3.3.17 \(\int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx\) [217]

Optimal. Leaf size=652 \[ -\frac {2 c (c+d) \cot (e+f x) \Pi \left (\frac {a (c+d)}{(a+b) c};\text {ArcSin}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))^{3/2} \sqrt {\frac {(a+b) (b c-a d) (-1+\sec (e+f x)) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}}}{a (a+b) f \sqrt {c+d \sec (e+f x)}}+\frac {2 d (c+d) \cot (e+f x) \Pi \left (\frac {b (c+d)}{(a+b) d};\text {ArcSin}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))^{3/2} \sqrt {-\frac {(a+b) (-b c+a d) (-1+\sec (e+f x)) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}}}{b (a+b) f \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \cot (e+f x) F\left (\text {ArcSin}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (-1+\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a b f \sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}} \]

[Out]

-2*c*(c+d)*cot(f*x+e)*EllipticPi(((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2),a*(c+d)/(a+b)/c,((a-b)*
(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec(f*x+e))^(3/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)*((a+
b)*(-a*d+b*c)*(-1+sec(f*x+e))*(c+d*sec(f*x+e))/(c+d)^2/(a+b*sec(f*x+e))^2)^(1/2)/a/(a+b)/f/(c+d*sec(f*x+e))^(1
/2)+2*d*(c+d)*cot(f*x+e)*EllipticPi(((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2),b*(c+d)/(a+b)/d,((a-
b)*(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec(f*x+e))^(3/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)*(
-(a+b)*(a*d-b*c)*(-1+sec(f*x+e))*(c+d*sec(f*x+e))/(c+d)^2/(a+b*sec(f*x+e))^2)^(1/2)/b/(a+b)/f/(c+d*sec(f*x+e))
^(1/2)+2*(-a*d+b*c)*cot(f*x+e)*EllipticF(((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2),((a-b)*(c+d)/(a
+b)/(c-d))^(1/2))*((-a*d+b*c)*(-1+sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(
a+b*sec(f*x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(c+d*sec(f*x+e))^(1/2)/a/b/f/((a+b)*(c+d*sec(f*x+e))/(c+d)/(a+b*
sec(f*x+e)))^(1/2)

________________________________________________________________________________________

Rubi [F]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx &=\int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 35.69, size = 49385, normalized size = 75.74 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [A]
time = 2.80, size = 491, normalized size = 0.75

method	result	size
default	\(\frac {2 \left (2 \EllipticPi \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) d^{2}+2 \EllipticPi \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) c^{2}-\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c^{2}+2 \EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c d -\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) d^{2}\right ) \cos \left (f x +e \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \left (c +d \right )}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}}{f \left (-1+\cos \left (f x +e \right )\right ) \left (d +c \cos \left (f x +e \right )\right ) \left (a \cos \left (f x +e \right )+b \right ) \sqrt {\frac {a -b}{a +b}}}\)	\(491\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________