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3.15.80 \(\int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1480]

Optimal. Leaf size=366 \[ \frac {5 a \sec (e+f x) (b+a \sin (e+f x)) \sqrt {a+b \sin (e+f x)}}{6 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}-\frac {5 a (a+b)^{3/2} \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{6 \sqrt {d} f}-\frac {5 a b (a+b) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {b+a \csc (e+f x)}{-a+b}} E\left (\sin ^{-1}\left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {-a+b}{a+b}\right ) (1+\sin (e+f x)) \tan (e+f x)}{6 f \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]

[Out]

1/3*sec(f*x+e)^3*(a+b*sin(f*x+e))^(5/2)*(d*sin(f*x+e))^(1/2)/d/f+5/6*a*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)-5/6*a*(a+b)^(3/2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*s
in(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)-5/6*a*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+csc(f*x+e))/(a-b))^(1/2)/(d*s
in(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)

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Rubi [F]
time = 0.23, 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 ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2))/(3*d*f) + (5*a*Defer[Int][(Sec[e + f*x]^2*(a
+ b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x])/6

Rubi steps

\begin {align*} \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx &=\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}+\frac {1}{6} (5 a) \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. \(4665\) vs. \(2(366)=732\).
time = 32.70, size = 4665, normalized size = 12.75 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]^3*(a^2 + b^2 + 2*a*b*Sin[e + f*x]))/3 + (Sec[e + f*x]*(5
*a^2 - 2*b^2 + 5*a*b*Sin[e + f*x]))/6))/(f*Sqrt[d*Sin[e + f*x]]) + (5*a*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*
Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]]*((5*a^2*Sqrt[a + b*Sin[e + f*x]])/(12*Sqrt[Sin[e + f*x]]) - (5*a*b*Sqr
t[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/6)*(-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*Elliptic
F[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])])))/(96*f*Sqrt[d*Sin[
e + f*x]]*((5*a*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[S
qrt[(-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])])))/(192*Sqrt[a + b*Sin[e + f*x]]) + (35*a*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*Ell
ipticF[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])])))/192 - (5*a*C
sc[(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*S
qrt[-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])])))/48 + (5*a*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*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])])))/96 + (5*a*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*S
qrt[-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]]/Sq
rt[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])]*S...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2425\) vs. \(2(329)=658\).
time = 0.40, size = 2426, normalized size = 6.63

method result size
default \(\text {Expression too large to display}\) \(2426\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/12/f*(10*(-a^2+b^2)^(1/2)*cos(f*x+e)^4*((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^2-5*(-a^2+b^2)^(1/2)*cos(f*x+e)^4*((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^2-10*cos(f*x+e)^4*((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+10*cos(f*x+e)^4*((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+10*(-a^2+b^2)^(1/2)*cos(f*x+e)^3*((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)*Ell
ipticE((((-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^2-5*(-a^2+b^2)^(1/2)*cos(f*x+e)^3*((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)*El
lipticF((((-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^2-10*cos(f*x+e)^3*((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+10*cos(f*x+e)^3*((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+5*2^(1/2)*sin(f*x+e)*cos(f*x+e)^3*a*b^2+5*2^(1/2)*cos(f
*x+e)^4*a^2*b-2*2^(1/2)*cos(f*x+e)^4*b^3-5*2^(1/2)*sin(f*x+e)*cos(f*x+e)^2*a^3+2^(1/2)*sin(f*x+e)*cos(f*x+e)^2
*a*b^2+5*2^(1/2)*cos(f*x+e)^3*a^2*b-4*2^(1/2)*cos(f*x+e)^2*a^2*b+4*2^(1/2)*cos(f*x+e)^2*b^3-2*2^(1/2)*sin(f*x+
e)*a^3-6*2^(1/2)*sin(f*x+e)*a*b^2-6*2^(1/2)*a^2*b-2*2^(1/2)*b^3)/cos(f*x+e)^3/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*
x+e))^(1/2)*2^(1/2)

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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)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/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)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((2*a*b*sec(f*x + e)^4*sin(f*x + e) - (b^2*cos(f*x + e)^2 - a^2 - b^2)*sec(f*x + e)^4)*sqrt(b*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e))/(d*sin(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)**4*(a+b*sin(f*x+e))**(5/2)/(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)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/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 )}^{5/2}}{{\cos \left (e+f\,x\right )}^4\,\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))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)),x)

[Out]

int((a + b*sin(e + f*x))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)), x)

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