\(\int \cot ^5(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [311]
Optimal result
Integrand size = 25, antiderivative size = 161 \[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (8 a^2-12 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}
\]
[Out]
(a-b)^(3/2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f-1/8*(8*a^2-12*a*b+3*b^2)*arctanh((a+b*tan(f*x+e)^2
)^(1/2)/a^(1/2))/f/a^(1/2)+1/8*(4*a-5*b)*cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2)/f-1/4*a*cot(f*x+e)^4*(a+b*tan(f
*x+e)^2)^(1/2)/f
Rubi [A] (verified)
Time = 0.39 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3751, 457, 100, 156, 162, 65,
214} \[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (8 a^2-12 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}+\frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}
\]
[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 + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(Sqrt[a]*f) + ((a - b)^(3/2)*ArcTa
nh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + ((4*a - 5*b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*f)
- (a*Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)
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 100
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])
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 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*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^5 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3 (1+x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = -\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (4 a-5 b)+\frac {1}{2} (3 a-4 b) b x}{x^2 (1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{4 f} \\ & = \frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} a \left (8 a^2-12 a b+3 b^2\right )+\frac {1}{4} a (4 a-5 b) b x}{x (1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{4 a f} \\ & = \frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}+\frac {\left (8 a^2-12 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{16 f} \\ & = \frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f}-\frac {(a-b)^2 \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 )}{b f}+\frac {\left (8 a^2-12 a b+3 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 b f} \\ & = -\frac {\left (8 a^2-12 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87
\[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\left (-8 a^2+12 a b-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+\cot ^2(e+f x) \left (4 a-5 b-2 a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}\right )}{8 \sqrt {a} f}
\]
[In]
Integrate[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
((-8*a^2 + 12*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(8*(a - b)^(3/2)*ArcTanh[Sqrt
[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Cot[e + f*x]^2*(4*a - 5*b - 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^
2]))/(8*Sqrt[a]*f)
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(2241\) vs. \(2(139)=278\).
Time = 0.91 (sec) , antiderivative size = 2242, normalized size of antiderivative =
13.93
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\(\text {Expression too large to display}\) |
\(2242\) |
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[In]
int(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/64/f/(a-b)^(1/2)/a^(1/2)*((a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*
x+e)+1)^2*csc(f*x+e)^2+a)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^2)^(3/2)*((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^3/(a*
(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(3/2)/
(-cos(f*x+e)+1)^4*(a^(3/2)*(-cos(f*x+e)+1)^6*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e
)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(a-b)^(1/2)*csc(f*x+e)^2-32*a^2*ln(2/(-cos(f*x+e)+1)^2*(-a*(-c
os(f*x+e)+1)^2+2*b*(-cos(f*x+e)+1)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*
(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))*(a-b)^(1/2)*(-cos(f*x+e)+1)^4+32
*a^2*ln((a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2
+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*(a-b)^(1/2)*(-cos(f*x+e)+1)^4+48*a*ln(2/(
-cos(f*x+e)+1)^2*(-a*(-cos(f*x+e)+1)^2+2*b*(-cos(f*x+e)+1)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e
)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))*b*(a-b)^
(1/2)*(-cos(f*x+e)+1)^4-48*a*b*ln((a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-co
s(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*(a-b)^(1/2)*(-c
os(f*x+e)+1)^4-11*a^(3/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e
)+1)^2*csc(f*x+e)^2+a)^(1/2)*(a-b)^(1/2)*(-cos(f*x+e)+1)^4+10*b*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*
x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(a-b)^(1/2)*(-cos(f*x+e)+1)^4*a^(1/2)-12*ln
(2/(-cos(f*x+e)+1)^2*(-a*(-cos(f*x+e)+1)^2+2*b*(-cos(f*x+e)+1)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f
*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))*b^2*
(a-b)^(1/2)*(-cos(f*x+e)+1)^4+12*b^2*ln((a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*
a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*(a-b)^(1/
2)*(-cos(f*x+e)+1)^4+64*ln(4*(-a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a-b)^(1/2)*(
a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2
)+a-b)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2+1))*a^(5/2)*(-cos(f*x+e)+1)^4-128*ln(4*(-a*(-cos(f*x+e)+1)^2*csc(f*x+e)
^2+b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a-b)^(1/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*
x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)+a-b)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2+1))*a^(3/2)*b*(-cos(f*
x+e)+1)^4+64*ln(4*(-a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a-b)^(1/2)*(a*(-cos(f*x
+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)+a-b)/((-c
os(f*x+e)+1)^2*csc(f*x+e)^2+1))*b^2*(-cos(f*x+e)+1)^4*a^(1/2)+11*a^(3/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a
*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(a-b)^(1/2)*(-cos(f*x+e)+1)^2*sin(
f*x+e)^2-10*b*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f
