3.2.46 \(\int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx\) [146]

Optimal. Leaf size=319 \[ \frac {2 \sqrt {c+d} \cot (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {c+d}{c};\text {ArcSin}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\text {ArcSin}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \]

[Out]

2*cot(f*x+e)*EllipticF((c+d*sec(f*x+e))^(1/2)/(c+d)^(1/2),((c+d)/(c-d))^(1/2))*(c+d)^(1/2)*(d*(1-sec(f*x+e))/(
c+d))^(1/2)*(-d*(1+sec(f*x+e))/(c-d))^(1/2)/a/(c-d)/f-2*cot(f*x+e)*EllipticPi((c+d*sec(f*x+e))^(1/2)/(c+d)^(1/
2),(c+d)/c,((c+d)/(c-d))^(1/2))*(c+d)^(1/2)*(d*(1-sec(f*x+e))/(c+d))^(1/2)*(-d*(1+sec(f*x+e))/(c-d))^(1/2)/a/c
/f-EllipticE(tan(f*x+e)/(1+sec(f*x+e)),((c-d)/(c+d))^(1/2))*(1/(1+sec(f*x+e)))^(1/2)*(c+d*sec(f*x+e))^(1/2)/a/
(c-d)/f/((c+d*sec(f*x+e))/(c+d)/(1+sec(f*x+e)))^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 319, 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 = {4014, 4006, 3869, 3917, 4053} \begin {gather*} \frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} F\left (\text {ArcSin}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a f (c-d)}-\frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} \Pi \left (\frac {c+d}{c};\text {ArcSin}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a c f}-\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\text {ArcSin}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f (c-d) \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

(2*Sqrt[c + d]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(
1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*(c - d)*f) - (2*Sqrt[c + d]*Cot[e + f*
x]*EllipticPi[(c + d)/c, ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f
*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*c*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e +
f*x])], (c - d)/(c + d)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sec[
e + f*x])/((c + d)*(1 + Sec[e + f*x]))])

Rule 3869

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3917

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

Rule 4006

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

Rule 4014

Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Dist[1
/(c*(b*c - a*d)), Int[(b*c - a*d - b*d*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d^2/(c*(b*c - a*d
)), 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 4053

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

Rubi steps

\begin {align*} \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx &=-\frac {\int \frac {-a c+a d-a d \sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a^2 (c-d)}+\frac {a \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{-a c+a d}\\ &=-\frac {E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}+\frac {\int \frac {1}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}+\frac {d \int \frac {\sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a (c-d)}\\ &=\frac {2 \sqrt {c+d} \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {c+d}{c};\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}\\ \end {align*}

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Mathematica [A]
time = 12.32, size = 187, normalized size = 0.59 \begin {gather*} \frac {2 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}} \left ((c+d) E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+2 (c-2 d) F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+4 (-c+d) \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )\right ) \sec (e+f x)}{a (-c+d) f \sqrt {c+d \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

(2*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*
x]))]*((c + d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 2*(c - 2*d)*EllipticF[ArcSin[Tan[(e + f*
x)/2]], (c - d)/(c + d)] + 4*(-c + d)*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sec[e + f*x])
/(a*(-c + d)*f*Sqrt[c + d*Sec[e + f*x]])

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Maple [A]
time = 0.31, size = 327, normalized size = 1.03

method	result	size
default	\(-\frac {\sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \left (c +d \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right ) \left (2 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) c -4 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) d +c \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )+d \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )-4 c \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {c -d}{c +d}}\right )+4 \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{a f \left (d +c \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2} \left (c -d \right )}\)	\(327\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/f*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/
(c+d))^(1/2)*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))*(2*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1/2))*c-4
*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1/2))*d+c*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+
d))^(1/2))+d*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1/2))-4*c*EllipticPi((-1+cos(f*x+e))/sin(f*x+
e),-1,((c-d)/(c+d))^(1/2))+4*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((c-d)/(c+d))^(1/2))*d)/(d+c*cos(f*x+e))
/sin(f*x+e)^2/(c-d)

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

[Out]

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

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

[Out]

integral(sqrt(d*sec(f*x + e) + c)/(a*d*sec(f*x + e)^2 + a*c + (a*c + a*d)*sec(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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