Integrand size = 26, antiderivative size = 116 \[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {2 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {3 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}-\frac {A \tan (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]
2*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(3/2)/d-3/2*A*arct an(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a-a*sec(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2 )-A*tan(d*x+c)/d/(a-a*sec(d*x+c))^(3/2)
Time = 0.55 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.09 \[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {A \left (4 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) (-1+\sec (c+d x))-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right ) (-1+\sec (c+d x))+2 \sqrt {1+\sec (c+d x)}\right ) \tan (c+d x)}{2 a d (-1+\sec (c+d x)) \sqrt {1+\sec (c+d x)} \sqrt {a-a \sec (c+d x)}} \]
(A*(4*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*(-1 + Sec[c + d*x]) - 3*Sqrt[2]*ArcT anh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]]*(-1 + Sec[c + d*x]) + 2*Sqrt[1 + Sec[c + d*x]])*Tan[c + d*x])/(2*a*d*(-1 + Sec[c + d*x])*Sqrt[1 + Sec[c + d*x]]* Sqrt[a - a*Sec[c + d*x]])
Time = 0.42 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3042, 4392, 3042, 4375, 373, 397, 216}
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.
| \(\displaystyle \int \frac {A \sec (c+d x)+A}{(a-a \sec (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+A}{\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle -a A \int \frac {\tan ^2(c+d x)}{(a-a \sec (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a A \int \frac {\cot \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle \frac {2 A \int \frac {\tan ^2(c+d x)}{(a-a \sec (c+d x)) \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+1\right ) \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {2 A \left (-\frac {\int \frac {1-\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}}{\left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+1\right ) \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{2 a}-\frac {\tan (c+d x)}{2 a \sqrt {a-a \sec (c+d x)} \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2\right )}\right )}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {2 A \left (-\frac {2 \int \frac {1}{\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )-3 \int \frac {1}{\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{2 a}-\frac {\tan (c+d x)}{2 a \sqrt {a-a \sec (c+d x)} \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2\right )}\right )}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 A \left (-\frac {\frac {3 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a}}-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a}}}{2 a}-\frac {\tan (c+d x)}{2 a \sqrt {a-a \sec (c+d x)} \left (\frac {a \tan ^2(c+d x)}{a-a \sec (c+d x)}+2\right )}\right )}{d}\) |
(2*A*(-1/2*((-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/S qrt[a] + (3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x] ])])/(Sqrt[2]*Sqrt[a]))/a - Tan[c + d*x]/(2*a*Sqrt[a - a*Sec[c + d*x]]*(2 + (a*Tan[c + d*x]^2)/(a - a*Sec[c + d*x])))))/d
3.2.71.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d) Subst[Int[x^m*((2 + a*x^2 )^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] ]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I ntegerQ[n - 1/2]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(99)=198\).
Time = 4.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.39
| method | result | size |
| default | \(\frac {A \sqrt {2}\, \left (2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}{2}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+3 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-\left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}} \sin \left (d x +c \right )+\left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )\right )}{2 a d \sqrt {\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (1-\cos \left (d x +c \right )\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\) | \(277\) |
| parts | \(\frac {A \sqrt {2}\, \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )-4 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+4 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-5 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+5 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \tan \left (d x +c \right )}{4 d \left (\cos \left (d x +c \right )+1\right ) a \left (\sec \left (d x +c \right )-1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}-\frac {A \sqrt {2}\, \left (1-\cos \left (d x +c \right )\right ) \left (\left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}-\left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{2}-\arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}\right ) \csc \left (d x +c \right )}{4 d \left (\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right )^{\frac {3}{2}} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}}\) | \(468\) |
1/2*A/a/d*2^(1/2)/(a*(1-cos(d*x+c))^2/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*cs c(d*x+c)^2)^(1/2)/(1-cos(d*x+c))/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*( 2*2^(1/2)*arctan(1/2*2^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2))*(1-c os(d*x+c))^2*csc(d*x+c)+3*arctan(1/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2) )*(1-cos(d*x+c))^2*csc(d*x+c)-((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(3/2)*sin( d*x+c)+(1-cos(d*x+c))^2*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*csc(d*x+c) )
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 502, normalized size of antiderivative = 4.33 \[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} + {\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \, {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 4 \, {\left (A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{4 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {3 \, \sqrt {2} {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 4 \, {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 2 \, {\left (A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{2 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]
[-1/4*(3*sqrt(2)*(A*cos(d*x + c) - A)*sqrt(-a)*log((2*sqrt(2)*(cos(d*x + c )^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) + (3* a*cos(d*x + c) + a)*sin(d*x + c))/((cos(d*x + c) - 1)*sin(d*x + c)))*sin(d *x + c) + 4*(A*cos(d*x + c) - A)*sqrt(-a)*log((2*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) - (2*a*cos(d*x + c ) + a)*sin(d*x + c))/sin(d*x + c))*sin(d*x + c) - 4*(A*cos(d*x + c)^2 + A* cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((a^2*d*cos(d*x + c ) - a^2*d)*sin(d*x + c)), 1/2*(3*sqrt(2)*(A*cos(d*x + c) - A)*sqrt(a)*arct an(sqrt(2)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*s in(d*x + c)))*sin(d*x + c) - 4*(A*cos(d*x + c) - A)*sqrt(a)*arctan(sqrt((a *cos(d*x + c) - a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin( d*x + c) + 2*(A*cos(d*x + c)^2 + A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a) /cos(d*x + c)))/((a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))]
\[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=A \left (\int \frac {\sec {\left (c + d x \right )}}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {1}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \]
A*(Integral(sec(c + d*x)/(-a*sqrt(-a*sec(c + d*x) + a)*sec(c + d*x) + a*sq rt(-a*sec(c + d*x) + a)), x) + Integral(1/(-a*sqrt(-a*sec(c + d*x) + a)*se c(c + d*x) + a*sqrt(-a*sec(c + d*x) + a)), x))
\[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\int { \frac {A \sec \left (d x + c\right ) + A}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Time = 1.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {3 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {4 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{2 \, d} \]
1/2*(3*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(3/2 ) - 4*A*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^( 3/2) - sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)*A/(a^2*tan(1/2*d*x + 1/2 *c)^2))/d
Timed out. \[ \int \frac {A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx=\int \frac {A+\frac {A}{\cos \left (c+d\,x\right )}}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]