Integrand size = 21, antiderivative size = 108 \[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{21 b \sqrt {d \tan (a+b x)}}-\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b} \]
2/21*d^2*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4 *Pi+b*x),2^(1/2))*sec(b*x+a)*sin(2*b*x+2*a)^(1/2)/b/(d*tan(b*x+a))^(1/2)-2 /21*d*sec(b*x+a)*(d*tan(b*x+a))^(1/2)/b+2/7*d*sec(b*x+a)^3*(d*tan(b*x+a))^ (1/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.74 \[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {d \sec ^3(a+b x) \left (-5+\cos (2 (a+b x))+4 \cos ^4(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {d \tan (a+b x)}}{21 b} \]
-1/21*(d*Sec[a + b*x]^3*(-5 + Cos[2*(a + b*x)] + 4*Cos[a + b*x]^4*Hypergeo metric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan [a + b*x]])/b
Time = 0.60 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3091, 3042, 3093, 3042, 3094, 3042, 3053, 3042, 3120}
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.
\(\displaystyle \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (a+b x)^3 (d \tan (a+b x))^{3/2}dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \int \frac {\sec ^3(a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \int \frac {\sec (a+b x)^3}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3093 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2}{3} \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}}dx+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2}{3} \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}}dx+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3094 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2 \sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{3 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2 \sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{3 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 \sqrt {d \tan (a+b x)}}+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 \sqrt {d \tan (a+b x)}}+\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{7 b}-\frac {1}{7} d^2 \left (\frac {2 \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b d}+\frac {2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{3 b \sqrt {d \tan (a+b x)}}\right )\) |
(2*d*Sec[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/(7*b) - (d^2*((2*EllipticF[a - P i/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*Sqrt[d*Tan[a + b*x ]]) + (2*Sec[a + b*x]*Sqrt[d*Tan[a + b*x]])/(3*b*d)))/7
3.3.42.3.1 Defintions of rubi rules used
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/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[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && 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[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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. \(243\) vs. \(2(119)=238\).
Time = 1.47 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {d \sqrt {d \tan \left (b x +a \right )}\, \left (2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sin \left (b x +a \right ) \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sin \left (b x +a \right ) \tan \left (b x +a \right ) \sqrt {2}-3 \left (\tan ^{2}\left (b x +a \right )\right ) \sec \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{21 b \left (\cos ^{2}\left (b x +a \right )-1\right )}\) | \(244\) |
1/21/b*d*(d*tan(b*x+a))^(1/2)/(cos(b*x+a)^2-1)*(2*sin(b*x+a)*cos(b*x+a)*(1 +csc(b*x+a)-cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)+1 +cot(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+ 2*sin(b*x+a)*(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))+sin(b*x+a)*tan(b*x+a)*2^(1/2)-3*tan(b*x+a)^2*sec(b*x+a)*2^( 1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.06 \[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {i \, d} d \cos \left (b x + a\right )^{3} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {-i \, d} d \cos \left (b x + a\right )^{3} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - {\left (d \cos \left (b x + a\right )^{2} - 3 \, d\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{21 \, b \cos \left (b x + a\right )^{3}} \]
2/21*(sqrt(I*d)*d*cos(b*x + a)^3*elliptic_f(arcsin(cos(b*x + a) + I*sin(b* x + a)), -1) + sqrt(-I*d)*d*cos(b*x + a)^3*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - (d*cos(b*x + a)^2 - 3*d)*sqrt(d*sin(b*x + a)/cos( b*x + a)))/(b*cos(b*x + a)^3)
\[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \sec ^{3}{\left (a + b x \right )}\, dx \]
\[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sec \left (b x + a\right )^{3} \,d x } \]
\[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sec \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}}{{\cos \left (a+b\,x\right )}^3} \,d x \]