3.1479 \(\int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx\)
Optimal. Leaf size=312 \[ \frac{\sec (e+f x) (a \sin (e+f x)+b) \sqrt{a+b \sin (e+f x)}}{f \sqrt{d \sin (e+f x)}}-\frac{(a+b)^{3/2} \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{\sqrt{d} f}-\frac{b (a+b) (\sin (e+f x)+1) \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a \csc (e+f x)+b}{b-a}} E\left (\sin ^{-1}\left (\sqrt{-\frac{b+a \csc (e+f x)}{a-b}}\right )|\frac{b-a}{a+b}\right )}{f \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}}} \]
[Out]
(Sec[e + f*x]*(b + a*Sin[e + f*x])*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[d*Sin[e + f*x]]) - ((a + b)^(3/2)*Sqrt[-(
(a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Si
n[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f) - (b*(a + b)*S
qrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(b + a*Csc[e + f*x])/(-a + b)]*EllipticE[ArcSin[Sqrt[-((b + a*Csc
[e + f*x])/(a - b))]], (-a + b)/(a + b)]*(1 + Sin[e + f*x])*Tan[e + f*x])/(f*Sqrt[(a*(1 + Csc[e + f*x]))/(a -
b)]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])
________________________________________________________________________________________
Rubi [F] time = 0.18996, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]
[Out]
Defer[Int][(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x]
Rubi steps
\begin{align*} \int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx &=\int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx\\ \end{align*}
Mathematica [B] time = 26.654, size = 6059, normalized size = 19.42 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]
[Out]
Result too large to show
________________________________________________________________________________________
Maple [B] time = 0.381, size = 2313, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x)
[Out]
-1/2/f*2^(1/2)/a*(2*(-a^2+b^2)^(1/2)*cos(f*x+e)^2*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(
b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1
/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((((-a^2+b^2)^(1/2)*s
in(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2)
)/(-a^2+b^2)^(1/2))^(1/2))*b^2-(-a^2+b^2)^(1/2)*cos(f*x+e)^2*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*
x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-
a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticF((((-a^2+b
^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2
+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2-2*cos(f*x+e)^2*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)
*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+
b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((((-a^2+b^2)^
(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2
)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b+2*cos(f*x+e)^2*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a
+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^
2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((((-a^2+b^2)^(1
/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^
(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^3+2*(-a^2+b^2)^(1/2)*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-c
os(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-
a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((((-
a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+
(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2-(-a^2+b^2)^(1/2)*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin
(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(
f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*Ellip
ticF((((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(
1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2-2*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e
)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)
*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((
((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*(
(b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b+2*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-co
s(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a
)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((((-a
^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(
-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^3+cos(f*x+e)^2*2^(1/2)*a^2*b+cos(f*x+e)*sin(f*x+e)*2^(1/2)*a*b^2+c
os(f*x+e)*2^(1/2)*a^2*b-sin(f*x+e)*2^(1/2)*a^3-sin(f*x+e)*2^(1/2)*a*b^2-2*2^(1/2)*a^2*b)/cos(f*x+e)/(d*sin(f*x
+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \sec \left (f x + e\right )^{2} \sin \left (f x + e\right ) + a \sec \left (f x + e\right )^{2}\right )} \sqrt{b \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right )}}{d \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
integral((b*sec(f*x + e)^2*sin(f*x + e) + a*sec(f*x + e)^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))/(d*s
in(f*x + e)), x)
________________________________________________________________________________________
Sympy [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)**2*(a+b*sin(f*x+e))**(3/2)/(d*sin(f*x+e))**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), x)