Integrand size = 23, antiderivative size = 88 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=-\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f} \] Output:
-2*a*d^2*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(d*sec(f *x+e))^(1/2)+2/3*b*(d*sec(f*x+e))^(3/2)/f+2*a*d*(d*sec(f*x+e))^(1/2)*sin(f *x+e)/f
Time = 0.41 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\frac {(d \sec (e+f x))^{3/2} \left (2 b-6 a \cos ^{\frac {3}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+3 a \sin (2 (e+f x))\right )}{3 f} \] Input:
Integrate[(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]),x]
Output:
((d*Sec[e + f*x])^(3/2)*(2*b - 6*a*Cos[e + f*x]^(3/2)*EllipticE[(e + f*x)/ 2, 2] + 3*a*Sin[2*(e + f*x)]))/(3*f)
Time = 0.46 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3967, 3042, 4255, 3042, 4258, 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.
\(\displaystyle \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle a \int (d \sec (e+f x))^{3/2}dx+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle a \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \sec (e+f x)}}dx\right )+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle a \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle a \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {2 d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}\) |
Input:
Int[(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]),x]
Output:
(2*b*(d*Sec[e + f*x])^(3/2))/(3*f) + a*((-2*d^2*EllipticE[(e + f*x)/2, 2]) /(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*d*Sqrt[d*Sec[e + f*x]]*S in[e + f*x])/f)
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]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Result contains complex when optimal does not.
Time = 12.40 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {2 d \sqrt {d \sec \left (f x +e \right )}\, \left (i \left (3 \cos \left (f x +e \right )^{2}+6 \cos \left (f x +e \right )+3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, a +i \left (-3 \cos \left (f x +e \right )^{2}-6 \cos \left (f x +e \right )-3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticE}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, a +3 a \sin \left (f x +e \right )+b \left (1+\sec \left (f x +e \right )\right )\right )}{3 f \left (1+\cos \left (f x +e \right )\right )}\) | \(199\) |
parts | \(\frac {2 a \left (i \left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+i \left (-\cos \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticE}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+\sin \left (f x +e \right )\right ) \sqrt {d \sec \left (f x +e \right )}\, d}{f \left (1+\cos \left (f x +e \right )\right )}+\frac {2 b \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}\) | \(200\) |
Input:
int((d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e)),x,method=_RETURNVERBOSE)
Output:
2/3/f*d*(d*sec(f*x+e))^(1/2)/(1+cos(f*x+e))*(I*(3*cos(f*x+e)^2+6*cos(f*x+e )+3)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)) ,I)*(1/(1+cos(f*x+e)))^(1/2)*a+I*(-3*cos(f*x+e)^2-6*cos(f*x+e)-3)*(cos(f*x +e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos (f*x+e)))^(1/2)*a+3*a*sin(f*x+e)+b*(1+sec(f*x+e)))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\frac {-3 i \, \sqrt {2} a d^{\frac {3}{2}} \cos \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {2} a d^{\frac {3}{2}} \cos \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (3 \, a d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b d\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\right )} \] Input:
integrate((d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e)),x, algorithm="fricas")
Output:
1/3*(-3*I*sqrt(2)*a*d^(3/2)*cos(f*x + e)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*I*sqrt(2)*a*d^(3/2)*c os(f*x + e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2*(3*a*d*cos(f*x + e)*sin(f*x + e) + b*d)*sqrt(d/cos (f*x + e)))/(f*cos(f*x + e))
\[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \] Input:
integrate((d*sec(f*x+e))**(3/2)*(a+b*tan(f*x+e)),x)
Output:
Integral((d*sec(e + f*x))**(3/2)*(a + b*tan(e + f*x)), x)
\[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e)),x, algorithm="maxima")
Output:
integrate((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e) + a), x)
\[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e)),x, algorithm="giac")
Output:
integrate((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e) + a), x)
Timed out. \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \] Input:
int((d/cos(e + f*x))^(3/2)*(a + b*tan(e + f*x)),x)
Output:
int((d/cos(e + f*x))^(3/2)*(a + b*tan(e + f*x)), x)
\[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx=\frac {\sqrt {d}\, d \left (2 \sqrt {\sec \left (f x +e \right )}\, \sec \left (f x +e \right ) b +3 \left (\int \sqrt {\sec \left (f x +e \right )}\, \sec \left (f x +e \right )d x \right ) a f \right )}{3 f} \] Input:
int((d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e)),x)
Output:
(sqrt(d)*d*(2*sqrt(sec(e + f*x))*sec(e + f*x)*b + 3*int(sqrt(sec(e + f*x)) *sec(e + f*x),x)*a*f))/(3*f)