3.15.79 \(\int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1479]
Optimal. Leaf size=312 \[ \frac {\sec (e+f x) (b+a \sin (e+f x)) \sqrt {a+b \sin (e+f x)}}{f \sqrt {d \sin (e+f x)}}-\frac {(a+b)^{3/2} \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f}-\frac {b (a+b) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {b+a \csc (e+f x)}{-a+b}} E\left (\sin ^{-1}\left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {-a+b}{a+b}\right ) (1+\sin (e+f x)) \tan (e+f x)}{f \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]
[Out]
sec(f*x+e)*(b+a*sin(f*x+e))*(a+b*sin(f*x+e))^(1/2)/f/(d*sin(f*x+e))^(1/2)-(a+b)^(3/2)*EllipticF(d^(1/2)*(a+b*s
in(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1
+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/f/d^(1/2)-b*(a+b)*EllipticE(((-b-a*csc(f*x+e))/(a-b))^(1/2),((-a+b)/(a+b)
)^(1/2))*(1+sin(f*x+e))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*((b+a*csc(f*x+e))/(-a+b))^(1/2)*tan(f*x+e)/f/(a*(1+cs
c(f*x+e))/(a-b))^(1/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)
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Rubi [F]
time = 0.11, 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 {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx \end {gather*}
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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4593\) vs. \(2(312)=624\).
time = 32.90, size = 4593, normalized size = 14.72 \begin {gather*} \text {Result too large to show} \end {gather*}
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]
(Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]]*((a*Sqrt[a + b*Sin[e + f*x]])/(
2*Sqrt[Sin[e + f*x]]) - b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2
+ b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a
^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[
(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]]
, (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Ta
n[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b
^2])])))/(16*f*Sqrt[d*Sin[e + f*x]]*((b*Cos[e + f*x]*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^(7/2)*
(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(
-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^
2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(
e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2]
)/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(
a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(32*Sqrt[a + b*Sin[e + f*x]]) + (7*Cos[e + f*x]*Csc[(e + f*x)/
2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^(5/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 +
b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2
+ b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e
+ f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (
2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(
e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2]
)])))/32 - (Csc[(e + f*x)/2]^5*Sec[(e + f*x)/2]*Sin[e + f*x]^(7/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Tan[(e + f*x
)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcS
in[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sq
rt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[
-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2
+ b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2]
)/(-b + Sqrt[-a^2 + b^2])])))/8 + (Csc[(e + f*x)/2]^3*Sec[(e + f*x)/2]^3*Sin[e + f*x]^(7/2)*Sqrt[a + b*Sin[e +
f*x]]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 -
b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*S
qrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] +
a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e +
f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])
*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/16 + (Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x
]^(7/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] - (a*Sqrt[-a^2 + b^2]*Sec[(e + f*x)
/2]^2*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2
+ b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e
+ f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]],
(2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[
(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(2*(-b + Sqrt[-a^2 + b^2])*(a + b*Sin[e + f*x])*((a*Tan[(e + f*x)/2])
/(-b + Sqrt[-a^2 + b^2]))^(3/2)) - (2*b*Sqrt[-a^2 + b^2]*Cos[e + f*x]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e
+ f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]
]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqr
t[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*S
qrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a +
b*Sin[e + f*x])^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) + (Sqrt[-a^2 + b^2]*((a*b*Cos[e + f*x]*
Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f...
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2312\) vs.
\(2(287)=574\).
time = 0.32, size = 2313, normalized size = 7.41
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2313\) |
| | |
|
|
|
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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,method=_RETURNVERBOSE)
[Out]
-1/2/f*(2*cos(f*x+e)^2*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))
^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2)*
(-a^2+b^2)^(1/2)*b^2-2*cos(f*x+e)^2*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2
)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2)^(
1/2)))^(1/2)*a^2*b+2*cos(f*x+e)^2*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/
sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/
2)))^(1/2)*b^3-cos(f*x+e)^2*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/sin(f*
x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2)*EllipticF((((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)
-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*a^2+2*cos(f*x+e)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(
1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2
)^(1/2)))^(1/2)*(-a^2+b^2)^(1/2)*b^2-2*cos(f*x+e)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(
-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(
b+(-a^2+b^2)^(1/2)))^(1/2)*a^2*b+2*cos(f*x+e)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2
+b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-
a^2+b^2)^(1/2)))^(1/2)*b^3-cos(f*x+e)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1
/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(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)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2)*EllipticF((((-a^2+b^2)^(1/2)*sin(f*x+e)+b*
sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(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)^(1/2)*a^2+2^(1/2)*sin(f*x+e)*cos(f*x+e)*a*b^2+2^(1/2)*cos(f*x+e)^2*a^2*b-2^(1/2)*sin
(f*x+e)*a^3-2^(1/2)*sin(f*x+e)*a*b^2+2^(1/2)*cos(f*x+e)*a^2*b-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)*2^(1/2)/a
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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(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)
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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(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)
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Sympy [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(sec(f*x+e)**2*(a+b*sin(f*x+e))**(3/2)/(d*sin(f*x+e))**(1/2),x)
[Out]
Timed out
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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(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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^2\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sin(e + f*x))^(3/2)/(cos(e + f*x)^2*(d*sin(e + f*x))^(1/2)),x)
[Out]
int((a + b*sin(e + f*x))^(3/2)/(cos(e + f*x)^2*(d*sin(e + f*x))^(1/2)), x)
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