\(\int \cot ^4(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [317]

Optimal result

Integrand size = 25, antiderivative size = 115 \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f} \]

[Out]

(a-b)^(3/2)*arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/f+1/3*(3*a-4*b)*cot(f*x+e)*(a+b*tan(f*x+e)
^2)^(1/2)/f-1/3*a*cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/f

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3751, 485, 597, 12, 385, 209} \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}+\frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f} \]

[In]

Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 4*b)*Cot[e + f*x]*Sq
rt[a + b*Tan[e + f*x]^2])/(3*f) - (a*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 385

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

Rule 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 597

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

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^4 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}+\frac {\text {Subst}\left (\int \frac {-a (3 a-4 b)-(2 a-3 b) b x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 f} \\ & = \frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {\text {Subst}\left (\int -\frac {3 a (a-b)^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a f} \\ & = \frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f} \\ & = \frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {(3 a-4 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f}-\frac {a \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.43 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68 \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {\cot (e+f x) \left (b+a \cot ^2(e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {(a-b) \tan ^2(e+f x)}{a+b \tan ^2(e+f x)}\right ) \sqrt {a+b \tan ^2(e+f x)}}{3 f} \]

[In]

Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

-1/3*(Cot[e + f*x]*(b + a*Cot[e + f*x]^2)*Hypergeometric2F1[-3/2, 1, -1/2, -(((a - b)*Tan[e + f*x]^2)/(a + b*T
an[e + f*x]^2))]*Sqrt[a + b*Tan[e + f*x]^2])/f

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(101)=202\).

Time = 4.79 (sec) , antiderivative size = 667, normalized size of antiderivative = 5.80

method	result	size
default	\(-\frac {\left (4 \sin \left (f x +e \right )^{2} \sqrt {a -b}\, \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b -3 \sin \left (f x +e \right ) a^{2} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) \cos \left (f x +e \right )+6 \sin \left (f x +e \right ) a b \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) \cos \left (f x +e \right )-3 \sin \left (f x +e \right ) b^{2} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) \cos \left (f x +e \right )+4 \cos \left (f x +e \right )^{2} \sqrt {a -b}\, \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, a +3 \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) a^{2} \sin \left (f x +e \right )-6 \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) a b \sin \left (f x +e \right )+3 \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) b^{2} \sin \left (f x +e \right )-3 \sqrt {a -b}\, \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, a \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \cot \left (f x +e \right )^{3}}{3 f \sqrt {a -b}\, \left (a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}\right ) \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\)	\(667\)

[In]

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/f/(a-b)^(1/2)*(4*sin(f*x+e)^2*(a-b)^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b-3*
sin(f*x+e)*a^2*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f
*x+e)))*cos(f*x+e)+6*sin(f*x+e)*a*b*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1
/2)*(cot(f*x+e)+csc(f*x+e)))*cos(f*x+e)-3*sin(f*x+e)*b^2*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)
/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*cos(f*x+e)+4*cos(f*x+e)^2*(a-b)^(1/2)*((a*cos(f*x+e)^2-b*cos
(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a+3*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^
2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a^2*sin(f*x+e)-6*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(
f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*b*sin(f*x+e)+3*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e
)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*b^2*sin(f*x+e)-3*(a-b)^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+
e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a)*(a+b*tan(f*x+e)^2)^(3/2)/(a*cos(f*x+e)^2+b*sin(f*x+e)^2)/((a*cos(f*x+e)^2-b
*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*cot(f*x+e)^3

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.68 \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {3 \, {\left (a - b\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} - 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} - 4 \, {\left ({\left (3 \, a - 4 \, b\right )} \tan \left (f x + e\right )^{2} - a\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, f \tan \left (f x + e\right )^{3}}, \frac {3 \, {\left (a - b\right )}^{\frac {3}{2}} \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left ({\left (3 \, a - 4 \, b\right )} \tan \left (f x + e\right )^{2} - a\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, f \tan \left (f x + e\right )^{3}}\right ] \]

[In]

integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(a - b)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 +
a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4
+ 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^3 - 4*((3*a - 4*b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(f*ta
n(f*x + e)^3), 1/6*(3*(a - b)^(3/2)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*t
an(f*x + e)^2 - a))*tan(f*x + e)^3 + 2*((3*a - 4*b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(f*tan(f*x
+ e)^3)]

Sympy [F]

\[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot ^{4}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**4*(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)**4, x)

Maxima [F]

\[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(cot(f*x+e)^4*(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)^4, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]

[In]

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

[Out]

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