\(\int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx\) [595]

Optimal result

Integrand size = 25, antiderivative size = 176 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx=-\frac {2 a \left (5 a^2-6 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{5 f \sec ^2(e+f x)^{3/4}}+\frac {2 a \left (5 a^2-6 b^2\right ) \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}+\frac {2 b (d \sec (e+f x))^{3/2} \left (10 \left (9 a^2-2 b^2\right )+33 a b \tan (e+f x)\right )}{105 f} \]

[Out]

-2/5*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)))*EllipticE(sin(1/2*arcta
n(tan(f*x+e))),2^(1/2))*(d*sec(f*x+e))^(3/2)/f/(sec(f*x+e)^2)^(3/4)+2/5*a*(5*a^2-6*b^2)*cos(f*x+e)*(d*sec(f*x+
e))^(3/2)*sin(f*x+e)/f+2/7*b*(d*sec(f*x+e))^(3/2)*(a+b*tan(f*x+e))^2/f+2/105*b*(d*sec(f*x+e))^(3/2)*(90*a^2-20
*b^2+33*a*b*tan(f*x+e))/f

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3593, 757, 794, 233, 202} \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx=-\frac {2 a \left (5 a^2-6 b^2\right ) (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{5 f \sec ^2(e+f x)^{3/4}}+\frac {2 b (d \sec (e+f x))^{3/2} \left (10 \left (9 a^2-2 b^2\right )+33 a b \tan (e+f x)\right )}{105 f}+\frac {2 a \left (5 a^2-6 b^2\right ) \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f} \]

[In]

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

[Out]

(-2*a*(5*a^2 - 6*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(d*Sec[e + f*x])^(3/2))/(5*f*(Sec[e + f*x]^2)^(3/4)
) + (2*a*(5*a^2 - 6*b^2)*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(5*f) + (2*b*(d*Sec[e + f*x])^(3/2)
*(a + b*Tan[e + f*x])^2)/(7*f) + (2*b*(d*Sec[e + f*x])^(3/2)*(10*(9*a^2 - 2*b^2) + 33*a*b*Tan[e + f*x]))/(105*
f)

Rule 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)), x] - Dist[a, Int[1/(a + b*x^2)^(5
/4), x], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 757

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

Rule 794

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

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 {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {(a+x)^3}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}+\frac {\left (2 b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {(a+x) \left (\frac {1}{2} \left (-4+\frac {7 a^2}{b^2}\right )+\frac {11 a x}{2 b^2}\right )}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{7 f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}+\frac {2 b (d \sec (e+f x))^{3/2} \left (10 \left (9 a^2-2 b^2\right )+33 a b \tan (e+f x)\right )}{105 f}-\frac {\left (a \left (6-\frac {5 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 a \left (5 a^2-6 b^2\right ) \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}+\frac {2 b (d \sec (e+f x))^{3/2} \left (10 \left (9 a^2-2 b^2\right )+33 a b \tan (e+f x)\right )}{105 f}+\frac {\left (a \left (6-\frac {5 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{5 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {2 a \left (5 a^2-6 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{5 f \sec ^2(e+f x)^{3/4}}+\frac {2 a \left (5 a^2-6 b^2\right ) \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}+\frac {2 b (d \sec (e+f x))^{3/2} \left (10 \left (9 a^2-2 b^2\right )+33 a b \tan (e+f x)\right )}{105 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.86 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx=-\frac {d \sqrt {d \sec (e+f x)} \left (70 b \left (-3 a^2+b^2\right ) \cos ^2(e+f x)+42 a \left (5 a^2-6 b^2\right ) \cos ^{\frac {7}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )-42 a \left (5 a^2-6 b^2\right ) \cos ^3(e+f x) \sin (e+f x)-3 b^2 (10 b+21 a \sin (2 (e+f x)))\right ) (a+b \tan (e+f x))^3}{105 f (a \cos (e+f x)+b \sin (e+f x))^3} \]

[In]

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

[Out]

-1/105*(d*Sqrt[d*Sec[e + f*x]]*(70*b*(-3*a^2 + b^2)*Cos[e + f*x]^2 + 42*a*(5*a^2 - 6*b^2)*Cos[e + f*x]^(7/2)*E
llipticE[(e + f*x)/2, 2] - 42*a*(5*a^2 - 6*b^2)*Cos[e + f*x]^3*Sin[e + f*x] - 3*b^2*(10*b + 21*a*Sin[2*(e + f*
x)]))*(a + b*Tan[e + f*x])^3)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 22.07 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.95

method	result	size
parts	\(\text {Expression too large to display}\)	\(872\)
default	\(\text {Expression too large to display}\)	\(899\)

[In]

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

[Out]

-2*a^3/f*(I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*
cos(f*x+e)^2-I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/
2)*cos(f*x+e)^2+2*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1)
)^(1/2)*cos(f*x+e)-2*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)
+1))^(1/2)*cos(f*x+e)+I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot
(f*x+e)),I)-I*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2
)-sin(f*x+e))*(d*sec(f*x+e))^(1/2)*d/(cos(f*x+e)+1)+2*b^3/f/d^2*(1/7*(d*sec(f*x+e))^(7/2)-1/3*d^2*(d*sec(f*x+e
))^(3/2))+2*a^2*b/f*(d*sec(f*x+e))^(3/2)+6/5*a*b^2/f*(d*sec(f*x+e))^(1/2)*d/(cos(f*x+e)+1)*(2*I*EllipticE(I*(c
sc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2-2*I*EllipticF
(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2+4*I*cos(
f*x+e)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-4*I*c
os(f*x+e)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+2*
I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-2*I*(cos(f
*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)-2*sin(f*x+e)+tan(f
*x+e)+sec(f*x+e)*tan(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 = 200, normalized size of antiderivative = 1.14 \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx=\frac {-21 i \, \sqrt {2} {\left (5 \, a^{3} - 6 \, a b^{2}\right )} d^{\frac {3}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, a^{3} - 6 \, a b^{2}\right )} d^{\frac {3}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (15 \, b^{3} d + 35 \, {\left (3 \, a^{2} b - b^{3}\right )} d \cos \left (f x + e\right )^{2} + 21 \, {\left (3 \, a b^{2} d \cos \left (f x + e\right ) + {\left (5 \, a^{3} - 6 \, a b^{2}\right )} d \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{105 \, f \cos \left (f x + e\right )^{3}} \]

[In]

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

[Out]

1/105*(-21*I*sqrt(2)*(5*a^3 - 6*a*b^2)*d^(3/2)*cos(f*x + e)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0
, cos(f*x + e) + I*sin(f*x + e))) + 21*I*sqrt(2)*(5*a^3 - 6*a*b^2)*d^(3/2)*cos(f*x + e)^3*weierstrassZeta(-4,
0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2*(15*b^3*d + 35*(3*a^2*b - b^3)*d*cos(f*x + e
)^2 + 21*(3*a*b^2*d*cos(f*x + e) + (5*a^3 - 6*a*b^2)*d*cos(f*x + e)^3)*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(f*
cos(f*x + e)^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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