Integrand size = 21, antiderivative size = 51 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {a+b \sec (c+d x)}}{d} \] Output:
-2*a^(1/2)*arctanh((a+b*sec(d*x+c))^(1/2)/a^(1/2))/d+2*(a+b*sec(d*x+c))^(1 /2)/d
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sec (c+d x)}}{d} \] Input:
Integrate[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]] + 2*Sqrt[a + b*Sec[c + d*x]])/d
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4373, 60, 73, 220}
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.
\(\displaystyle \int \tan (c+d x) \sqrt {a+b \sec (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\cot \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot \left (\frac {1}{2} (2 c+\pi )+d x\right ) \sqrt {a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle \frac {\int \frac {\cos (c+d x) \sqrt {a+b \sec (c+d x)}}{b}d(b \sec (c+d x))}{d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \int \frac {\cos (c+d x)}{b \sqrt {a+b \sec (c+d x)}}d(b \sec (c+d x))+2 \sqrt {a+b \sec (c+d x)}}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 a \int \frac {1}{b^2 \sec ^2(c+d x)-a}d\sqrt {a+b \sec (c+d x)}+2 \sqrt {a+b \sec (c+d x)}}{d}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {2 \sqrt {a+b \sec (c+d x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}\) |
Input:
Int[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]] + 2*Sqrt[a + b*Sec[c + d*x]])/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {2 \sqrt {a +b \sec \left (d x +c \right )}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(42\) |
default | \(\frac {2 \sqrt {a +b \sec \left (d x +c \right )}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(42\) |
Input:
int((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/d*(2*(a+b*sec(d*x+c))^(1/2)-2*a^(1/2)*arctanh((a+b*sec(d*x+c))^(1/2)/a^( 1/2)))
Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.76 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\left [\frac {\sqrt {a} \log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} + 4 \, {\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right ) + 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{2 \, d}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right ) + 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{d}\right ] \] Input:
integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x, algorithm="fricas")
Output:
[1/2*(sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2* a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d *x + c))) + 4*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/d, (sqrt(-a)*arctan (2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos( d*x + c) + b)) + 2*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/d]
\[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\int \sqrt {a + b \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx \] Input:
integrate((a+b*sec(d*x+c))**(1/2)*tan(d*x+c),x)
Output:
Integral(sqrt(a + b*sec(c + d*x))*tan(c + d*x), x)
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\frac {\sqrt {a} \log \left (\frac {\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 2 \, \sqrt {a + \frac {b}{\cos \left (d x + c\right )}}}{d} \] Input:
integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x, algorithm="maxima")
Output:
(sqrt(a)*log((sqrt(a + b/cos(d*x + c)) - sqrt(a))/(sqrt(a + b/cos(d*x + c) ) + sqrt(a))) + 2*sqrt(a + b/cos(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (43) = 86\).
Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 3.63 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\frac {2 \, {\left (\frac {a \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2 \, b}{\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} - \sqrt {a - b}}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{d} \] Input:
integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x, algorithm="giac")
Output:
2*(a*arctan(-1/2*(sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b ) + sqrt(a - b))/sqrt(-a))/sqrt(-a) - 2*b/(sqrt(a - b)*tan(1/2*d*x + 1/2*c )^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1 /2*d*x + 1/2*c)^2 + a + b) - sqrt(a - b)))*sgn(cos(d*x + c))/d
Time = 11.82 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\frac {2\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{d}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{d} \] Input:
int(tan(c + d*x)*(a + b/cos(c + d*x))^(1/2),x)
Output:
(2*(a + b/cos(c + d*x))^(1/2))/d - (2*a^(1/2)*atanh((a + b/cos(c + d*x))^( 1/2)/a^(1/2)))/d
\[ \int \sqrt {a+b \sec (c+d x)} \tan (c+d x) \, dx=\frac {2 \sqrt {\sec \left (d x +c \right ) b +a}+\left (\int \frac {\sqrt {\sec \left (d x +c \right ) b +a}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right ) b +a}d x \right ) a d}{d} \] Input:
int((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x)
Output:
(2*sqrt(sec(c + d*x)*b + a) + int((sqrt(sec(c + d*x)*b + a)*tan(c + d*x))/ (sec(c + d*x)*b + a),x)*a*d)/d