\(\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx\) [235]
Optimal result
Integrand size = 35, antiderivative size = 140 \[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {\sqrt {2} \sqrt {c-d} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}
\]
[Out]
arctan(1/2*a^(1/2)*(c-d)^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2))*2^(1/2)*(c-d)
^(1/2)/f/a^(1/2)+2*arctanh(a^(1/2)*d^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2))*d^(1/2)/f
/a^(1/2)
Rubi [A] (verified)
Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4066, 4068, 209, 4065, 212}
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {\sqrt {2} \sqrt {c-d} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}
\]
[In]
Int[(Sec[e + f*x]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + a*Sec[e + f*x]],x]
[Out]
(Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Se
c[e + f*x]])])/(Sqrt[a]*f) + (2*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[
c + d*Sec[e + f*x]])])/(Sqrt[a]*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 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 4065
Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Dist[-2*(b/f), Subst[Int[1/(1 - b*d*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sq
rt[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 4066
Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Dist[-(b*c - a*d)/d, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]),
x], x] + Dist[b/d, Int[Csc[e + f*x]*(Sqrt[c + d*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] && EqQ[c^2 - d^2, 0]
Rule 4068
Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (
c_)]), x_Symbol] :> Dist[-2*(a/(b*f)), Subst[Int[1/(2 + (a*c - b*d)*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]
Rubi steps \begin{align*}
\text {integral}& = (c-d) \int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx+\frac {d \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a} \\ & = -\frac {(2 (c-d)) \text {Subst}\left (\int \frac {1}{2+(a c-a d) x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{1-a d x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \\ & = \frac {\sqrt {2} \sqrt {c-d} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f} \\
\end{align*}
Mathematica [A] (verified)
Time = 14.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.34
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {\sqrt {c} \left (-\sqrt {2} \sqrt {c-d} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+c \cos (e+f x)}}{\sqrt {c-d} \sqrt {c-c \cos (e+f x)}}\right )+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+c \cos (e+f x)}}{\sqrt {d} \sqrt {c-c \cos (e+f x)}}\right )\right ) \sqrt {c+d \sec (e+f x)} \sin (e+f x)}{f \sqrt {c-c \cos (e+f x)} \sqrt {d+c \cos (e+f x)} \sqrt {a (1+\sec (e+f x))}}
\]
[In]
Integrate[(Sec[e + f*x]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + a*Sec[e + f*x]],x]
[Out]
(Sqrt[c]*(-(Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[c - d]*Sqrt[c - c*Cos[
e + f*x]])]) + 2*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])])*Sqrt[
c + d*Sec[e + f*x]]*Sin[e + f*x])/(f*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x
])])
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(477\) vs. \(2(113)=226\).
Time = 5.30 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.41
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| method | result | size |
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| default |
\(-\frac {\sqrt {2}\, \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\ln \left (\frac {\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}\, \sqrt {c -d}-c \cot \left (f x +e \right )+d \cot \left (f x +e \right )+c \csc \left (f x +e \right )-d \csc \left (f x +e \right )}{\sqrt {c -d}}\right ) \sqrt {2}\, \sqrt {-d}\, c -\ln \left (\frac {\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}\, \sqrt {c -d}-c \cot \left (f x +e \right )+d \cot \left (f x +e \right )+c \csc \left (f x +e \right )-d \csc \left (f x +e \right )}{\sqrt {c -d}}\right ) \sqrt {2}\, \sqrt {-d}\, d +d \ln \left (\frac {-2 \sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )+2 \sin \left (f x +e \right ) c +2 \sin \left (f x +e \right ) d -2 c +2 d}{\cos \left (f x +e \right )-1+\sin \left (f x +e \right )}\right ) \sqrt {c -d}-d \ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-\sin \left (f x +e \right ) c -\sin \left (f x +e \right ) d +c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d \right )}{\cos \left (f x +e \right )-1-\sin \left (f x +e \right )}\right ) \sqrt {c -d}\right ) \cos \left (f x +e \right )}{f a \sqrt {c -d}\, \sqrt {-d}\, \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}}\) |
