8 a^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (a \cot ^2(e+f x) \left (-4 a-5 b+2 a \cot ^2(e+f x)\right )-\left (8 a^2+12 a b+15 b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \tan ^2(e+f x)}{a}\right )\right )}{8 a^3 (-a+b) f \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)^(3/2),x]

[Out]

(8*a^3*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a*Cot[e + f*x]^2*(-4*a - 5*b
+ 2*a*Cot[e + f*x]^2) - (8*a^2 + 12*a*b + 15*b^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a]))
/(8*a^3*(-a + b)*f*Sqrt[a + b*Tan[e + f*x]^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79933\) vs. \(2(189)=378\).
time = 2.10, size = 79934, normalized size = 371.79

method	result	size
default	\(\text {Expression too large to display}\)	\(79934\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),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)^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]
time = 4.44, size = 1574, normalized size = 7.32 \begin {gather*} \left [-\frac {8 \, {\left (a^{4} b \tan \left (f x + e\right )^{6} + a^{5} \tan \left (f x + e\right )^{4}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left ({\left (8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + {\left (8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) + 2 \, {\left (2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left (4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{6} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{4}\right )}}, \frac {16 \, {\left (a^{4} b \tan \left (f x + e\right )^{6} + a^{5} \tan \left (f x + e\right )^{4}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) + {\left ({\left (8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + {\left (8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) - 2 \, {\left (2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left (4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{6} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{4}\right )}}, \frac {{\left ({\left (8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + {\left (8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) - 4 \, {\left (a^{4} b \tan \left (f x + e\right )^{6} + a^{5} \tan \left (f x + e\right )^{4}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left (4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{6} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{4}\right )}}, \frac {{\left ({\left (8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + {\left (8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) + 8 \, {\left (a^{4} b \tan \left (f x + e\right )^{6} + a^{5} \tan \left (f x + e\right )^{4}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) - {\left (2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left (4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{6} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{4}\right )}}\right ] \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(8*(a^4*b*tan(f*x + e)^6 + a^5*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)) - ((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*t
an(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*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*(2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b
- a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*
sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(
f*x + e)^4), 1/16*(16*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 +
a)*sqrt(-a + b)/(a - b)) + ((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a
^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*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*(2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*
a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/
((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/8*(((8*a^4*b
- 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4
)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - 4*(a^4*b*tan(f*x + e)^6 + a^5*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)) - (2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^
5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3
)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/8*(((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*
b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(-a)*ar
ctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 8*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(-a + b)*arcta
n(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a
^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x +
e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*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 {3}{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)**(3/2),x)

[Out]

Integral(cot(e + f*x)**5/(a + b*tan(e + f*x)**2)**(3/2), 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)^(3/2),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 = 13.13, size = 2118, normalized size = 9.85 \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)^(3/2),x)

[Out]

