3.2.57 \(\int \frac {(a+a \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx\) [157]

Optimal. Leaf size=110 \[ \frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c f}+\frac {2 a^{3/2} (c-d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{c \sqrt {d} \sqrt {c+d} f} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4012, 3859, 209, 4052, 211} \begin {gather*} \frac {2 a^{3/2} (c-d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}\right )}{c \sqrt {d} f \sqrt {c+d}}+\frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 211

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

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 4012

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[a/c
, Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[Csc[e + f*x]*(Sqrt[a + b*Csc[e + f*x]]/(c + d
*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2
- d^2, 0])

Rule 4052

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
), x_Symbol] :> Dist[-2*(b/f), Subst[Int[1/(b*c + a*d + d*x^2), x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]
])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.52, size = 135, normalized size = 1.23 \begin {gather*} \frac {\sqrt {2} a \left (\sqrt {d} \sqrt {c+d} \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )+(c-d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d} \sqrt {\cos (e+f x)}}\right )\right ) \sqrt {\cos (e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{c \sqrt {d} \sqrt {c+d} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*a*(Sqrt[d]*Sqrt[c + d]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] + (c - d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*
x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])])*Sqrt[Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(c*S
qrt[d]*Sqrt[c + d]*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(867\) vs. \(2(90)=180\).
time = 4.65, size = 868, normalized size = 7.89

method	result	size
default	\(-\frac {\left (2 \sqrt {\frac {d}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right )+\ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\right )}{c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )-c +d}\right ) c -\ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\right )}{c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )-c +d}\right ) d -\ln \left (\frac {-2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )+2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )+2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )-2 \sqrt {\left (c +d \right ) \left (c -d \right )}}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )+c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d}\right ) c +\ln \left (\frac {-2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )+2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )+2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )-2 \sqrt {\left (c +d \right ) \left (c -d \right )}}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )+c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d}\right ) d \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \sqrt {2}\, a}{2 f \sqrt {\left (c +d \right ) \left (c -d \right )}\, c \sqrt {\frac {d}{c -d}}}\)	\(868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/f*(2*(d/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(
f*x+e)*2^(1/2))+ln(-2*(2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c
-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*d*sin(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)+((c+d)*(c-d))^(1/2)*cos
(f*x+e)-((c+d)*(c-d))^(1/2))/(c*cos(f*x+e)-d*cos(f*x+e)-((c+d)*(c-d))^(1/2)*sin(f*x+e)-c+d))*c-ln(-2*(2^(1/2)*
(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(
f*x+e)+1))^(1/2)*d*sin(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)+((c+d)*(c-d))^(1/2)*cos(f*x+e)-((c+d)*(c-d))^(1/2))/(c
*cos(f*x+e)-d*cos(f*x+e)-((c+d)*(c-d))^(1/2)*sin(f*x+e)-c+d))*d-ln(2*(-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)*c*sin(f*x+e)+2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*d*sin(f*x+e)+(
(c+d)*(c-d))^(1/2)*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c
*cos(f*x+e)-d*cos(f*x+e)-c+d))*c+ln(2*(-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*c*sin(f*x
+e)+2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*d*sin(f*x+e)+((c+d)*(c-d))^(1/2)*cos(f*x+e)+c
*sin(f*x+e)-d*sin(f*x+e)-((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d))*
d)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*2^(1/2)*a/((c+d)*(c-d))^(1/2)/c/(d
/(c-d))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.01, size = 777, normalized size = 7.06 \begin {gather*} \left [-\frac {{\left (a c - a d\right )} \sqrt {-\frac {a}{c d + d^{2}}} \log \left (\frac {2 \, {\left (c d + d^{2}\right )} \sqrt {-\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c + 2 \, a d\right )} \cos \left (f x + e\right )^{2} - a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{c \cos \left (f x + e\right )^{2} + {\left (c + d\right )} \cos \left (f x + e\right ) + d}\right ) - \sqrt {-a} a \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right )}{c f}, -\frac {2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) + {\left (a c - a d\right )} \sqrt {-\frac {a}{c d + d^{2}}} \log \left (\frac {2 \, {\left (c d + d^{2}\right )} \sqrt {-\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c + 2 \, a d\right )} \cos \left (f x + e\right )^{2} - a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{c \cos \left (f x + e\right )^{2} + {\left (c + d\right )} \cos \left (f x + e\right ) + d}\right )}{c f}, -\frac {2 \, {\left (a c - a d\right )} \sqrt {\frac {a}{c d + d^{2}}} \arctan \left (\frac {{\left (c + d\right )} \sqrt {\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right ) - \sqrt {-a} a \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right )}{c f}, -\frac {2 \, {\left ({\left (a c - a d\right )} \sqrt {\frac {a}{c d + d^{2}}} \arctan \left (\frac {{\left (c + d\right )} \sqrt {\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right ) + a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right )\right )}}{c f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((a*c - a*d)*sqrt(-a/(c*d + d^2))*log((2*(c*d + d^2)*sqrt(-a/(c*d + d^2))*sqrt((a*cos(f*x + e) + a)/cos(f*x
+ e))*cos(f*x + e)*sin(f*x + e) + (a*c + 2*a*d)*cos(f*x + e)^2 - a*d + (a*c + a*d)*cos(f*x + e))/(c*cos(f*x +
e)^2 + (c + d)*cos(f*x + e) + d)) - sqrt(-a)*a*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/
cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)))/(c*f), -(2*a^(3/2)*arctan(s
qrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) + (a*c - a*d)*sqrt(-a/(c*d + d^2))
*log((2*(c*d + d^2)*sqrt(-a/(c*d + d^2))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + (
a*c + 2*a*d)*cos(f*x + e)^2 - a*d + (a*c + a*d)*cos(f*x + e))/(c*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + d)))/
(c*f), -(2*(a*c - a*d)*sqrt(a/(c*d + d^2))*arctan((c + d)*sqrt(a/(c*d + d^2))*sqrt((a*cos(f*x + e) + a)/cos(f*
x + e))*cos(f*x + e)/(a*sin(f*x + e))) - sqrt(-a)*a*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*cos(f*x + e)
+ a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)))/(c*f), -2*((a*c - a*d)
*sqrt(a/(c*d + d^2))*arctan((c + d)*sqrt(a/(c*d + d^2))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(
a*sin(f*x + e))) + a^(3/2)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e)))
)/(c*f)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sec(e + f*x) + 1))**(3/2)/(c + d*sec(e + f*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (90) = 180\).
time = 1.19, size = 285, normalized size = 2.59 \begin {gather*} \frac {\sqrt {2} \sqrt {-a} a^{3} {\left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} d + a c + 3 \, a d\right )}}{4 \, \sqrt {-c d - d^{2}} a}\right )}{\sqrt {-c d - d^{2}} a^{2} c} - \frac {\sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{a c {\left | a \right |}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*sqrt(-a)*a^3*(2*sqrt(2)*(c - d)*arctan(1/4*sqrt(2)*((sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1
/2*f*x + 1/2*e)^2 + a))^2*c - (sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2*d + a*c
+ 3*a*d)/(sqrt(-c*d - d^2)*a))/(sqrt(-c*d - d^2)*a^2*c) - sqrt(2)*log(abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - s
qrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a
*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 4*sqrt(2)*abs(a) - 6*a))/(a*c*abs(a)))*sgn(cos(f*x + e))/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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