Integrand size = 11, antiderivative size = 54 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=-\frac {\log (1-\sin (x))}{2 (a+b)}+\frac {\log (1+\sin (x))}{2 (a-b)}-\frac {b \log (b+a \sin (x))}{a^2-b^2} \] Output:
-1/2*ln(1-sin(x))/(a+b)+ln(1+sin(x))/(2*a-2*b)-b*ln(b+a*sin(x))/(a^2-b^2)
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=\frac {(-a+b) \log (1-\sin (x))+(a+b) \log (1+\sin (x))-2 b \log (b+a \sin (x))}{2 (a-b) (a+b)} \] Input:
Integrate[Sec[x]/(a + b*Csc[x]),x]
Output:
((-a + b)*Log[1 - Sin[x]] + (a + b)*Log[1 + Sin[x]] - 2*b*Log[b + a*Sin[x] ])/(2*(a - b)*(a + b))
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 4359, 3042, 3200, 587, 16, 452, 219, 240}
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.
\(\displaystyle \int \frac {\sec (x)}{a+b \csc (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (x) (a+b \csc (x))}dx\) |
\(\Big \downarrow \) 4359 |
\(\displaystyle \int \frac {\tan (x)}{a \sin (x)+b}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (x)}{a \sin (x)+b}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle \int \frac {a \sin (x)}{\left (a^2-a^2 \sin ^2(x)\right ) (a \sin (x)+b)}d(a \sin (x))\) |
\(\Big \downarrow \) 587 |
\(\displaystyle \frac {\int \frac {a^2-a b \sin (x)}{a^2-a^2 \sin ^2(x)}d(a \sin (x))}{a^2-b^2}-\frac {b \int \frac {1}{b+a \sin (x)}d(a \sin (x))}{a^2-b^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\int \frac {a^2-a b \sin (x)}{a^2-a^2 \sin ^2(x)}d(a \sin (x))}{a^2-b^2}-\frac {b \log (a \sin (x)+b)}{a^2-b^2}\) |
\(\Big \downarrow \) 452 |
\(\displaystyle \frac {a^2 \int \frac {1}{a^2-a^2 \sin ^2(x)}d(a \sin (x))-b \int \frac {a \sin (x)}{a^2-a^2 \sin ^2(x)}d(a \sin (x))}{a^2-b^2}-\frac {b \log (a \sin (x)+b)}{a^2-b^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \text {arctanh}(\sin (x))-b \int \frac {a \sin (x)}{a^2-a^2 \sin ^2(x)}d(a \sin (x))}{a^2-b^2}-\frac {b \log (a \sin (x)+b)}{a^2-b^2}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\frac {1}{2} b \log \left (a^2-a^2 \sin ^2(x)\right )+a \text {arctanh}(\sin (x))}{a^2-b^2}-\frac {b \log (a \sin (x)+b)}{a^2-b^2}\) |
Input:
Int[Sec[x]/(a + b*Csc[x]),x]
Output:
-((b*Log[b + a*Sin[x]])/(a^2 - b^2)) + (a*ArcTanh[Sin[x]] + (b*Log[a^2 - a ^2*Sin[x]^2])/2)/(a^2 - b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_.)/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[(- c)*(d/(b*c^2 + a*d^2)) Int[1/(c + d*x), x], x] + Simp[1/(b*c^2 + a*d^2) Int[(a*d + b*c*x)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c ^2 + a*d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m _.), x_Symbol] :> Int[Cot[e + f*x]^p*(b + a*Sin[e + f*x])^m, x] /; FreeQ[{a , b, e, f, p}, x] && IntegerQ[m] && EqQ[m, p]
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\ln \left (1+\sin \left (x \right )\right )}{2 a -2 b}-\frac {b \ln \left (a \sin \left (x \right )+b \right )}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\sin \left (x \right )-1\right )}{2 a +2 b}\) | \(55\) |
parallelrisch | \(\frac {-b \ln \left (2 a \tan \left (\frac {x}{2}\right )+b \sec \left (\frac {x}{2}\right )^{2}\right )+\left (-a +b \right ) \ln \left (\tan \left (\frac {x}{2}\right )-1\right )+\ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \left (a +b \right )}{a^{2}-b^{2}}\) | \(58\) |
