Integrand size = 15, antiderivative size = 75 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-b \sqrt {a+b \cot ^2(x)} \] Output:
a^(3/2)*arctanh((a+b*cot(x)^2)^(1/2)/a^(1/2))-(a-b)^(3/2)*arctanh((a+b*cot (x)^2)^(1/2)/(a-b)^(1/2))-b*(a+b*cot(x)^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-b \sqrt {a+b \cot ^2(x)} \] Input:
Integrate[(a + b*Cot[x]^2)^(3/2)*Tan[x],x]
Output:
a^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - (a - b)^(3/2)*ArcTanh[Sqrt [a + b*Cot[x]^2]/Sqrt[a - b]] - b*Sqrt[a + b*Cot[x]^2]
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 25, 4153, 25, 354, 95, 174, 73, 221}
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.
| \(\displaystyle \int \tan (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}}{\tan \left (x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}}{\tan \left (x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int -\frac {\tan (x) \left (a+b \cot ^2(x)\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b \cot ^2(x)+a\right )^{3/2} \tan (x)}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {1}{2} \int \frac {\left (b \cot ^2(x)+a\right )^{3/2} \tan (x)}{\cot ^2(x)+1}d\cot ^2(x)\) |
| \(\Big \downarrow \) 95 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\left (a^2+(2 a-b) b \cot ^2(x)\right ) \tan (x)}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)-2 b \sqrt {a+b \cot ^2(x)}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (a^2 \left (-\int \frac {\tan (x)}{\sqrt {b \cot ^2(x)+a}}d\cot ^2(x)\right )+(a-b)^2 \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)-2 b \sqrt {a+b \cot ^2(x)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 a^2 \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \cot ^2(x)+a}}{b}+\frac {2 (a-b)^2 \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}-2 b \sqrt {a+b \cot ^2(x)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )-2 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-2 b \sqrt {a+b \cot ^2(x)}\right )\) |
Input:
Int[(a + b*Cot[x]^2)^(3/2)*Tan[x],x]
Output:
(2*a^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - 2*(a - b)^(3/2)*ArcTanh [Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - 2*b*Sqrt[a + b*Cot[x]^2])/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p - 1)/(b*d*(p - 1))), x] + Simp[1/(b*d) Int[(b *d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*((e + f*x)^(p - 2)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(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]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(61)=122\).
Time = 3.64 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.99
| method | result | size |
| default | \(-\frac {\sqrt {4}\, \left (\left (-1+\cos \left (x \right )\right ) \sin \left (x \right ) \arctan \left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {-a +b}\, \left (-1+\cos \left (x \right )\right )}\right ) a^{2}+\left (2-2 \cos \left (x \right )\right ) \sin \left (x \right ) \arctan \left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {-a +b}\, \left (-1+\cos \left (x \right )\right )}\right ) a b +\left (-1+\cos \left (x \right )\right ) \sin \left (x \right ) \arctan \left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {-a +b}\, \left (-1+\cos \left (x \right )\right )}\right ) b^{2}+\left (-1+\cos \left (x \right )\right ) \sin \left (x \right ) a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {a}\, \left (-1+\cos \left (x \right )\right )}\right ) \sqrt {-a +b}-b \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, \sin \left (x \right )^{2}\right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{2 \sqrt {-a +b}\, \left (-\cos \left (x \right )^{2} b -a \sin \left (x \right )^{2}\right ) \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(299\) |
Input:
int((a+b*cot(x)^2)^(3/2)*tan(x),x,method=_RETURNVERBOSE)
Output:
-1/2*4^(1/2)/(-a+b)^(1/2)*((-1+cos(x))*sin(x)*arctan(1/(-a+b)^(1/2)*((cos( x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*sin(x)/(-1+cos(x)))*a^2+(2-2*cos(x) )*sin(x)*arctan(1/(-a+b)^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2 )*sin(x)/(-1+cos(x)))*a*b+(-1+cos(x))*sin(x)*arctan(1/(-a+b)^(1/2)*((cos(x )^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*sin(x)/(-1+cos(x)))*b^2+(-1+cos(x))* sin(x)*a^(3/2)*arctanh(1/a^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1 /2)*sin(x)/(-1+cos(x)))*(-a+b)^(1/2)-b*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1) ^2)^(1/2)*(-a+b)^(1/2)*sin(x)^2)*(a+b*cot(x)^2)^(3/2)/(-cos(x)^2*b-a*sin(x )^2)/((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)
Time = 0.58 (sec) , antiderivative size = 565, normalized size of antiderivative = 7.53 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cot(x)^2)^(3/2)*tan(x),x, algorithm="fricas")
Output:
