Integrand size = 19, antiderivative size = 59 \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}-\frac {\text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f} \]
-arctan((b*sec(f*x+e))^(1/2)/b^(1/2))/f/b^(1/2)-arctanh((b*sec(f*x+e))^(1/ 2)/b^(1/2))/f/b^(1/2)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\left (2 \arctan \left (\sqrt {\sec (e+f x)}\right )-\log \left (1-\sqrt {\sec (e+f x)}\right )+\log \left (1+\sqrt {\sec (e+f x)}\right )\right ) \sqrt {\sec (e+f x)}}{2 f \sqrt {b \sec (e+f x)}} \]
-1/2*((2*ArcTan[Sqrt[Sec[e + f*x]]] - Log[1 - Sqrt[Sec[e + f*x]]] + Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3102, 25, 27, 266, 756, 216, 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.
\(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}}dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {\int -\frac {b^2}{\sqrt {b \sec (e+f x)} \left (b^2-b^2 \sec ^2(e+f x)\right )}d(b \sec (e+f x))}{b f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {b^2}{\sqrt {b \sec (e+f x)} \left (b^2-b^2 \sec ^2(e+f x)\right )}d(b \sec (e+f x))}{b f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \int \frac {1}{\sqrt {b \sec (e+f x)} \left (b^2-b^2 \sec ^2(e+f x)\right )}d(b \sec (e+f x))}{f}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {2 b \int \frac {1}{b^2-b^4 \sec ^4(e+f x)}d\sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle -\frac {2 b \left (\frac {\int \frac {1}{b-b^2 \sec ^2(e+f x)}d\sqrt {b \sec (e+f x)}}{2 b}+\frac {\int \frac {1}{b^2 \sec ^2(e+f x)+b}d\sqrt {b \sec (e+f x)}}{2 b}\right )}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2 b \left (\frac {\int \frac {1}{b-b^2 \sec ^2(e+f x)}d\sqrt {b \sec (e+f x)}}{2 b}+\frac {\arctan \left (\sqrt {b} \sec (e+f x)\right )}{2 b^{3/2}}\right )}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 b \left (\frac {\arctan \left (\sqrt {b} \sec (e+f x)\right )}{2 b^{3/2}}+\frac {\text {arctanh}\left (\sqrt {b} \sec (e+f x)\right )}{2 b^{3/2}}\right )}{f}\) |
(-2*b*(ArcTan[Sqrt[b]*Sec[e + f*x]]/(2*b^(3/2)) + ArcTanh[Sqrt[b]*Sec[e + f*x]]/(2*b^(3/2))))/f
3.5.14.3.1 Defintions of rubi rules used
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] :> 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(47)=94\).
Time = 0.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.41
method | result | size |
default | \(\frac {\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )+\ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right )}{2 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b \sec \left (f x +e \right )}}\) | \(142\) |
1/2/f*(arctan(1/2/(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))+ln((2*cos(f*x+e)*( -cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2) -cos(f*x+e)+1)/(cos(f*x+e)+1)))/(cos(f*x+e)+1)/(-cos(f*x+e)/(cos(f*x+e)+1) ^2)^(1/2)/(b*sec(f*x+e))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (47) = 94\).
Time = 0.37 (sec) , antiderivative size = 253, normalized size of antiderivative = 4.29 \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\left [\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) - \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, b f}, \frac {2 \, \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) + \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, b f}\right ] \]
[1/4*(2*sqrt(-b)*arctan(1/2*sqrt(-b)*sqrt(b/cos(f*x + e))*(cos(f*x + e) + 1)/b) - sqrt(-b)*log((b*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 - cos(f*x + e)) *sqrt(-b)*sqrt(b/cos(f*x + e)) - 6*b*cos(f*x + e) + b)/(cos(f*x + e)^2 + 2 *cos(f*x + e) + 1)))/(b*f), 1/4*(2*sqrt(b)*arctan(1/2*sqrt(b/cos(f*x + e)) *(cos(f*x + e) - 1)/sqrt(b)) + sqrt(b)*log((b*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + cos(f*x + e))*sqrt(b)*sqrt(b/cos(f*x + e)) + 6*b*cos(f*x + e) + b )/(cos(f*x + e)^2 - 2*cos(f*x + e) + 1)))/(b*f)]
\[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}} - \frac {\log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{b^{\frac {3}{2}}}\right )}}{2 \, f} \]
-1/2*b*(2*arctan(sqrt(b/cos(f*x + e))/sqrt(b))/b^(3/2) - log(-(sqrt(b) - s qrt(b/cos(f*x + e)))/(sqrt(b) + sqrt(b/cos(f*x + e))))/b^(3/2))/f
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b}} + \frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{\sqrt {b}}}{f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
(arctan(sqrt(b*cos(f*x + e))/sqrt(-b))/sqrt(-b) + arctan(sqrt(b*cos(f*x + e))/sqrt(b))/sqrt(b))/(f*sgn(cos(f*x + e)))
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]