*x+e)^2+a)^(1/2)*(a-b)^(1/2)*(-cos(f*x+e)+1)^2*a^(1/2)*sin(f*x+e)^2-a^(3/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-
2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(a-b)^(1/2)*sin(f*x+e)^4)
Fricas [A] (verification not implemented)
none
Time = 0.32 (sec) , antiderivative size = 748, normalized size of antiderivative = 4.65
\[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {8 \, {\left (a^{2} - a b\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 ) \tan \left (f x + e\right )^{4} - {\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\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 ) \tan \left (f x + e\right )^{4} - 2 \, {\left ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a f \tan \left (f x + e\right )^{4}}, \frac {16 \, {\left (a^{2} - a b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{4} + {\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\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 ) \tan \left (f x + e\right )^{4} + 2 \, {\left ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a f \tan \left (f x + e\right )^{4}}, \frac {{\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{4} - 4 \, {\left (a^{2} - a b\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 ) \tan \left (f x + e\right )^{4} + {\left ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a f \tan \left (f x + e\right )^{4}}, \frac {{\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{4} + 8 \, {\left (a^{2} - a b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{4} + {\left ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a f \tan \left (f x + e\right )^{4}}\right ]
\]
[In]
integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
[-1/16*(8*(a^2 - a*b)*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))*tan(f*x + e)^4 - (8*a^2 - 12*a*b + 3*b^2)*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)*tan(f*x + e)^4 - 2*((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt
(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/16*(16*(a^2 - a*b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2
+ a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^2 - 12*a*b + 3*b^2)*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)*tan(f*x + e)^4 + 2*((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)
*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*
x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - 4*(a^2 - a*b)*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))*tan(f*x + e)^4 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*
a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(b*ta
n(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + 8*(a^2 - a*b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*s
qrt(-a + b)/(a - b))*tan(f*x + e)^4 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*
f*tan(f*x + e)^4)]
Sympy [F]
\[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot ^{5}{\left (e + f x \right )}\, dx
\]
[In]
integrate(cot(f*x+e)**5*(a+b*tan(f*x+e)**2)**(3/2),x)
[Out]
Integral((a + b*tan(e + f*x)**2)**(3/2)*cot(e + f*x)**5, x)
Maxima [F]
\[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{5} \,d x }
\]
[In]
integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*tan(f*x + e)^2 + a)^(3/2)*cot(f*x + e)^5, x)
Giac [F(-1)]
Timed out. \[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 11.41 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.59
\[
\int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (\frac {3\,a\,b^2}{8}-\frac {a^2\,b}{2}\right )+\frac {b\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}\,\left (4\,a-5\,b\right )}{8}}{f\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^2+a^2\,f-2\,a\,f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}-\frac {\mathrm {atanh}\left (\frac {9\,b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{32\,\left (\frac {a^3\,b^5}{4}-\frac {25\,a^2\,b^6}{32}+\frac {13\,a\,b^7}{16}-\frac {9\,b^8}{32}\right )}-\frac {a\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{4\,\left (\frac {a^3\,b^5}{4}-\frac {25\,a^2\,b^6}{32}+\frac {13\,a\,b^7}{16}-\frac {9\,b^8}{32}\right )}\right )\,\sqrt {{\left (a-b\right )}^3}}{f}-\frac {\mathrm {atanh}\left (\frac {75\,\sqrt {a}\,b^7\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{64\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}-\frac {159\,b^8\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,\sqrt {a}\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}-\frac {29\,a^{3/2}\,b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{32\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}+\frac {a^{5/2}\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{4\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}+\frac {27\,b^9\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,a^{3/2}\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}\right )\,\left (8\,a^2-12\,a\,b+3\,b^2\right )}{8\,\sqrt {a}\,f}
\]
[In]
int(cot(e + f*x)^5*(a + b*tan(e + f*x)^2)^(3/2),x)
[Out]
((a + b*tan(e + f*x)^2)^(1/2)*((3*a*b^2)/8 - (a^2*b)/2) + (b*(a + b*tan(e + f*x)^2)^(3/2)*(4*a - 5*b))/8)/(f*(
a + b*tan(e + f*x)^2)^2 + a^2*f - 2*a*f*(a + b*tan(e + f*x)^2)) - (atanh((9*b^6*(a + b*tan(e + f*x)^2)^(1/2)*(
3*a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(32*((13*a*b^7)/16 - (9*b^8)/32 - (25*a^2*b^6)/32 + (a^3*b^5)/4)) - (a*b
^5*(a + b*tan(e + f*x)^2)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(4*((13*a*b^7)/16 - (9*b^8)/32 - (25*a^
2*b^6)/32 + (a^3*b^5)/4)))*((a - b)^3)^(1/2))/f - (atanh((75*a^(1/2)*b^7*(a + b*tan(e + f*x)^2)^(1/2))/(64*((7
5*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))) - (159*b^8*(a + b*tan(e + f*x
)^2)^(1/2))/(256*a^(1/2)*((75*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))) -
(29*a^(3/2)*b^6*(a + b*tan(e + f*x)^2)^(1/2))/(32*((75*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5
)/4 + (27*b^9)/(256*a))) + (a^(5/2)*b^5*(a + b*tan(e + f*x)^2)^(1/2))/(4*((75*a*b^7)/64 - (159*b^8)/256 - (29*
a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))) + (27*b^9*(a + b*tan(e + f*x)^2)^(1/2))/(256*a^(3/2)*((75*a*b^7
)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))))*(8*a^2 - 12*a*b + 3*b^2))/(8*a^(1/2
)*f)