Integrand size = 25, antiderivative size = 149 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=-\frac {2 b \left (3 a^2-b^2\right )}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 b^3 \sec ^2(e+f x)}{f (d \sec (e+f x))^{3/2}}+\frac {2 a \left (a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 a \left (a^2-3 b^2\right ) \tan (e+f x)}{3 f (d \sec (e+f x))^{3/2}} \] Output:
-2/3*b*(3*a^2-b^2)/f/(d*sec(f*x+e))^(3/2)+2*b^3*sec(f*x+e)^2/f/(d*sec(f*x+ e))^(3/2)+2/3*a*(a^2+6*b^2)*InverseJacobiAM(1/2*arctan(tan(f*x+e)),2^(1/2) )*(sec(f*x+e)^2)^(3/4)/f/(d*sec(f*x+e))^(3/2)+2/3*a*(a^2-3*b^2)*tan(f*x+e) /f/(d*sec(f*x+e))^(3/2)
Time = 2.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {\sec ^2(e+f x) \left (-3 a^2 b+7 b^3+\left (-3 a^2 b+b^3\right ) \cos (2 (e+f x))+2 a \left (a^2+6 b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+a^3 \sin (2 (e+f x))-3 a b^2 \sin (2 (e+f x))\right )}{3 f (d \sec (e+f x))^{3/2}} \] Input:
Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(3/2),x]
Output:
(Sec[e + f*x]^2*(-3*a^2*b + 7*b^3 + (-3*a^2*b + b^3)*Cos[2*(e + f*x)] + 2* a*(a^2 + 6*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + a^3*Sin[2*( e + f*x)] - 3*a*b^2*Sin[2*(e + f*x)]))/(3*f*(d*Sec[e + f*x])^(3/2))
Time = 0.36 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3994, 495, 27, 676, 229}
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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3994 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \int \frac {(a+b \tan (e+f x))^3}{\left (\tan ^2(e+f x)+1\right )^{7/4}}d(b \tan (e+f x))}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {2}{3} b^2 \int \frac {(a+b \tan (e+f x)) \left (\left (\frac {a^2}{b^2}+4\right ) b^2-3 a b \tan (e+f x)\right )}{2 b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))-\frac {2 (a+b \tan (e+f x))^2 \left (b^2-a b \tan (e+f x)\right )}{3 \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {1}{3} \int \frac {(a+b \tan (e+f x)) \left (a^2-3 b \tan (e+f x) a+4 b^2\right )}{\left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))-\frac {2 (a+b \tan (e+f x))^2 \left (b^2-a b \tan (e+f x)\right )}{3 \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {1}{3} \left (a \left (a^2+6 b^2\right ) \int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))-4 b^2 \left (a^2-2 b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1}-2 a b^3 \tan (e+f x) \sqrt [4]{\tan ^2(e+f x)+1}\right )-\frac {2 (a+b \tan (e+f x))^2 \left (b^2-a b \tan (e+f x)\right )}{3 \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 229 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {1}{3} \left (2 a b \left (a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )-4 b^2 \left (a^2-2 b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1}-2 a b^3 \tan (e+f x) \sqrt [4]{\tan ^2(e+f x)+1}\right )-\frac {2 (a+b \tan (e+f x))^2 \left (b^2-a b \tan (e+f x)\right )}{3 \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
Input:
Int[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(3/2),x]
Output:
((Sec[e + f*x]^2)^(3/4)*((-2*(a + b*Tan[e + f*x])^2*(b^2 - a*b*Tan[e + f*x ]))/(3*(1 + Tan[e + f*x]^2)^(3/4)) + (2*a*b*(a^2 + 6*b^2)*EllipticF[ArcTan [Tan[e + f*x]]/2, 2] - 4*b^2*(a^2 - 2*b^2)*(1 + Tan[e + f*x]^2)^(1/4) - 2* a*b^3*Tan[e + f*x]*(1 + Tan[e + f*x]^2)^(1/4))/3))/(b*f*(d*Sec[e + f*x])^( 3/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (a*d - b*c*x)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[d^(2*IntPart[m/2])*((d*Sec[e + f*x])^(2*FracP art[m/2])/(b*f*(Sec[e + f*x]^2)^FracPart[m/2])) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] && IntegerQ[n]
Result contains complex when optimal does not.
