Integrand size = 23, antiderivative size = 85 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}-\frac {\text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d}+\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}} \]
-arcsinh(tan(d*x+c)/(1+sec(d*x+c))^(1/2))/d+arcsinh(tan(d*x+c)/(1+sec(d*x+ c)))*2^(1/2)/d+sec(d*x+c)^(3/2)*sin(d*x+c)/d/(1+sec(d*x+c))^(1/2)
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\left (\arcsin \left (\sqrt {1-\sec (c+d x)}\right )+2 \arcsin \left (\sqrt {\sec (c+d x)}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )+\sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}\right ) \tan (c+d x)}{d \sqrt {-\tan ^2(c+d x)}} \]
((ArcSin[Sqrt[1 - Sec[c + d*x]]] + 2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]* ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(d*Sqrt[-Tan[c + d*x]^2])
Time = 0.60 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4309, 3042, 4511, 3042, 4288, 222, 4294, 222}
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.
| \(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {\sec (c+d x)+1}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx\) |
| \(\Big \downarrow \) 4309 |
| \(\displaystyle \frac {1}{2} \int \frac {(1-\sec (c+d x)) \sqrt {\sec (c+d x)}}{\sqrt {\sec (c+d x)+1}}dx+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (1-\csc \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4511 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {\sec (c+d x)+1}}dx-\int \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x)+1}dx\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx-\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}dx\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4288 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx+\frac {2 \int \frac {1}{\sqrt {\frac {\tan ^2(c+d x)}{\sec (c+d x)+1}+1}}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx-\frac {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4294 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \sqrt {2} \int \frac {1}{\sqrt {\frac {\tan ^2(c+d x)}{(\sec (c+d x)+1)^2}+1}}d\left (-\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\) |
((2*Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d - (2*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/d)/2 + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/ (d*Sqrt[1 + Sec[c + d*x]])
3.3.66.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(a/(b*f))*Sqrt[a*(d/b)] Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a , b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-Sqrt[2])*(Sqrt[a]/(b*f)) Subst[Int[1/Sqrt[1 + x^2], x], x, b*(Cot[e + f*x]/(a + b*Csc[e + f*x]))], x] /; FreeQ[{a, b, d , e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d - a/b, 0] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*d^2*Cot[e + f*x]*((d*Csc[e + f*x])^(n - 2)/( f*(2*n - 3)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[d^2/(b*(2*n - 3)) Int[( d*Csc[e + f*x])^(n - 2)*((2*b*(n - 2) - a*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)/b Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Simp[B/b Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b , d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(77)=154\).
Time = 1.51 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.72
| method | result | size |
| default | \(\frac {\sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {1+\sec \left (d x +c \right )}\, \left (-2 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{3} \sqrt {2}+2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right )}{2 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(231\) |
1/2/d*sec(d*x+c)^(5/2)*(1+sec(d*x+c))^(1/2)/(cos(d*x+c)+1)/(-1/(cos(d*x+c) +1))^(1/2)*(-2*arctan(1/2*sin(d*x+c)*2^(1/2)/(cos(d*x+c)+1)/(-1/(cos(d*x+c )+1))^(1/2))*cos(d*x+c)^3*2^(1/2)+2*(-1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*c os(d*x+c)^2-cos(d*x+c)^3*arctan(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(cos(d*x+c)+ 1)/(-1/(cos(d*x+c)+1))^(1/2))+cos(d*x+c)^3*arctan(1/2*(cos(d*x+c)+sin(d*x+ c)+1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.49 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {2 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) - {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
1/4*(2*(sqrt(2)*cos(d*x + c) + sqrt(2))*log((2*sqrt(2)*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - cos(d*x + c)^2 + 2*co s(d*x + c) + 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (cos(d*x + c) + 1 )*log(-(cos(d*x + c)^2 + 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos( d*x + c))*sin(d*x + c) - cos(d*x + c) - 2)/(cos(d*x + c) + 1)) - (cos(d*x + c) + 1)*log(-(cos(d*x + c)^2 - 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*s qrt(cos(d*x + c))*sin(d*x + c) - cos(d*x + c) - 2)/(cos(d*x + c) + 1)) + 4 *sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d *cos(d*x + c) + d)
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (77) = 154\).
Time = 0.46 (sec) , antiderivative size = 873, normalized size of antiderivative = 10.27 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Too large to display} \]
-1/4*(4*sqrt(2)*cos(3/2*arctan2(sin(d*x + c), cos(d*x + c)))*sin(2*d*x + 2 *c) - 4*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))*sin(2*d*x + 2 *c) + (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*l og(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(si n(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos (d*x + c))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*log(2 *cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d* x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) + (c os(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*log(2*cos (1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c ))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - (cos(2 *d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2 *arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 2*(sqrt(2) *cos(2*d*x + 2*c)^2 + sqrt(2)*sin(2*d*x + 2*c)^2 + 2*sqrt(2)*cos(2*d*x + 2 *c) + sqrt(2))*log(cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + sin...
\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right ) + 1}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]