3.83 \(\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{\sqrt{c-c \sec (e+f x)}} \, dx\)

Optimal. Leaf size=164 \[ -\frac{8 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{c} f}+\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{4 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 f \sqrt{c-c \sec (e+f x)}}+\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}} \]

[Out]

(-8*Sqrt[2]*a^3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f) + (8*a^3*Tan[e
+ f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) + (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[c - c*Sec[e + f*x]]
) + (4*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])

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Rubi [A]  time = 0.325402, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {3956, 3795, 203} \[ -\frac{8 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{c} f}+\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{4 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 f \sqrt{c-c \sec (e+f x)}}+\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[2]*a^3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f) + (8*a^3*Tan[e
+ f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) + (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[c - c*Sec[e + f*x]]
) + (4*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])

Rule 3956

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

Rule 3795

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{\sqrt{c-c \sec (e+f x)}} \, dx &=\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}+(2 a) \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt{c-c \sec (e+f x)}} \, dx\\ &=\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}+\frac{4 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}+\left (4 a^2\right ) \int \frac{\sec (e+f x) (a+a \sec (e+f x))}{\sqrt{c-c \sec (e+f x)}} \, dx\\ &=\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}+\frac{4 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}+\left (8 a^3\right ) \int \frac{\sec (e+f x)}{\sqrt{c-c \sec (e+f x)}} \, dx\\ &=\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}+\frac{4 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{\left (16 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 c+x^2} \, dx,x,\frac{c \tan (e+f x)}{\sqrt{c-c \sec (e+f x)}}\right )}{f}\\ &=-\frac{8 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{c} f}+\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}+\frac{4 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 1.53253, size = 185, normalized size = 1.13 \[ \frac{4 a^3 e^{-\frac{1}{2} i (e+f x)} \sin \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )+i \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right ) \left (3 \sec ^2(e+f x)+16 \sec (e+f x)+73\right )-30 \sqrt{2} e^{-\frac{1}{2} i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right )\right )}{15 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*Sec[e + f*x]*((-30*Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1
+ E^((2*I)*(e + f*x))])])/E^((I/2)*(e + f*x)) + Cos[(e + f*x)/2]*(73 + 16*Sec[e + f*x] + 3*Sec[e + f*x]^2))*(C
os[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/(15*E^((I/2)*(e + f*x))*f*Sqrt[c - c*Sec[e + f*x]])

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Maple [A]  time = 0.273, size = 206, normalized size = 1.3 \begin{align*} -{\frac{2\,{a}^{3}\sin \left ( fx+e \right ) }{15\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( 15\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{5/2}+30\,\cos \left ( fx+e \right ) \arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{5/2}+15\,\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{5/2}-73\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-16\,\cos \left ( fx+e \right ) -3 \right ){\frac{1}{\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(1/2),x)

[Out]

-2/15*a^3/f*(15*cos(f*x+e)^2*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(5/
2)+30*cos(f*x+e)*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(5/2)+15*arctan
(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(5/2)-73*cos(f*x+e)^2-16*cos(f*x+e)-3)
*sin(f*x+e)/cos(f*x+e)^3/(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0.616898, size = 942, normalized size = 5.74 \begin{align*} \left [\frac{2 \,{\left (30 \, \sqrt{2} a^{3} c \sqrt{-\frac{1}{c}} \cos \left (f x + e\right )^{2} \log \left (-\frac{2 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sqrt{-\frac{1}{c}} -{\left (3 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) -{\left (73 \, a^{3} \cos \left (f x + e\right )^{3} + 89 \, a^{3} \cos \left (f x + e\right )^{2} + 19 \, a^{3} \cos \left (f x + e\right ) + 3 \, a^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{15 \, c f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}, \frac{2 \,{\left (60 \, \sqrt{2} a^{3} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt{c} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) -{\left (73 \, a^{3} \cos \left (f x + e\right )^{3} + 89 \, a^{3} \cos \left (f x + e\right )^{2} + 19 \, a^{3} \cos \left (f x + e\right ) + 3 \, a^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{15 \, c f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/15*(30*sqrt(2)*a^3*c*sqrt(-1/c)*cos(f*x + e)^2*log(-(2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e))*sqrt((c*cos(
f*x + e) - c)/cos(f*x + e))*sqrt(-1/c) - (3*cos(f*x + e) + 1)*sin(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))
*sin(f*x + e) - (73*a^3*cos(f*x + e)^3 + 89*a^3*cos(f*x + e)^2 + 19*a^3*cos(f*x + e) + 3*a^3)*sqrt((c*cos(f*x
+ e) - c)/cos(f*x + e)))/(c*f*cos(f*x + e)^2*sin(f*x + e)), 2/15*(60*sqrt(2)*a^3*sqrt(c)*arctan(sqrt(2)*sqrt((
c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(sqrt(c)*sin(f*x + e)))*cos(f*x + e)^2*sin(f*x + e) - (73*a^3*c
os(f*x + e)^3 + 89*a^3*cos(f*x + e)^2 + 19*a^3*cos(f*x + e) + 3*a^3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/
(c*f*cos(f*x + e)^2*sin(f*x + e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{\sec{\left (e + f x \right )}}{\sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{3 \sec ^{2}{\left (e + f x \right )}}{\sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{3 \sec ^{3}{\left (e + f x \right )}}{\sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{\sec ^{4}{\left (e + f x \right )}}{\sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*(Integral(sec(e + f*x)/sqrt(-c*sec(e + f*x) + c), x) + Integral(3*sec(e + f*x)**2/sqrt(-c*sec(e + f*x) +
c), x) + Integral(3*sec(e + f*x)**3/sqrt(-c*sec(e + f*x) + c), x) + Integral(sec(e + f*x)**4/sqrt(-c*sec(e + f
*x) + c), x))

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Giac [C]  time = 2.16939, size = 308, normalized size = 1.88 \begin{align*} -\frac{2 \,{\left (4 \, a^{3} c^{3}{\left (\frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{\sqrt{c}}\right )}{c^{\frac{7}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} + \frac{\sqrt{2}{\left (15 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{2} - 5 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} c + 3 \, c^{2}\right )}}{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{5}{2}} c^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}\right )} + \frac{{\left (60 i \, \sqrt{2} a^{3} \sqrt{-c} \arctan \left (-i\right ) - 92 \, \sqrt{2} a^{3} \sqrt{-c}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}{c}\right )}}{15 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(4*a^3*c^3*(15*sqrt(2)*arctan(sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)/sqrt(c))/(c^(7/2)*sgn(tan(1/2*f*x + 1/2
*e)^2 - 1)*sgn(tan(1/2*f*x + 1/2*e))) + sqrt(2)*(15*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2 - 5*(c*tan(1/2*f*x + 1/2*
e)^2 - c)*c + 3*c^2)/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(5/2)*c^3*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*sgn(tan(1/2*f*x
+ 1/2*e)))) + (60*I*sqrt(2)*a^3*sqrt(-c)*arctan(-I) - 92*sqrt(2)*a^3*sqrt(-c))*sgn(tan(1/2*f*x + 1/2*e))/c)/f