Integrand size = 17, antiderivative size = 32 \[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=\frac {\log (\sin (c+d x)) \tan (c+d x)}{d \sqrt {-a \tan ^2(c+d x)}} \] Output:
ln(sin(d*x+c))*tan(d*x+c)/d/(-a*tan(d*x+c)^2)^(1/2)
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=\frac {\log (\sin (c+d x)) \tan (c+d x)}{d \sqrt {-a \tan ^2(c+d x)}} \] Input:
Integrate[1/Sqrt[a - a*Sec[c + d*x]^2],x]
Output:
(Log[Sin[c + d*x]]*Tan[c + d*x])/(d*Sqrt[-(a*Tan[c + d*x]^2)])
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 4609, 3042, 4141, 3042, 25, 3956}
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.
\(\displaystyle \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a-a \sec (c+d x)^2}}dx\) |
\(\Big \downarrow \) 4609 |
\(\displaystyle \int \frac {1}{\sqrt {-a \tan ^2(c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {-a \tan (c+d x)^2}}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\tan (c+d x) \int \cot (c+d x)dx}{\sqrt {-a \tan ^2(c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{\sqrt {-a \tan ^2(c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\tan (c+d x) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{\sqrt {-a \tan ^2(c+d x)}}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {\tan (c+d x) \log (-\sin (c+d x))}{d \sqrt {-a \tan ^2(c+d x)}}\) |
Input:
Int[1/Sqrt[a - a*Sec[c + d*x]^2],x]
Output:
(Log[-Sin[c + d*x]]*Tan[c + d*x])/(d*Sqrt[-(a*Tan[c + d*x]^2)])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[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]])
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Time = 0.52 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {\tan \left (d x +c \right ) \left (2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (1+\tan \left (d x +c \right )^{2}\right )\right )}{2 d \sqrt {-a \tan \left (d x +c \right )^{2}}}\) | \(48\) |
risch | \(\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) x}{\sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (d x +c \right )}{\sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{\sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(194\) |
Input:
int(1/(a-sec(d*x+c)^2*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/d*tan(d*x+c)*(2*ln(tan(d*x+c))-ln(1+tan(d*x+c)^2))/(-a*tan(d*x+c)^2)^( 1/2)
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=-\frac {\sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{a d \sin \left (d x + c\right )} \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(1/2),x, algorithm="fricas")
Output:
-sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)*cos(d*x + c)*log(1/2*sin(d*x + c))/(a*d*sin(d*x + c))
\[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {- a \sec ^{2}{\left (c + d x \right )} + a}}\, dx \] Input:
integrate(1/(a-a*sec(d*x+c)**2)**(1/2),x)
Output:
Integral(1/sqrt(-a*sec(c + d*x)**2 + a), x)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=-\frac {\frac {\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a}} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a}}}{2 \, d} \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(1/2),x, algorithm="maxima")
Output:
-1/2*(log(tan(d*x + c)^2 + 1)/sqrt(-a) - 2*log(tan(d*x + c))/sqrt(-a))/d
\[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=\int { \frac {1}{\sqrt {-a \sec \left (d x + c\right )^{2} + a}} \,d x } \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(-a*sec(d*x + c)^2 + a), x)
Timed out. \[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {a-\frac {a}{{\cos \left (c+d\,x\right )}^2}}} \,d x \] Input:
int(1/(a - a/cos(c + d*x)^2)^(1/2),x)
Output:
int(1/(a - a/cos(c + d*x)^2)^(1/2), x)
\[ \int \frac {1}{\sqrt {a-a \sec ^2(c+d x)}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sec \left (d x +c \right )^{2}+1}}{\sec \left (d x +c \right )^{2}-1}d x \right )}{a} \] Input:
int(1/(a-a*sec(d*x+c)^2)^(1/2),x)
Output:
( - sqrt(a)*int(sqrt( - sec(c + d*x)**2 + 1)/(sec(c + d*x)**2 - 1),x))/a