[1/2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)* tan(x)^2 + b) - 1/4*(a - b)^(3/2)*log(-((8*a^2 - 8*a*b + b^2)*tan(x)^4 + 2 *(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sqrt (a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - b* sqrt((a*tan(x)^2 + b)/tan(x)^2), -sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x )^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) - 1/4*(a - b)^(3/2)*log(-((8 *a^2 - 8*a*b + b^2)*tan(x)^4 + 2*(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/( tan(x)^4 + 2*tan(x)^2 + 1)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), 1/2*(-a + b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/ ((2*a - b)*tan(x)^2 + b)) + 1/2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt( (a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - b*sqrt((a*tan(x)^2 + b)/tan(x)^ 2), -sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/( a*tan(x)^2 + b)) + 1/2*(-a + b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x )^2 + b)/tan(x)^2)*tan(x)^2/((2*a - b)*tan(x)^2 + b)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2)]
\[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=\int \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \tan {\left (x \right )}\, dx \] Input:
integrate((a+b*cot(x)**2)**(3/2)*tan(x),x)
Output:
Integral((a + b*cot(x)**2)**(3/2)*tan(x), x)
\[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=\int { {\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (x\right ) \,d x } \] Input:
integrate((a+b*cot(x)^2)^(3/2)*tan(x),x, algorithm="maxima")
Output:
integrate((b*cot(x)^2 + a)^(3/2)*tan(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.28 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=\frac {1}{2} \, {\left (\frac {2 \, \sqrt {a - b} a^{2} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b}} + {\left (a - b\right )}^{\frac {3}{2}} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) + \frac {4 \, \sqrt {a - b} b^{2}}{{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((a+b*cot(x)^2)^(3/2)*tan(x),x, algorithm="giac")
Output:
1/2*(2*sqrt(a - b)*a^2*arctan(1/2*((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - 2*a + b)/sqrt(-a^2 + a*b))/sqrt(-a^2 + a*b) + (a - b )^(3/2)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2) + 4*sqrt(a - b)*b^2/((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b) )^2 - b))*sgn(sin(x))
Time = 9.17 (sec) , antiderivative size = 506, normalized size of antiderivative = 6.75 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=\mathrm {atanh}\left (\frac {2\,b^6\,\sqrt {a^3}\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{-6\,a^5\,b^3+12\,a^4\,b^4-8\,a^3\,b^5+2\,a^2\,b^6}-\frac {8\,a\,b^5\,\sqrt {a^3}\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{-6\,a^5\,b^3+12\,a^4\,b^4-8\,a^3\,b^5+2\,a^2\,b^6}+\frac {12\,a^2\,b^4\,\sqrt {a^3}\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{-6\,a^5\,b^3+12\,a^4\,b^4-8\,a^3\,b^5+2\,a^2\,b^6}-\frac {6\,a^3\,b^3\,\sqrt {a^3}\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{-6\,a^5\,b^3+12\,a^4\,b^4-8\,a^3\,b^5+2\,a^2\,b^6}\right )\,\sqrt {a^3}-\mathrm {atanh}\left (\frac {2\,a\,b^5\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{6\,a^5\,b^3-18\,a^4\,b^4+20\,a^3\,b^5-10\,a^2\,b^6+2\,a\,b^7}-\frac {6\,a^2\,b^4\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{6\,a^5\,b^3-18\,a^4\,b^4+20\,a^3\,b^5-10\,a^2\,b^6+2\,a\,b^7}+\frac {6\,a^3\,b^3\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{6\,a^5\,b^3-18\,a^4\,b^4+20\,a^3\,b^5-10\,a^2\,b^6+2\,a\,b^7}\right )\,\sqrt {{\left (a-b\right )}^3}-b\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}} \] Input:
int(tan(x)*(a + b*cot(x)^2)^(3/2),x)
Output:
atanh((2*b^6*(a^3)^(1/2)*(a + b/tan(x)^2)^(1/2))/(2*a^2*b^6 - 8*a^3*b^5 + 12*a^4*b^4 - 6*a^5*b^3) - (8*a*b^5*(a^3)^(1/2)*(a + b/tan(x)^2)^(1/2))/(2* a^2*b^6 - 8*a^3*b^5 + 12*a^4*b^4 - 6*a^5*b^3) + (12*a^2*b^4*(a^3)^(1/2)*(a + b/tan(x)^2)^(1/2))/(2*a^2*b^6 - 8*a^3*b^5 + 12*a^4*b^4 - 6*a^5*b^3) - ( 6*a^3*b^3*(a^3)^(1/2)*(a + b/tan(x)^2)^(1/2))/(2*a^2*b^6 - 8*a^3*b^5 + 12* a^4*b^4 - 6*a^5*b^3))*(a^3)^(1/2) - atanh((2*a*b^5*(a + b/tan(x)^2)^(1/2)* (3*a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(2*a*b^7 - 10*a^2*b^6 + 20*a^3*b^5 - 18*a^4*b^4 + 6*a^5*b^3) - (6*a^2*b^4*(a + b/tan(x)^2)^(1/2)*(3*a*b^2 - 3 *a^2*b + a^3 - b^3)^(1/2))/(2*a*b^7 - 10*a^2*b^6 + 20*a^3*b^5 - 18*a^4*b^4 + 6*a^5*b^3) + (6*a^3*b^3*(a + b/tan(x)^2)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(2*a*b^7 - 10*a^2*b^6 + 20*a^3*b^5 - 18*a^4*b^4 + 6*a^5*b^3 ))*((a - b)^3)^(1/2) - b*(a + b/tan(x)^2)^(1/2)
\[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan (x) \, dx=\left (\int \sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{2} \tan \left (x \right )d x \right ) b +\left (\int \sqrt {\cot \left (x \right )^{2} b +a}\, \tan \left (x \right )d x \right ) a \] Input:
int((a+b*cot(x)^2)^(3/2)*tan(x),x)
Output:
int(sqrt(cot(x)**2*b + a)*cot(x)**2*tan(x),x)*b + int(sqrt(cot(x)**2*b + a )*tan(x),x)*a