Integrand size = 15, antiderivative size = 36 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a \csc ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {1}{\sqrt {a \csc ^2(x)}} \] Output:
arctanh((a*csc(x)^2)^(1/2)/a^(1/2))/a^(1/2)-1/(a*csc(x)^2)^(1/2)
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.53 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {-1+\text {arctanh}(\sin (x)) \csc (x)}{\sqrt {a \csc ^2(x)}} \] Input:
Integrate[Tan[x]/Sqrt[a + a*Cot[x]^2],x]
Output:
(-1 + ArcTanh[Sin[x]]*Csc[x])/Sqrt[a*Csc[x]^2]
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 25, 4140, 25, 3042, 25, 4612, 25, 61, 73, 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.
\(\displaystyle \int \frac {\tan (x)}{\sqrt {a \cot ^2(x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \sqrt {a \tan \left (x+\frac {\pi }{2}\right )^2+a}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \sqrt {a \tan \left (x+\frac {\pi }{2}\right )^2+a}}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle -\int -\frac {\tan (x)}{\sqrt {a \csc ^2(x)}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\tan (x)}{\sqrt {a \csc ^2(x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \sqrt {a \sec \left (x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\sqrt {a \sec \left (x+\frac {\pi }{2}\right )^2} \tan \left (x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4612 |
\(\displaystyle -\frac {1}{2} a \int -\frac {1}{\left (a \csc ^2(x)\right )^{3/2} \left (1-\csc ^2(x)\right )}d\csc ^2(x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} a \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2} \left (1-\csc ^2(x)\right )}d\csc ^2(x)\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{a \sqrt {a \csc ^2(x)}}-\frac {\int \frac {1}{\sqrt {a \csc ^2(x)} \left (1-\csc ^2(x)\right )}d\csc ^2(x)}{a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{a \sqrt {a \csc ^2(x)}}-\frac {2 \int \frac {1}{1-\frac {\csc ^4(x)}{a}}d\sqrt {a \csc ^2(x)}}{a^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{a \sqrt {a \csc ^2(x)}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \csc ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}\right )\) |
Input:
Int[Tan[x]/Sqrt[a + a*Cot[x]^2],x]
Output:
-1/2*(a*((-2*ArcTanh[Sqrt[a*Csc[x]^2]/Sqrt[a]])/a^(3/2) + 2/(a*Sqrt[a*Csc[ x]^2])))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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[((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[(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[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Simp[b/(2*f) Subst[Int[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x ], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p] && Int egerQ[(m - 1)/2]
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\left (\ln \left (\csc \left (x \right )-\cot \left (x \right )+1\right )-\ln \left (\csc \left (x \right )-\cot \left (x \right )-1\right )-\sin \left (x \right )\right ) \csc \left (x \right )}{\sqrt {a \csc \left (x \right )^{2}}}\) | \(37\) |
risch | \(-\frac {{\mathrm e}^{2 i x}}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}+\frac {1}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}-i\right )}{\sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}+i\right )}{\sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}\) | \(157\) |
Input:
int(tan(x)/(a+a*cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
(ln(csc(x)-cot(x)+1)-ln(csc(x)-cot(x)-1)-sin(x))/(a*csc(x)^2)^(1/2)*csc(x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {{\left (\tan \left (x\right )^{2} + 1\right )} \sqrt {a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a} \sqrt {\frac {a \tan \left (x\right )^{2} + a}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + a\right ) - 2 \, \sqrt {\frac {a \tan \left (x\right )^{2} + a}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2}}{2 \, {\left (a \tan \left (x\right )^{2} + a\right )}} \] Input:
integrate(tan(x)/(a+a*cot(x)^2)^(1/2),x, algorithm="fricas")
Output:
1/2*((tan(x)^2 + 1)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + a)/tan(x)^2)*tan(x)^2 + a) - 2*sqrt((a*tan(x)^2 + a)/tan(x)^2)*tan(x)^2) /(a*tan(x)^2 + a)
\[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\int \frac {\tan {\left (x \right )}}{\sqrt {a \left (\cot ^{2}{\left (x \right )} + 1\right )}}\, dx \] Input:
integrate(tan(x)/(a+a*cot(x)**2)**(1/2),x)
Output:
Integral(tan(x)/sqrt(a*(cot(x)**2 + 1)), x)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {1}{2} \, a {\left (\frac {\log \left (-\frac {\sqrt {a} - \sqrt {\frac {a}{\sin \left (x\right )^{2}}}}{\sqrt {a} + \sqrt {\frac {a}{\sin \left (x\right )^{2}}}}\right )}{a^{\frac {3}{2}}} + \frac {2}{a \sqrt {\frac {a}{\sin \left (x\right )^{2}}}}\right )} \] Input:
integrate(tan(x)/(a+a*cot(x)^2)^(1/2),x, algorithm="maxima")
Output:
-1/2*a*(log(-(sqrt(a) - sqrt(a/sin(x)^2))/(sqrt(a) + sqrt(a/sin(x)^2)))/a^ (3/2) + 2/(a*sqrt(a/sin(x)^2)))
Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.33 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {\sin \left (x\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:
integrate(tan(x)/(a+a*cot(x)^2)^(1/2),x, algorithm="giac")
Output:
-sin(x)/(sqrt(a)*sgn(sin(x)))
Time = 9.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\mathrm {atanh}\left (\sqrt {\frac {1}{{\sin \left (x\right )}^2}}\right )-\sqrt {{\sin \left (x\right )}^2}}{\sqrt {a}} \] Input:
int(tan(x)/(a + a*cot(x)^2)^(1/2),x)
Output:
(atanh((1/sin(x)^2)^(1/2)) - (sin(x)^2)^(1/2))/a^(1/2)
\[ \int \frac {\tan (x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cot \left (x \right )^{2}+1}\, \tan \left (x \right )}{\cot \left (x \right )^{2}+1}d x \right )}{a} \] Input:
int(tan(x)/(a+a*cot(x)^2)^(1/2),x)
Output:
(sqrt(a)*int((sqrt(cot(x)**2 + 1)*tan(x))/(cot(x)**2 + 1),x))/a