\(\int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx\) [192]

Optimal result

Integrand size = 21, antiderivative size = 214 \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]

[Out]

2/3*d^2/f/(d*cot(f*x+e))^(3/2)-1/2*arctan(1-2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/f*2^(1/2)+1/2*arctan
(1+2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/f*2^(1/2)-1/4*ln(d^(1/2)+cot(f*x+e)*d^(1/2)-2^(1/2)*(d*cot(f*
x+e))^(1/2))*d^(1/2)/f*2^(1/2)+1/4*ln(d^(1/2)+cot(f*x+e)*d^(1/2)+2^(1/2)*(d*cot(f*x+e))^(1/2))*d^(1/2)/f*2^(1/
2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} f}+\frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\sqrt {d} \log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f} \]

[In]

Int[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^3,x]

[Out]

-((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqr
t[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^2)/(3*f*(d*Cot[e + f*x])^(3/2)) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[
d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] +
Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 210

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

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = d^3 \int \frac {1}{(d \cot (e+f x))^{5/2}} \, dx \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-d \int \frac {1}{\sqrt {d \cot (e+f x)}} \, dx \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}+\frac {d^2 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}+\frac {d \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f}+\frac {d \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f}+\frac {d \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f} \\ & = \frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f} \\ & = -\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.43 \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=-\frac {\sqrt {d \cot (e+f x)} \left (-2+3 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}\right ) \tan ^2(e+f x)}{3 f} \]

[In]

Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^3,x]

[Out]

-1/3*(Sqrt[d*Cot[e + f*x]]*(-2 + 3*ArcTan[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(3/4) + 3*ArcTanh[(-Cot[e
+ f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(3/4))*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. \(567\) vs. \(2(163)=326\).

Time = 17.70 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.65

method	result	size
default	\(-\frac {\left (\sec ^{2}\left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right ) \left (-6 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )-6 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )+3 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-3 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )-2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )+4 \sqrt {2}\, \cos \left (f x +e \right )-4 \sqrt {2}\right ) \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}}{12 f}\)	\(568\)

[In]

int((cot(f*x+e)*d)^(1/2)*tan(f*x+e)^3,x,method=_RETURNVERBOSE)

[Out]

-1/12/f*sec(f*x+e)^2*(cos(f*x+e)+1)*(-6*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)*arctan((2^(
1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1))-6*(-sin(f*x+e)*c
os(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*s
in(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))+3*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)*ln(-(cot(
f*x+e)*cos(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)
+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))-3*(-sin(f*x+e
)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)*ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)-2*sin(f*x+e)*(-cot(f*x
+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+
e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))+4*2^(1/2)*cos(f*x+e)-4*2^(1/2))*(cot(f*x+e)*d)^(1/2)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=\frac {4 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )^{2} + 3 \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 3 i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 3 i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 3 \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}{6 \, f} \]

[In]

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^3,x, algorithm="fricas")

[Out]

1/6*(4*sqrt(d/tan(f*x + e))*tan(f*x + e)^2 + 3*f*(-d^2/f^4)^(1/4)*log(f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e)
)) + 3*I*f*(-d^2/f^4)^(1/4)*log(I*f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e))) - 3*I*f*(-d^2/f^4)^(1/4)*log(-I*f
*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e))) - 3*f*(-d^2/f^4)^(1/4)*log(-f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e)
)))/f

Sympy [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=\int \sqrt {d \cot {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx \]

[In]

integrate((d*cot(f*x+e))**(1/2)*tan(f*x+e)**3,x)

[Out]

Integral(sqrt(d*cot(e + f*x))*tan(e + f*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.89 \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=\frac {d^{4} {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{d^{2}} + \frac {8}{d^{2} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}\right )}}{12 \, f} \]

[In]

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^3,x, algorithm="maxima")

[Out]

1/12*d^4*(3*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d/tan(f*x + e)))/sqrt(d))/d^(3/2) + 2*sqrt
(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d/tan(f*x + e)))/sqrt(d))/d^(3/2) + sqrt(2)*log(sqrt(2)*sqrt
(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/d^(3/2) - sqrt(2)*log(-sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d
+ d/tan(f*x + e))/d^(3/2))/d^2 + 8/(d^2*(d/tan(f*x + e))^(3/2)))/f

Giac [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=\int { \sqrt {d \cot \left (f x + e\right )} \tan \left (f x + e\right )^{3} \,d x } \]

[In]

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^3,x, algorithm="giac")

[Out]

integrate(sqrt(d*cot(f*x + e))*tan(f*x + e)^3, x)

Mupad [B] (verification not implemented)

Time = 2.88 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.39 \[ \int \sqrt {d \cot (e+f x)} \tan ^3(e+f x) \, dx=\frac {2\,d^2}{3\,f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f}-\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f} \]

[In]

int(tan(e + f*x)^3*(d*cot(e + f*x))^(1/2),x)

[Out]

(2*d^2)/(3*f*(d/tan(e + f*x))^(3/2)) - ((-1)^(1/4)*d^(1/2)*atan(((-1)^(1/4)*(d/tan(e + f*x))^(1/2))/d^(1/2))*1
i)/f - ((-1)^(1/4)*d^(1/2)*atanh(((-1)^(1/4)*(d/tan(e + f*x))^(1/2))/d^(1/2))*1i)/f