Integrand size = 21, antiderivative size = 155 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}+\sqrt {d} \cot (e+f x)}\right )}{\sqrt {2} f}+\frac {2 d}{f \sqrt {d \cot (e+f x)}} \] Output:
-1/2*d^(1/2)*arctan(1-2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/f+1/2* d^(1/2)*arctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/f-1/2*d^(1/ 2)*arctanh(2^(1/2)*(d*cot(f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*cot(f*x+e)))*2^(1 /2)/f+2*d/f/(d*cot(f*x+e))^(1/2)
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.52 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {d \left (2+\arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}\right )}{f \sqrt {d \cot (e+f x)}} \] Input:
Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^2,x]
Output:
(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]] )
Time = 0.49 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.30, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 2030, 3955, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \tan ^2(e+f x) \sqrt {d \cot (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}}{\tan \left (e+f x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d^2 \int \frac {1}{\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle d^2 \left (\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {d \cot (e+f x)}dx}{d^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^2 \left (\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{d^2}\right )\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle d^2 \left (\frac {\int \frac {\sqrt {d \cot (e+f x)}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 266 |
\(\displaystyle d^2 \left (\frac {2 \int \frac {d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 826 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle d^2 \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}\right )\) |
Input:
Int[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^2,x]
Output:
d^2*(2/(d*f*Sqrt[d*Cot[e + f*x]]) + (2*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Cot[ e + f*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sqrt[d]*Cot[e + f*x]]/(S qrt[2]*Sqrt[d]))/2 + (Log[d - Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f *x]^2]/(2*Sqrt[2]*Sqrt[d]) - Log[d + Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Co t[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]))/2))/(d*f))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(122)=244\).
Time = 7.26 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.94
method | result | size |
default | \(-\frac {\sec \left (f x +e \right ) \csc \left (f x +e \right )^{2} \sqrt {d \cot \left (f x +e \right )}\, \left (1-\cos \left (f x +e \right )\right ) \left (\ln \left (\frac {\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-\ln \left (-\frac {-\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-2+2 \cos \left (f x +e \right )+\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-8 \csc \left (f x +e \right )+8 \cot \left (f x +e \right )\right ) \left (1+\cos \left (f x +e \right )\right )^{2}}{4 f}\) | \(456\) |
Input:
int((d*cot(f*x+e))^(1/2)*tan(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
-1/4/f*sec(f*x+e)*csc(f*x+e)^2*(d*cot(f*x+e))^(1/2)*(1-cos(f*x+e))*(ln(1/( 1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2+2*sin(f*x+e)*(-2*sin(f*x+e)*cos (f*x+e)/(1+cos(f*x+e))^2)^(1/2)+2-2*cos(f*x+e)-sin(f*x+e)))*(-2*sin(f*x+e) *cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+2*arctan((sin(f*x+e)*(-2*sin(f*x+e)*co s(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)-1))*(-2*sin(f*x +e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-ln(-1/(1-cos(f*x+e))*(-csc(f*x+e)*( 1-cos(f*x+e))^2+2*sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^( 1/2)-2+2*cos(f*x+e)+sin(f*x+e)))*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^ 2)^(1/2)+2*arctan((sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^ (1/2)+cos(f*x+e)-1)/(cos(f*x+e)-1))*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e ))^2)^(1/2)-8*csc(f*x+e)+8*cot(f*x+e))*(1+cos(f*x+e))^2
Time = 0.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.32 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d}{d}\right ) + 2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - d}{d}\right ) - \sqrt {2} \sqrt {d} \log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right ) + d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )}\right ) + \sqrt {2} \sqrt {d} \log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right ) - d \tan \left (f x + e\right ) - d}{\tan \left (f x + e\right )}\right ) + 8 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{4 \, f} \] Input:
integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^2,x, algorithm="fricas")
Output:
1/4*(2*sqrt(2)*sqrt(d)*arctan((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d)/d ) + 2*sqrt(2)*sqrt(d)*arctan((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) - d)/d) - sqrt(2)*sqrt(d)*log((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e))*tan(f*x + e) + d*tan(f*x + e) + d)/tan(f*x + e)) + sqrt(2)*sqrt(d)*log(-(sqrt(2)*sqrt(d )*sqrt(d/tan(f*x + e))*tan(f*x + e) - d*tan(f*x + e) - d)/tan(f*x + e)) + 8*sqrt(d/tan(f*x + e))*tan(f*x + e))/f
\[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\int \sqrt {d \cot {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate((d*cot(f*x+e))**(1/2)*tan(f*x+e)**2,x)
Output:
Integral(sqrt(d*cot(e + f*x))*tan(e + f*x)**2, x)
Time = 0.14 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.22 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {d^{3} {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}}{d^{2}} + \frac {8}{d^{2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}\right )}}{4 \, f} \] Input:
integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^2,x, algorithm="maxima")
Output:
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)*s qrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d) + sqrt(2)*log(-sqrt(2)*s qrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d))/d^2 + 8/(d^2*sq rt(d/tan(f*x + e))))/f
\[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\int { \sqrt {d \cot \left (f x + e\right )} \tan \left (f x + e\right )^{2} \,d x } \] Input:
integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^2,x, algorithm="giac")
Output:
integrate(sqrt(d*cot(f*x + e))*tan(f*x + e)^2, x)
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.52 \[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\frac {2\,d}{f\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}+\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f}-\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f} \] Input:
int(tan(e + f*x)^2*(d*cot(e + f*x))^(1/2),x)
Output:
(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
\[ \int \sqrt {d \cot (e+f x)} \tan ^2(e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\cot \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) \] Input:
int((d*cot(f*x+e))^(1/2)*tan(f*x+e)^2,x)
Output:
sqrt(d)*int(sqrt(cot(e + f*x))*tan(e + f*x)**2,x)