3.2.15 \(\int \frac {1}{(a+b \sec ^2(e+f x))^{3/2}} \, dx\) [115]

Optimal. Leaf size=77 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^{3/2} f}-\frac {b \tan (e+f x)}{a (a+b) f \sqrt {a+b+b \tan ^2(e+f x)}} \]

[Out]

arctan(a^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))/a^(3/2)/f-b*tan(f*x+e)/a/(a+b)/f/(a+b+b*tan(f*x+e)^2)^(1
/2)

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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4213, 390, 385, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{a^{3/2} f}-\frac {b \tan (e+f x)}{a f (a+b) \sqrt {a+b \tan ^2(e+f x)+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x]^2)^(-3/2),x]

[Out]

ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt
[a + b + b*Tan[e + f*x]^2])

Rule 209

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(77)=154\).
time = 1.46, size = 168, normalized size = 2.18 \begin {gather*} \frac {(a+2 b+a \cos (2 (e+f x))) \sec ^3(e+f x) \left (\sqrt {a+b} \text {ArcSin}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right ) (a+2 b+a \cos (2 (e+f x)))-\sqrt {2} \sqrt {a} b \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}} \sin (e+f x)\right )}{4 a^{3/2} (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*
b + a*Cos[2*(e + f*x)]) - Sqrt[2]*Sqrt[a]*b*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(4*a^(
3/2)*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.15, size = 1007, normalized size = 13.08

method	result	size
default	\(-\frac {\left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \left (\sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {-\frac {4 i a^{\frac {3}{2}} \sqrt {b}-4 i \sqrt {a}\, b^{\frac {3}{2}}-a^{2}+6 a b -b^{2}}{\left (a +b \right )^{2}}}\right ) a \sin \left (f x +e \right )+\sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {-\frac {4 i a^{\frac {3}{2}} \sqrt {b}-4 i \sqrt {a}\, b^{\frac {3}{2}}-a^{2}+6 a b -b^{2}}{\left (a +b \right )^{2}}}\right ) b \sin \left (f x +e \right )-2 \sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticPi \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{2 i \sqrt {a}\, \sqrt {b}+a -b}, \frac {\sqrt {-\frac {2 i \sqrt {a}\, \sqrt {b}-a +b}{a +b}}}{\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}\right ) a \sin \left (f x +e \right )-2 \sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticPi \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{2 i \sqrt {a}\, \sqrt {b}+a -b}, \frac {\sqrt {-\frac {2 i \sqrt {a}\, \sqrt {b}-a +b}{a +b}}}{\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}\right ) b \sin \left (f x +e \right )+\cos \left (f x +e \right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, b -\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, b \right ) \sin \left (f x +e \right )}{f \left (\cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right )^{3} \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}} \left (a +b \right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, a}\)	\(1007\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/f*(b+a*cos(f*x+e)^2)*(2^(1/2)*((I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e
))/(a+b))^(1/2)*(-2*(I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e))/(a+b))^(1/2
)*EllipticF((cos(f*x+e)-1)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/sin(f*x+e),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/
2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*a*sin(f*x+e)+2^(1/2)*((I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2
)+cos(f*x+e)*a+b)/(1+cos(f*x+e))/(a+b))^(1/2)*(-2*(I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a
-b)/(1+cos(f*x+e))/(a+b))^(1/2)*EllipticF((cos(f*x+e)-1)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/sin(f*x+e),(-
(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*b*sin(f*x+e)-2*2^(1/2)*((I*cos(f*x+e)*
a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e))/(a+b))^(1/2)*(-2*(I*cos(f*x+e)*a^(1/2)*b^(1/2
)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e))/(a+b))^(1/2)*EllipticPi((cos(f*x+e)-1)*((2*I*a^(1/2)*b^(1/2
)+a-b)/(a+b))^(1/2)/sin(f*x+e),-1/(2*I*a^(1/2)*b^(1/2)+a-b)*(a+b),(-(2*I*a^(1/2)*b^(1/2)-a+b)/(a+b))^(1/2)/((2
*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2))*a*sin(f*x+e)-2*2^(1/2)*((I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)+
cos(f*x+e)*a+b)/(1+cos(f*x+e))/(a+b))^(1/2)*(-2*(I*cos(f*x+e)*a^(1/2)*b^(1/2)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b
)/(1+cos(f*x+e))/(a+b))^(1/2)*EllipticPi((cos(f*x+e)-1)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/sin(f*x+e),-1/
(2*I*a^(1/2)*b^(1/2)+a-b)*(a+b),(-(2*I*a^(1/2)*b^(1/2)-a+b)/(a+b))^(1/2)/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/
2))*b*sin(f*x+e)+cos(f*x+e)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*b-((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*
b)*sin(f*x+e)/(cos(f*x+e)-1)/cos(f*x+e)^3/((b+a*cos(f*x+e)^2)/cos(f*x+e)^2)^(3/2)/(a+b)/((2*I*a^(1/2)*b^(1/2)+
a-b)/(a+b))^(1/2)/a

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2169 vs. \(2 (73) = 146\).
time = 0.70, size = 2169, normalized size = 28.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(2*a*b*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b
)*cos(2*f*x + 2*e) + a))*sin(2*f*x + 2*e) - 2*(a^2 + a*b)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin
(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^3 - 2*(a*b*cos(2*f*x + 2*e) + (a^2 + a*
b)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f
*x + 2*e) + a))^2 - a^2 - 2*a*b)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*
x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)) - (a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*
a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*
sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2
*f*x + 2*e))^(1/4)*(((a + b)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x +
4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^2 + (a + b)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f
*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^2)*arctan2(2*a*sin(2*f*x + 2*e) + 2*(a^2*co
s(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4
*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos
(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*f*x
+ 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)), 2*a*cos(2*f*x
+ 2*e) + 2*(a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(
a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2
*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*cos(1/2*ar
ctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a
)) + 2*a + 4*b) - ((a + b)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*
e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^2 + (a + b)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x
+ 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^2)*arctan2(2*(a^2*cos(4*f*x + 4*e)^2 + a^2*si
n(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2
*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x
+ 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(
2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)), 2*(a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f
*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) +
4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e
) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x
+ 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)) + 4*a + 4*b))*sqrt(a))/((a^2*cos(4*f*x + 4*e)
^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*si
n(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))
*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*((a^3 + a^2*b)*cos(1/2*arctan2(a*sin(4*f*x + 4*e)
+ 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a))^2 + (a^3 + a^2*b)*sin(
1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*
e) + a))^2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (73) = 146\).
time = 3.02, size = 632, normalized size = 8.21 \begin {gather*} \left [-\frac {8 \, a b \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a^{4} - a^{3} b\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )\right )}{8 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}, -\frac {4 \, a b \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} - a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right )}{4 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(8*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)
^2 + a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b
+ 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a
*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*
b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f
*x + e)^2)*sin(f*x + e)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), -1/4*(4*a*b*sqrt((a*cos(f*x
+ e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(a)*arcta
n(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a
*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*
sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(f*x+e)**2)**(3/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/cos(e + f*x)^2)^(3/2),x)

[Out]

int(1/(a + b/cos(e + f*x)^2)^(3/2), x)

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