3.186 \(\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=111 \[ \frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{c^{3/2} f}-\frac{2 a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.357118, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3939, 3934, 203, 3987, 37} \[ \frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{c^{3/2} f}-\frac{2 a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3939

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

Rule 3934

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

Rule 3987

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[(a^2*g*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x
]]), Subst[Int[((g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; Free
Q[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p,
1] || IntegerQ[m - 1/2])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.955859, size = 135, normalized size = 1.22 \[ -\frac{\sec \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \left (2 \sqrt{c} d \sin \left (\frac{1}{2} (e+f x)\right )-\sqrt{2} (c+d)^{3/2} \sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sin \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right ) \sqrt{\frac{c \cos (e+f x)+d}{c+d}}\right )}{c^{3/2} f (c+d) \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(-(Sqrt[2]*(c + d)^(3/2)*ArcSin[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/
2])/Sqrt[c + d]]*Sqrt[(d + c*Cos[e + f*x])/(c + d)]) + 2*Sqrt[c]*d*Sin[(e + f*x)/2]))/(c^(3/2)*(c + d)*f*Sqrt[
c + d*Sec[e + f*x]]))

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Maple [B]  time = 0.338, size = 377, normalized size = 3.4 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{f \left ( c+d \right ){c}^{2} \left ({c}^{2}-2\,cd+{d}^{2} \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) } \left ( \sqrt{2}\sqrt{- \left ( c-d \right ) ^{4}c}\arctan \left ({\frac{c \left ( c-d \right ) ^{2}\sqrt{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{- \left ( c-d \right ) ^{4}c}}}{\frac{1}{\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}}} \right ) \sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}c\sin \left ( fx+e \right ) +\sqrt{2}\sqrt{- \left ( c-d \right ) ^{4}c}\arctan \left ({\frac{c \left ( c-d \right ) ^{2}\sqrt{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{- \left ( c-d \right ) ^{4}c}}}{\frac{1}{\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}}} \right ) \sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}d\sin \left ( fx+e \right ) -2\,{c}^{3}d\cos \left ( fx+e \right ) +4\,{c}^{2}{d}^{2}\cos \left ( fx+e \right ) -2\,\cos \left ( fx+e \right ) c{d}^{3}+2\,{c}^{3}d-4\,{c}^{2}{d}^{2}+2\,c{d}^{3} \right ) \sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/f/(c+d)/c^2/(c^2-2*c*d+d^2)*(2^(1/2)*(-(c-d)^4*c)^(1/2)*arctan((c-d)^2*c*2^(1/2)/(-(c-d)^4*c)^(1/2)*(-1+cos
(f*x+e))/(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)/sin(f*x+e))*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*c*s
in(f*x+e)+2^(1/2)*(-(c-d)^4*c)^(1/2)*arctan((c-d)^2*c*2^(1/2)/(-(c-d)^4*c)^(1/2)*(-1+cos(f*x+e))/(-2*(d+c*cos(
f*x+e))/(1+cos(f*x+e)))^(1/2)/sin(f*x+e))*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)-2*c^3*d*cos(
f*x+e)+4*c^2*d^2*cos(f*x+e)-2*cos(f*x+e)*c*d^3+2*c^3*d-4*c^2*d^2+2*c*d^3)*cos(f*x+e)*(1/cos(f*x+e)*a*(1+cos(f*
x+e)))^(1/2)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)/(d+c*cos(f*x+e))/sin(f*x+e)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.919819, size = 1262, normalized size = 11.37 \begin{align*} \left [-\frac{2 \, d \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) -{\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right )^{2} + c d + d^{2} +{\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{-\frac{a}{c}} \log \left (-\frac{2 \, c \sqrt{-\frac{a}{c}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c \cos \left (f x + e\right )^{2} + a c - a d -{\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{{\left (c^{3} + c^{2} d\right )} f \cos \left (f x + e\right )^{2} +{\left (c^{3} + 2 \, c^{2} d + c d^{2}\right )} f \cos \left (f x + e\right ) +{\left (c^{2} d + c d^{2}\right )} f}, -\frac{2 \,{\left (d \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) +{\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right )^{2} + c d + d^{2} +{\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{\sqrt{\frac{a}{c}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right )\right )}}{{\left (c^{3} + c^{2} d\right )} f \cos \left (f x + e\right )^{2} +{\left (c^{3} + 2 \, c^{2} d + c d^{2}\right )} f \cos \left (f x + e\right ) +{\left (c^{2} d + c d^{2}\right )} f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out