Integrand size = 17, antiderivative size = 33 \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=a \text {arctanh}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-a \cot (x) \sqrt {a \sec ^2(x)} \] Output:
a*arctanh(sin(x))*cos(x)*(a*sec(x)^2)^(1/2)-a*cot(x)*(a*sec(x)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=-a \cot (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\sin ^2(x)\right ) \sqrt {a \sec ^2(x)} \] Input:
Integrate[Cot[x]^2*(a + a*Tan[x]^2)^(3/2),x]
Output:
-(a*Cot[x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[x]^2]*Sqrt[a*Sec[x]^2])
Time = 0.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4140, 3042, 4613, 3042, 3101, 25, 262, 219}
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.
\(\displaystyle \int \cot ^2(x) \left (a \tan ^2(x)+a\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \tan (x)^2+a\right )^{3/2}}{\tan (x)^2}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \cot ^2(x) \left (a \sec ^2(x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sec (x)^2\right )^{3/2}}{\tan (x)^2}dx\) |
\(\Big \downarrow \) 4613 |
\(\displaystyle a \cos (x) \sqrt {a \sec ^2(x)} \int \csc ^2(x) \sec (x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \cos (x) \sqrt {a \sec ^2(x)} \int \csc (x)^2 \sec (x)dx\) |
\(\Big \downarrow \) 3101 |
\(\displaystyle -a \cos (x) \sqrt {a \sec ^2(x)} \int -\frac {\csc ^2(x)}{1-\csc ^2(x)}d\csc (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \cos (x) \sqrt {a \sec ^2(x)} \int \frac {\csc ^2(x)}{1-\csc ^2(x)}d\csc (x)\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -a \cos (x) \sqrt {a \sec ^2(x)} \left (\csc (x)-\int \frac {1}{1-\csc ^2(x)}d\csc (x)\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -a \cos (x) \sqrt {a \sec ^2(x)} (\csc (x)-\text {arctanh}(\csc (x)))\) |
Input:
Int[Cot[x]^2*(a + a*Tan[x]^2)^(3/2),x]
Output:
-(a*Cos[x]*(-ArcTanh[Csc[x]] + Csc[x])*Sqrt[a*Sec[x]^2])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[(u_.)*((b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sec[e + f*x]^ n)^FracPart[p]/(Sec[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Se c[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.73 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64
method | result | size |
derivativedivides | \(-\frac {\left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}{\tan \left (x \right )}+a \tan \left (x \right ) \sqrt {a +a \tan \left (x \right )^{2}}+a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \left (x \right )+\sqrt {a +a \tan \left (x \right )^{2}}\right )\) | \(54\) |
default | \(-\frac {\left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}{\tan \left (x \right )}+a \tan \left (x \right ) \sqrt {a +a \tan \left (x \right )^{2}}+a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \left (x \right )+\sqrt {a +a \tan \left (x \right )^{2}}\right )\) | \(54\) |
risch | \(-\frac {2 i a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}{{\mathrm e}^{2 i x}-1}+2 a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )-2 a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )\) | \(104\) |
Input:
int(cot(x)^2*(a+a*tan(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/tan(x)*(a+a*tan(x)^2)^(3/2)+a*tan(x)*(a+a*tan(x)^2)^(1/2)+a^(3/2)*ln(a^ (1/2)*tan(x)+(a+a*tan(x)^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a \tan \left (x\right )^{2} + a} \sqrt {a} \tan \left (x\right ) + a\right ) \tan \left (x\right ) - 2 \, \sqrt {a \tan \left (x\right )^{2} + a} a}{2 \, \tan \left (x\right )} \] Input:
integrate(cot(x)^2*(a+a*tan(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/2*(a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a) *tan(x) - 2*sqrt(a*tan(x)^2 + a)*a)/tan(x)
\[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\int \left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac {3}{2}} \cot ^{2}{\left (x \right )}\, dx \] Input:
integrate(cot(x)**2*(a+a*tan(x)**2)**(3/2),x)
Output:
Integral((a*(tan(x)**2 + 1))**(3/2)*cot(x)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (29) = 58\).
Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.06 \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=-\frac {{\left (4 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) - {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) + 4 \, a \sin \left (x\right )\right )} \sqrt {a}}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} \] Input:
integrate(cot(x)^2*(a+a*tan(x)^2)^(3/2),x, algorithm="maxima")
Output:
-1/2*(4*a*cos(x)*sin(2*x) - 4*a*cos(2*x)*sin(x) - (a*cos(2*x)^2 + a*sin(2* x)^2 - 2*a*cos(2*x) + a)*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + (a*cos( 2*x)^2 + a*sin(2*x)^2 - 2*a*cos(2*x) + a)*log(cos(x)^2 + sin(x)^2 - 2*sin( x) + 1) + 4*a*sin(x))*sqrt(a)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
Time = 0.51 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \, {\left (\sqrt {a} \log \left ({\left (\sqrt {a} \tan \left (x\right ) - \sqrt {a \tan \left (x\right )^{2} + a}\right )}^{2}\right ) - \frac {4 \, a^{\frac {3}{2}}}{{\left (\sqrt {a} \tan \left (x\right ) - \sqrt {a \tan \left (x\right )^{2} + a}\right )}^{2} - a}\right )} a \] Input:
integrate(cot(x)^2*(a+a*tan(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/2*(sqrt(a)*log((sqrt(a)*tan(x) - sqrt(a*tan(x)^2 + a))^2) - 4*a^(3/2)/( (sqrt(a)*tan(x) - sqrt(a*tan(x)^2 + a))^2 - a))*a
Timed out. \[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\int {\mathrm {cot}\left (x\right )}^2\,{\left (a\,{\mathrm {tan}\left (x\right )}^2+a\right )}^{3/2} \,d x \] Input:
int(cot(x)^2*(a + a*tan(x)^2)^(3/2),x)
Output:
int(cot(x)^2*(a + a*tan(x)^2)^(3/2), x)
\[ \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\tan \left (x \right )^{2}+1}\, \cot \left (x \right )^{2} \tan \left (x \right )^{2}d x +\int \sqrt {\tan \left (x \right )^{2}+1}\, \cot \left (x \right )^{2}d x \right ) \] Input:
int(cot(x)^2*(a+a*tan(x)^2)^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(tan(x)**2 + 1)*cot(x)**2*tan(x)**2,x) + int(sqrt(tan(x )**2 + 1)*cot(x)**2,x))