\(478\) |
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[In]
int(sec(f*x+e)*(c+d*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/f/a/(c-d)^(1/2)*2^(1/2)/(-d)^(1/2)*(c+d*sec(f*x+e))^(1/2)*(a*(sec(f*x+e)+1))^(1/2)*(ln(1/(c-d)^(1/2)*((-2*(
d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c-d)^(1/2)-c*cot(f*x+e)+d*cot(f*x+e)+c*csc(f*x+e)-d*csc(f*x+e)))*2^(1/2
)*(-d)^(1/2)*c-ln(1/(c-d)^(1/2)*((-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c-d)^(1/2)-c*cot(f*x+e)+d*cot(f*x
+e)+c*csc(f*x+e)-d*csc(f*x+e)))*2^(1/2)*(-d)^(1/2)*d+d*ln(2*(-2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x
+e)+1))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)+sin(f*x+e)*c+sin(f*x+e)*d-c+d)/(cos(f*x+e)-1+sin(f*x+e)))*(
c-d)^(1/2)-d*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(
f*x+e)*d+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(cos(f*x+e)-1-sin(f*x+e)))*(c-d)^(1/2))*cos(f*x+e)/(cos(f*x+e)+1)/(-2*
(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.45 (sec) , antiderivative size = 1048, normalized size of antiderivative = 7.49
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Too large to display}
\]
[In]
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[1/2*(sqrt(2)*sqrt(-(c - d)/a)*log(-(2*sqrt(2)*sqrt(-(c - d)/a)*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) - (3*c - d)*cos(f*x + e)^2 - 2*(c + d)*cos(f*x + e
) + c - 3*d)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + sqrt(d/a)*log(-((c^2 - 6*c*d + d^2)*cos(f*x + e)^3 + 4*(
(c - d)*cos(f*x + e)^2 + 2*d*cos(f*x + e))*sqrt(d/a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x +
e) + d)/cos(f*x + e))*sin(f*x + e) + 8*c*d*cos(f*x + e) + (c^2 + 2*c*d - 7*d^2)*cos(f*x + e)^2 + 8*d^2)/(cos(
f*x + e)^3 + cos(f*x + e)^2)))/f, 1/2*(2*sqrt(2)*sqrt((c - d)/a)*arctan(-sqrt(2)*sqrt((c - d)/a)*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)/((c - d)*sin(f*x + e))) + sqrt
(d/a)*log(-((c^2 - 6*c*d + d^2)*cos(f*x + e)^3 + 4*((c - d)*cos(f*x + e)^2 + 2*d*cos(f*x + e))*sqrt(d/a)*sqrt(
(a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*sin(f*x + e) + 8*c*d*cos(f*x + e) +
(c^2 + 2*c*d - 7*d^2)*cos(f*x + e)^2 + 8*d^2)/(cos(f*x + e)^3 + cos(f*x + e)^2)))/f, 1/2*(sqrt(2)*sqrt(-(c -
d)/a)*log(-(2*sqrt(2)*sqrt(-(c - d)/a)*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) - (3*c - d)*cos(f*x + e)^2 - 2*(c + d)*cos(f*x + e) + c - 3*d)/(cos(f*x + e
)^2 + 2*cos(f*x + e) + 1)) + 2*sqrt(-d/a)*arctan(-2*sqrt(-d/a)*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 - d)*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + 2*
d)))/f, (sqrt(2)*sqrt((c - d)/a)*arctan(-sqrt(2)*sqrt((c - d)/a)*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)/((c - d)*sin(f*x + e))) + sqrt(-d/a)*arctan(-2*sqrt(-d/a)*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 - d)
*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + 2*d)))/f]
Sympy [F]
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {\sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}\, dx
\]
[In]
integrate(sec(f*x+e)*(c+d*sec(f*x+e))**(1/2)/(a+a*sec(f*x+e))**(1/2),x)
[Out]
Integral(sqrt(c + d*sec(e + f*x))*sec(e + f*x)/sqrt(a*(sec(e + f*x) + 1)), x)
Maxima [F]
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a}} \,d x }
\]
[In]
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(d*sec(f*x + e) + c)*sec(f*x + e)/sqrt(a*sec(f*x + e) + a), x)
Giac [F]
\[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a}} \,d x }
\]
[In]
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(d*sec(f*x + e) + c)*sec(f*x + e)/sqrt(a*sec(f*x + e) + a), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x
\]
[In]
int((c + d/cos(e + f*x))^(1/2)/(cos(e + f*x)*(a + a/cos(e + f*x))^(1/2)),x)
[Out]
int((c + d/cos(e + f*x))^(1/2)/(cos(e + f*x)*(a + a/cos(e + f*x))^(1/2)), x)