Integrand size = 21, antiderivative size = 104 \[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {4 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{b d^2 \sqrt {\sin (2 a+2 b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3} \]
-2*sec(b*x+a)/b/d/(d*tan(b*x+a))^(1/2)+4*cos(b*x+a)*(sin(a+1/4*Pi+b*x)^2)^ (1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*tan(b*x+a) )^(1/2)/b/d^2/sin(2*b*x+2*a)^(1/2)+4*cos(b*x+a)*(d*tan(b*x+a))^(3/2)/b/d^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \csc (a+b x) \sqrt {d \tan (a+b x)} \left (3 \cos (2 (a+b x)) \sqrt {\sec ^2(a+b x)}+4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \tan ^2(a+b x)\right )}{3 b d^2 \sqrt {\sec ^2(a+b x)}} \]
(-2*Csc[a + b*x]*Sqrt[d*Tan[a + b*x]]*(3*Cos[2*(a + b*x)]*Sqrt[Sec[a + b*x ]^2] + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Tan[a + b*x]^2) )/(3*b*d^2*Sqrt[Sec[a + b*x]^2])
Time = 0.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3088, 3042, 3093, 3042, 3095, 3042, 3052, 3042, 3119}
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.
| \(\displaystyle \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (a+b x)^3}{(d \tan (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3088 |
| \(\displaystyle \frac {2 \int \sec (a+b x) \sqrt {d \tan (a+b x)}dx}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \sec (a+b x) \sqrt {d \tan (a+b x)}dx}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3093 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-2 \int \cos (a+b x) \sqrt {d \tan (a+b x)}dx\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-2 \int \frac {\sqrt {d \tan (a+b x)}}{\sec (a+b x)}dx\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3095 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-\frac {2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\sin (a+b x)}}\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-\frac {2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\sin (a+b x)}}\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-\frac {2 \cos (a+b x) \sqrt {d \tan (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-\frac {2 \cos (a+b x) \sqrt {d \tan (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 \left (\frac {2 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d}-\frac {2 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}}\right )}{d^2}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
(-2*Sec[a + b*x])/(b*d*Sqrt[d*Tan[a + b*x]]) + (2*((-2*Cos[a + b*x]*Ellipt icE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(b*Sqrt[Sin[2*a + 2*b*x]]) + (2*Cos[a + b*x]*(d*Tan[a + b*x])^(3/2))/(b*d)))/d^2
3.3.64.3.1 Defintions of rubi rules used
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1))), x] - Simp[a^2*((m - 2)/(b^2*(n + 1))) Int[(a*Sec[e + f *x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, -3/2])) && IntegersQ[2*m, 2*n]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[a^2*((m - 2)/(m + n - 1)) Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && ( GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[ 2*m, 2*n]
Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[ Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(121)=242\).
Time = 1.48 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.53
| method | result | size |
| default | \(-\frac {\left (-4 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-4 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}-\sec \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{b \sqrt {d \tan \left (b x +a \right )}\, d}\) | \(367\) |
-1/b/(d*tan(b*x+a))^(1/2)/d*(-4*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+ a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+ a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+2*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b* x+a)+1+cot(b*x+a))^(1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*EllipticF((1+csc( b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-4*sec(b*x+a)*(1+csc(b*x+a)-cot(b*x+a ))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*El lipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+2*sec(b*x+a)*(1+csc(b *x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b *x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+2*2^(1 /2)-sec(b*x+a)*2^(1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \, {\left (i \, \sqrt {i \, d} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {-i \, d} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + i \, \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (2 \, \cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{b d^{2} \sin \left (b x + a\right )} \]
-2*(I*sqrt(I*d)*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin( b*x + a) - I*sqrt(-I*d)*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) - I*sqrt(I*d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + I*sqrt(-I*d)*elliptic_f(arcsin(cos(b*x + a) - I* sin(b*x + a)), -1)*sin(b*x + a) + (2*cos(b*x + a)^2 - 1)*sqrt(d*sin(b*x + a)/cos(b*x + a)))/(b*d^2*sin(b*x + a))
\[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {\sec ^{3}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\sec \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\sec \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]