\(\int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx\) [600]

Optimal result

Integrand size = 25, antiderivative size = 170 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\frac {2 a \left (5 a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}} \]

[Out]

2/21*a*(5*a^2+6*b^2)*(cos(1/2*arctan(tan(f*x+e)))^2)^(1/2)/cos(1/2*arctan(tan(f*x+e)))*EllipticF(sin(1/2*arcta
n(tan(f*x+e))),2^(1/2))*(sec(f*x+e)^2)^(3/4)/d^2/f/(d*sec(f*x+e))^(3/2)-2/7*cos(f*x+e)^2*(b-a*tan(f*x+e))*(a+b
*tan(f*x+e))^2/d^2/f/(d*sec(f*x+e))^(3/2)-2/21*(2*b*(3*a^2+2*b^2)-a*(5*a^2+3*b^2)*tan(f*x+e))/d^2/f/(d*sec(f*x
+e))^(3/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3593, 753, 792, 237} \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\frac {2 a \left (5 a^2+6 b^2\right ) \sec ^2(e+f x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}} \]

[In]

Int[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(7/2),x]

[Out]

(2*a*(5*a^2 + 6*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(21*d^2*f*(d*Sec[e + f*x])^(
3/2)) - (2*Cos[e + f*x]^2*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(7*d^2*f*(d*Sec[e + f*x])^(3/2)) - (2*(
2*b*(3*a^2 + 2*b^2) - a*(5*a^2 + 3*b^2)*Tan[e + f*x]))/(21*d^2*f*(d*Sec[e + f*x])^(3/2))

Rule 237

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]

Rule 753

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 792

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a*(e*f + d*g) - (
c*d*f - a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 3593

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[d^(2*
IntPart[m/2])*((d*Sec[e + f*x])^(2*FracPart[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/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {(a+x)^3}{\left (1+\frac {x^2}{b^2}\right )^{11/4}} \, dx,x,b \tan (e+f x)\right )}{b d^2 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {(a+x) \left (\frac {1}{2} \left (4+\frac {5 a^2}{b^2}\right )+\frac {a x}{2 b^2}\right )}{\left (1+\frac {x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{7 d^2 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (a \left (6+\frac {5 a^2}{b^2}\right ) b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \left (5 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}}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.49 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\frac {\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \left (4 \left (5 a^3+6 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+\sqrt {\cos (e+f x)} \left (-b \left (27 a^2+19 b^2\right ) \cos (e+f x)+\left (-9 a^2 b+3 b^3\right ) \cos (3 (e+f x))+2 a \left (13 a^2+3 b^2+3 \left (a^2-3 b^2\right ) \cos (2 (e+f x))\right ) \sin (e+f x)\right )\right )}{42 d^4 f} \]

[In]

Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(7/2),x]

[Out]

(Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]*(4*(5*a^3 + 6*a*b^2)*EllipticF[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(
-(b*(27*a^2 + 19*b^2)*Cos[e + f*x]) + (-9*a^2*b + 3*b^3)*Cos[3*(e + f*x)] + 2*a*(13*a^2 + 3*b^2 + 3*(a^2 - 3*b
^2)*Cos[2*(e + f*x)])*Sin[e + f*x])))/(42*d^4*f)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 23.65 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.15

method	result	size
default	\(\frac {\frac {10 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a^{3}}{21}+\frac {4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a \,b^{2}}{7}-\frac {6 \left (\cos ^{3}\left (f x +e \right )\right ) a^{2} b}{7}+\frac {2 \left (\cos ^{3}\left (f x +e \right )\right ) b^{3}}{7}+\frac {2 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a^{3}}{7}-\frac {6 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a \,b^{2}}{7}+\frac {10 i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a^{3}}{21}+\frac {4 i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a \,b^{2}}{7}-\frac {2 \cos \left (f x +e \right ) b^{3}}{3}+\frac {10 a^{3} \sin \left (f x +e \right )}{21}+\frac {4 a \,b^{2} \sin \left (f x +e \right )}{7}}{d^{3} f \sqrt {d \sec \left (f x +e \right )}}\)	\(366\)
parts	\(-\frac {2 a^{3} \left (5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+5 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}+\frac {2 b^{3} \left (3 \left (\cos ^{3}\left (f x +e \right )\right )-7 \cos \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}-\frac {6 a^{2} b}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {2 a \,b^{2} \left (2 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+2 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right )\right )}{7 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}\)	\(386\)

[In]

int((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/21/d^3/f/(d*sec(f*x+e))^(1/2)*(5*I*(1/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(cos(f*x+
e)/(cos(f*x+e)+1))^(1/2)*a^3+6*I*(1/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(cos(f*x+e)/(
cos(f*x+e)+1))^(1/2)*a*b^2-9*cos(f*x+e)^3*a^2*b+3*cos(f*x+e)^3*b^3+3*cos(f*x+e)^2*sin(f*x+e)*a^3-9*cos(f*x+e)^
2*sin(f*x+e)*a*b^2+5*I*sec(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)*a^3+6*I*sec(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(cos(
f*x+e)/(cos(f*x+e)+1))^(1/2)*a*b^2-7*cos(f*x+e)*b^3+5*a^3*sin(f*x+e)+6*a*b^2*sin(f*x+e))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\frac {\sqrt {2} {\left (-5 i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (5 i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (7 \, b^{3} \cos \left (f x + e\right )^{2} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (5 \, a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{21 \, d^{4} f} \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/21*(sqrt(2)*(-5*I*a^3 - 6*I*a*b^2)*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + sqrt(
2)*(5*I*a^3 + 6*I*a*b^2)*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) - 2*(7*b^3*cos(f*x
+ e)^2 + 3*(3*a^2*b - b^3)*cos(f*x + e)^4 - (3*(a^3 - 3*a*b^2)*cos(f*x + e)^3 + (5*a^3 + 6*a*b^2)*cos(f*x + e)
)*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(d^4*f)

Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((a+b*tan(f*x+e))**3/(d*sec(f*x+e))**(7/2),x)

[Out]

Integral((a + b*tan(e + f*x))**3/(d*sec(e + f*x))**(7/2), x)

Maxima [F]

\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^3/(d*sec(f*x + e))^(7/2), x)

Giac [F]

\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)^3/(d*sec(f*x + e))^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]

[In]

int((a + b*tan(e + f*x))^3/(d/cos(e + f*x))^(7/2),x)

[Out]

int((a + b*tan(e + f*x))^3/(d/cos(e + f*x))^(7/2), x)