(atan((((((a + b*tan(e + f*x)^2)^(1/2)*(230400*a^9*b^11*f^3 - 783360*a^10*b^10*f^3 + 854016*a^11*b^9*f^3 - 387
072*a^12*b^8*f^3 + 480256*a^13*b^7*f^3 - 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 327
680*a^17*b^3*f^3 - 131072*a^18*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(638976*a^13*b^9*f^4 - 122880*a^12*b^10*f^4 -
1318912*a^14*b^8*f^4 + 1376256*a^15*b^7*f^4 - 794624*a^16*b^6*f^4 + 311296*a^17*b^5*f^4 - 122880*a^18*b^4*f^4
+ 32768*a^19*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144*a^15*b^8*f^5 - 1835008*a^16*b^7
*f^5 + 5242880*a^17*b^6*f^5 - 7864320*a^18*b^5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21
*b^2*f^5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3) + ((((a + b*tan(e + f*x)^2)^
(1/2)*(230400*a^9*b^11*f^3 - 783360*a^10*b^10*f^3 + 854016*a^11*b^9*f^3 - 387072*a^12*b^8*f^3 + 480256*a^13*b^
7*f^3 - 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 327680*a^17*b^3*f^3 - 131072*a^18*b^
2*f^3))/2 + (((a - b)^3)^(1/2)*(122880*a^12*b^10*f^4 - 638976*a^13*b^9*f^4 + 1318912*a^14*b^8*f^4 - 1376256*a^
15*b^7*f^4 + 794624*a^16*b^6*f^4 - 311296*a^17*b^5*f^4 + 122880*a^18*b^4*f^4 - 32768*a^19*b^3*f^4 + ((a + b*ta
n(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144*a^15*b^8*f^5 - 1835008*a^16*b^7*f^5 + 5242880*a^17*b^6*f^5 - 786
4320*a^18*b^5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21*b^2*f^5))/(4*f*(a - b)^3)))/(2*f
*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3))/(230400*a^9*b^10*f^2 - 552960*a^10*b^9*f^2 + 301056*a^11*b^8
*f^2 + 36864*a^12*b^7*f^2 + 123904*a^13*b^6*f^2 - 147456*a^14*b^5*f^2 - 24576*a^15*b^4*f^2 + 32768*a^16*b^3*f^
2 - ((((a + b*tan(e + f*x)^2)^(1/2)*(230400*a^9*b^11*f^3 - 783360*a^10*b^10*f^3 + 854016*a^11*b^9*f^3 - 387072
*a^12*b^8*f^3 + 480256*a^13*b^7*f^3 - 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 327680
*a^17*b^3*f^3 - 131072*a^18*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(638976*a^13*b^9*f^4 - 122880*a^12*b^10*f^4 - 131
8912*a^14*b^8*f^4 + 1376256*a^15*b^7*f^4 - 794624*a^16*b^6*f^4 + 311296*a^17*b^5*f^4 - 122880*a^18*b^4*f^4 + 3
2768*a^19*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144*a^15*b^8*f^5 - 1835008*a^16*b^7*f^
5 + 5242880*a^17*b^6*f^5 - 7864320*a^18*b^5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21*b^
2*f^5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2))/(f*(a - b)^3) + ((((a + b*tan(e + f*x)^2)^(1/2)*
(230400*a^9*b^11*f^3 - 783360*a^10*b^10*f^3 + 854016*a^11*b^9*f^3 - 387072*a^12*b^8*f^3 + 480256*a^13*b^7*f^3
- 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 327680*a^17*b^3*f^3 - 131072*a^18*b^2*f^3)
)/2 + (((a - b)^3)^(1/2)*(122880*a^12*b^10*f^4 - 638976*a^13*b^9*f^4 + 1318912*a^14*b^8*f^4 - 1376256*a^15*b^7
*f^4 + 794624*a^16*b^6*f^4 - 311296*a^17*b^5*f^4 + 122880*a^18*b^4*f^4 - 32768*a^19*b^3*f^4 + ((a + b*tan(e +
f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144*a^15*b^8*f^5 - 1835008*a^16*b^7*f^5 + 5242880*a^17*b^6*f^5 - 7864320*a
^18*b^5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21*b^2*f^5))/(4*f*(a - b)^3)))/(2*f*(a -
b)^3))*((a - b)^3)^(1/2))/(f*(a - b)^3)))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3) - (b^3/(a*(a - b)) - (b*(a + b*t
an(e + f*x)^2)*(5*a*b + 4*a^2 - 25*b^2))/(8*(a^2*b - a^3)) + (b*(a + b*tan(e + f*x)^2)^2*(3*a*b + 4*a^2 - 15*b
^2))/(8*(a^3*b - a^4)))/(f*(a + b*tan(e + f*x)^2)^(5/2) + a^2*f*(a + b*tan(e + f*x)^2)^(1/2) - 2*a*f*(a + b*ta
n(e + f*x)^2)^(3/2)) - (atan((a^15*b^12*(a + b*tan(e + f*x)^2)^(1/2)*3375i - a^16*b^11*(a + b*tan(e + f*x)^2)^
(1/2)*12150i + a^17*b^10*(a + b*tan(e + f*x)^2)^(1/2)*13905i - a^18*b^9*(a + b*tan(e + f*x)^2)^(1/2)*6912i + a
^19*b^8*(a + b*tan(e + f*x)^2)^(1/2)*10953i - a^20*b^7*(a + b*tan(e + f*x)^2)^(1/2)*16542i + a^21*b^6*(a + b*t
an(e + f*x)^2)^(1/2)*7343i - a^22*b^5*(a + b*tan(e + f*x)^2)^(1/2)*1932i + a^23*b^4*(a + b*tan(e + f*x)^2)^(1/
2)*4200i - a^24*b^3*(a + b*tan(e + f*x)^2)^(1/2)*2240i)/(a^7*(a^7)^(1/2)*(a^7*(6912*a*b^9 - 13905*b^10 - 10953
*a^2*b^8 + 16542*a^3*b^7 - 7343*a^4*b^6 + 1932*a^5*b^5 - 4200*a^6*b^4 + 2240*a^7*b^3) - 3375*a^5*b^12 + 12150*
a^6*b^11)))*(12*a*b + 8*a^2 + 15*b^2)*1i)/(8*f*(a^7)^(1/2))

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