Time = 21.58 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\frac {\left (\frac {2 \cos \left (f x +e \right )}{3}+2 \sec \left (f x +e \right )\right ) b^{3}+\frac {2 a^{3} \sin \left (f x +e \right )}{3}+\frac {2 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, a \,b^{2} \operatorname {EllipticF}\left (i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ), i\right ) \left (6+6 \sec \left (f x +e \right )\right )}{3}-2 \cos \left (f x +e \right ) a^{2} b -2 a \,b^{2} \sin \left (f x +e \right )+\frac {2 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, a^{3} \operatorname {EllipticF}\left (i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ), i\right ) \left (1+\sec \left (f x +e \right )\right )}{3}}{d f \sqrt {d \sec \left (f x +e \right )}}\) | \(210\) |
| parts | \(\frac {a^{3} \left (-\frac {2 i \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 )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \left (1+\sec \left (f x +e \right )\right )}{3}+\frac {2 \sin \left (f x +e \right )}{3}\right )}{f \sqrt {d \sec \left (f x +e \right )}\, d}+\frac {b^{3} \left (-\frac {\ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-2 \cos \left (f x +e \right )+2}{1+\cos \left (f x +e \right )}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (3+3 \sec \left (f x +e \right )\right )}{6}-\frac {\ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-\cos \left (f x +e \right )+1}{1+\cos \left (f x +e \right )}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (-3-3 \sec \left (f x +e \right )\right )}{6}+\frac {2 \cos \left (f x +e \right )}{3}+2 \sec \left (f x +e \right )\right )}{f \sqrt {d \sec \left (f x +e \right )}\, d}+\frac {3 a \,b^{2} \left (-\frac {2 i \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 ) \left (2+2 \sec \left (f x +e \right )\right )}{3}-\frac {2 \sin \left (f x +e \right )}{3}\right )}{f \sqrt {d \sec \left (f x +e \right )}\, d}-\frac {2 a^{2} b}{f \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\) | \(455\) |
Input:
int((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d/f*((2/3*cos(f*x+e)+2*sec(f*x+e))*b^3+2/3*a^3*sin(f*x+e)+2/3*I*(cos(f*x +e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*a*b^2*EllipticF(I*(-csc (f*x+e)+cot(f*x+e)),I)*(6+6*sec(f*x+e))-2*cos(f*x+e)*a^2*b-2*a*b^2*sin(f*x +e)+2/3*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*a^3*E llipticF(I*(-csc(f*x+e)+cot(f*x+e)),I)*(1+sec(f*x+e)))/(d*sec(f*x+e))^(1/2 )
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (3 \, b^{3} - {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, d^{2} f} \] Input:
integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(3/2),x, algorithm="fricas")
Output:
1/3*(sqrt(2)*(-I*a^3 - 6*I*a*b^2)*sqrt(d)*weierstrassPInverse(-4, 0, cos(f *x + e) + I*sin(f*x + e)) + sqrt(2)*(I*a^3 + 6*I*a*b^2)*sqrt(d)*weierstras sPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) + 2*(3*b^3 - (3*a^2*b - b^ 3)*cos(f*x + e)^2 + (a^3 - 3*a*b^2)*cos(f*x + e)*sin(f*x + e))*sqrt(d/cos( f*x + e)))/(d^2*f)
\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*tan(f*x+e))**3/(d*sec(f*x+e))**(3/2),x)
Output:
Integral((a + b*tan(e + f*x))**3/(d*sec(e + f*x))**(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e) + a)^3/(d*sec(f*x + e))^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate((b*tan(f*x + e) + a)^3/(d*sec(f*x + e))^(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((a + b*tan(e + f*x))^3/(d/cos(e + f*x))^(3/2),x)
Output:
int((a + b*tan(e + f*x))^3/(d/cos(e + f*x))^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {d}\, \left (\left (\int \frac {\sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )^{2}}d x \right ) a^{3}+\left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \tan \left (f x +e \right )^{3}}{\sec \left (f x +e \right )^{2}}d x \right ) b^{3}+3 \left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}}{\sec \left (f x +e \right )^{2}}d x \right ) a \,b^{2}+3 \left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \tan \left (f x +e \right )}{\sec \left (f x +e \right )^{2}}d x \right ) a^{2} b \right )}{d^{2}} \] Input:
int((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(3/2),x)
Output:
(sqrt(d)*(int(sqrt(sec(e + f*x))/sec(e + f*x)**2,x)*a**3 + int((sqrt(sec(e + f*x))*tan(e + f*x)**3)/sec(e + f*x)**2,x)*b**3 + 3*int((sqrt(sec(e + f* x))*tan(e + f*x)**2)/sec(e + f*x)**2,x)*a*b**2 + 3*int((sqrt(sec(e + f*x)) *tan(e + f*x))/sec(e + f*x)**2,x)*a**2*b))/d**2