norman | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a -b}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{a +b}-\frac {b \ln \left (b \tan \left (\frac {x}{2}\right )^{2}+2 a \tan \left (\frac {x}{2}\right )+b \right )}{a^{2}-b^{2}}\) | \(63\) |
risch | \(\frac {i x}{a +b}-\frac {i x}{a -b}+\frac {2 i x b}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{a +b}+\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{a -b}-\frac {b \ln \left ({\mathrm e}^{2 i x}-1+\frac {2 i b \,{\mathrm e}^{i x}}{a}\right )}{a^{2}-b^{2}}\) | \(105\) |
Input:
int(sec(x)/(a+b*csc(x)),x,method=_RETURNVERBOSE)
Output:
ln(1+sin(x))/(2*a-2*b)-b/(a+b)/(a-b)*ln(a*sin(x)+b)-1/(2*a+2*b)*ln(sin(x)- 1)
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=-\frac {2 \, b \log \left (a \sin \left (x\right ) + b\right ) - {\left (a + b\right )} \log \left (\sin \left (x\right ) + 1\right ) + {\left (a - b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{2 \, {\left (a^{2} - b^{2}\right )}} \] Input:
integrate(sec(x)/(a+b*csc(x)),x, algorithm="fricas")
Output:
-1/2*(2*b*log(a*sin(x) + b) - (a + b)*log(sin(x) + 1) + (a - b)*log(-sin(x ) + 1))/(a^2 - b^2)
\[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=\int \frac {\sec {\left (x \right )}}{a + b \csc {\left (x \right )}}\, dx \] Input:
integrate(sec(x)/(a+b*csc(x)),x)
Output:
Integral(sec(x)/(a + b*csc(x)), x)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=-\frac {b \log \left (a \sin \left (x\right ) + b\right )}{a^{2} - b^{2}} + \frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, {\left (a - b\right )}} - \frac {\log \left (\sin \left (x\right ) - 1\right )}{2 \, {\left (a + b\right )}} \] Input:
integrate(sec(x)/(a+b*csc(x)),x, algorithm="maxima")
Output:
-b*log(a*sin(x) + b)/(a^2 - b^2) + 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(s in(x) - 1)/(a + b)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=-\frac {a b \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{3} - a b^{2}} + \frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, {\left (a - b\right )}} - \frac {\log \left (-\sin \left (x\right ) + 1\right )}{2 \, {\left (a + b\right )}} \] Input:
integrate(sec(x)/(a+b*csc(x)),x, algorithm="giac")
Output:
-a*b*log(abs(a*sin(x) + b))/(a^3 - a*b^2) + 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(-sin(x) + 1)/(a + b)
Time = 15.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=\frac {\ln \left (\sin \left (x\right )+1\right )}{2\,\left (a-b\right )}-\frac {\ln \left (\sin \left (x\right )-1\right )}{2\,\left (a+b\right )}-\frac {b\,\ln \left (b+a\,\sin \left (x\right )\right )}{a^2-b^2} \] Input:
int(1/(cos(x)*(a + b/sin(x))),x)
Output:
log(sin(x) + 1)/(2*(a - b)) - log(sin(x) - 1)/(2*(a + b)) - (b*log(b + a*s in(x)))/(a^2 - b^2)
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.31 \[ \int \frac {\sec (x)}{a+b \csc (x)} \, dx=\frac {-\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) a +\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) b +\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) a +\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) b -\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b +2 \tan \left (\frac {x}{2}\right ) a +b \right ) b}{a^{2}-b^{2}} \] Input:
int(sec(x)/(a+b*csc(x)),x)
Output:
( - log(tan(x/2) - 1)*a + log(tan(x/2) - 1)*b + log(tan(x/2) + 1)*a + log( tan(x/2) + 1)*b - log(tan(x/2)**2*b + 2*tan(x/2)*a + b)*b)/(a**2 - b**2)