\(\int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx\) [193]

Optimal result

Integrand size = 21, antiderivative size = 210 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d}{f \sqrt {d \cot (e+f x)}}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]

[Out]

-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)+2*d/f/(d*cot(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 210, 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, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} f}+\frac {2 d}{f \sqrt {d \cot (e+f x)}}+\frac {\sqrt {d} \log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f}-\frac {\sqrt {d} \log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f} \]

[In]

Int[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^2,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)/(f*Sqrt[d*Cot[e + f*x]]) + (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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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^2 \int \frac {1}{(d \cot (e+f x))^{3/2}} \, dx \\ & = \frac {2 d}{f \sqrt {d \cot (e+f x)}}-\int \sqrt {d \cot (e+f x)} \, dx \\ & = \frac {2 d}{f \sqrt {d \cot (e+f x)}}+\frac {d \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = \frac {2 d}{f \sqrt {d \cot (e+f x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2 d}{f \sqrt {d \cot (e+f x)}}-\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}{f \sqrt {d \cot (e+f x)}}+\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}{f \sqrt {d \cot (e+f x)}}+\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}{f \sqrt {d \cot (e+f x)}}+\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 = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.38 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {d \left (2+\arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}\right )}{f \sqrt {d \cot (e+f x)}} \]

[In]

Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^2,x]

[Out]

(d*(2 + ArcTan[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(1/4) - ArcTanh[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f
*x]^2)^(1/4)))/(f*Sqrt[d*Cot[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(655\) vs. \(2(161)=322\).

Time = 18.90 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.12

method	result	size
default	\(\frac {\csc \left (f x +e \right ) \sqrt {-\frac {d \left (\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}-\sin \left (f x +e \right )\right )}{1-\cos \left (f x +e \right )}}\, \left (1-\cos \left (f x +e \right )\right ) \left (\ln \left (\frac {\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+2-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+1-\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-\ln \left (-\frac {-\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2+2 \cos \left (f x +e \right )+\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-1+\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-8 \csc \left (f x +e \right )+8 \cot \left (f x +e \right )\right ) \sqrt {2}}{4 f \sqrt {\csc \left (f x +e \right ) \left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right ) \left (1-\cos \left (f x +e \right )\right )}\, \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}}\)	\(656\)

[In]

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

[Out]

1/4/f*csc(f*x+e)*(-d/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2-sin(f*x+e)))^(1/2)*(1-cos(f*x+e))*(ln(1/(1-co
s(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2+2*sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2
)+2-2*cos(f*x+e)-sin(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-2*arctan(1/(1-cos(f*
x+e))*(sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)+1-cos(f*x+e)))*(csc(f*x+e)^3*(1-
cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-ln(-1/(1-cos(f*x+e))*(-csc(f*x+e)*(1-cos(f*x+e))^2+2*sin(f*x+e)*(cs
c(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-2+2*cos(f*x+e)+sin(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+
e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-2*arctan(1/(1-cos(f*x+e))*(sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*
x+e)+cot(f*x+e))^(1/2)-1+cos(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-8*csc(f*x+e)
+8*cot(f*x+e))/(csc(f*x+e)*(csc(f*x+e)^2*(1-cos(f*x+e))^2-1)*(1-cos(f*x+e)))^(1/2)/(csc(f*x+e)^3*(1-cos(f*x+e)
)^3-csc(f*x+e)+cot(f*x+e))^(1/2)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.95 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (i \, f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 4 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{2 \, f} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\int \sqrt {d \cot {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx \]

[In]

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

[Out]

Integral(sqrt(d*cot(e + f*x))*tan(e + f*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {d^{3} {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}}{d^{2}} + \frac {8}{d^{2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}\right )}}{4 \, f} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\int { \sqrt {d \cot \left (f x + e\right )} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(sqrt(d*cot(f*x + e))*tan(f*x + e)^2, x)

Mupad [B] (verification not implemented)

Time = 3.60 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.38 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {2\,d}{f\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}+\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f}-\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f} \]

[In]

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

[Out]

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