\(\int \frac {(a+b \tan (e+f x))^2}{\sqrt {d \sec (e+f x)}} \, dx\) [589]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {(a+b \tan (e+f x))^2}{\sqrt {d \sec (e+f x)}} \, dx=-\frac {6 a b}{f \sqrt {d \sec (e+f x)}}+\frac {2 \left (a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (a+b \tan (e+f x))}{f \sqrt {d \sec (e+f x)}} \]

[Out]

-6*a*b/f/(d*sec(f*x+e))^(1/2)+2*(a^2-2*b^2)*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*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*b*(a+b*tan(f*x+e))/f/(d*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 95, 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 = {3589, 3567, 3856, 2719} \[ \int \frac {(a+b \tan (e+f x))^2}{\sqrt {d \sec (e+f x)}} \, dx=\frac {2 \left (a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}-\frac {6 a b}{f \sqrt {d \sec (e+f x)}}+\frac {2 b (a+b \tan (e+f x))}{f \sqrt {d \sec (e+f x)}} \]

[In]

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

[Out]

(-6*a*b)/(f*Sqrt[d*Sec[e + f*x]]) + (2*(a^2 - 2*b^2)*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*S
ec[e + f*x]]) + (2*b*(a + b*Tan[e + f*x]))/(f*Sqrt[d*Sec[e + f*x]])

Rule 2719

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]

Rule 3567

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] + Dist[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])

Rule 3589

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(d*Sec
[e + f*x])^m*((a + b*Tan[e + f*x])/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(d*Sec[e + f*x])^m*(a^2*(m + 1) - b^
2 + a*b*(m + 2)*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 + b^2, 0] && NeQ[m, -1]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

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

Mathematica [A] (verified)

Time = 2.65 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b \tan (e+f x))^2}{\sqrt {d \sec (e+f x)}} \, dx=\frac {\frac {2 \left (a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}+2 b (-2 a+b \tan (e+f x))}{f \sqrt {d \sec (e+f x)}} \]

[In]

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

[Out]

((2*(a^2 - 2*b^2)*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] + 2*b*(-2*a + b*Tan[e + f*x]))/(f*Sqrt[d*Sec[e
+ f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 16.29 (sec) , antiderivative size = 805, normalized size of antiderivative = 8.47

method	result	size
parts	\(\frac {2 a^{2} \left (i \cos \left (f x +e \right ) E\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}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-i \cos \left (f x +e \right ) 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}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-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}}+i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-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}}+\sin \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {d \sec \left (f x +e \right )}}-\frac {2 b^{2} \left (2 i \cos \left (f x +e \right ) E\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}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-2 i \cos \left (f x +e \right ) 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}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-4 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 {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-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}}+\sin \left (f x +e \right )-\tan \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {d \sec \left (f x +e \right )}}-\frac {4 a b}{f \sqrt {d \sec \left (f x +e \right )}}\)	\(805\)
default	\(\text {Expression too large to display}\)	\(2179\)

[In]

int((a+b*tan(f*x+e))^2/(d*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*a^2/f/(cos(f*x+e)+1)/(d*sec(f*x+e))^(1/2)*(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)-I*cos(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)+I*sec(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ellipti
cE(I*(csc(f*x+e)-cot(f*x+e)),I)-I*sec(f*x+e)*(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))-2*b^2/f/(cos(f*x+e)+1)/(d*sec(f*x+e))^(1/2)*(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)+4*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)-4*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*I*sec(f*x+e)*(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*sec(f*x+e)*(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)-tan(f*x+e))-4
*a*b/f/(d*sec(f*x+e))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b \tan (e+f x))^2}{\sqrt {d \sec (e+f x)}} \, dx=\frac {\sqrt {2} {\left (i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {2} {\left (-i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (2 \, a b \cos \left (f x + e\right ) - b^{2} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{d f} \]

[In]

integrate((a+b*tan(f*x+e))^2/(d*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

(sqrt(2)*(I*a^2 - 2*I*b^2)*sqrt(d)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x
+ e))) + sqrt(2)*(-I*a^2 + 2*I*b^2)*sqrt(d)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I
*sin(f*x + e))) - 2*(2*a*b*cos(f*x + e) - b^2*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(d*f)

Sympy [F]

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

[In]

integrate((a+b*tan(f*x+e))**2/(d*sec(f*x+e))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*tan(f*x+e))^2/(d*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*tan(f*x+e))^2/(d*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*tan(e + f*x))^2/(d/cos(e + f*x))^(1/2),x)

[Out]

int((a + b*tan(e + f*x))^2/(d/cos(e + f*x))^(1/2), x)