3.3.52 \(\int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx\) [252]

Optimal. Leaf size=228 \[ -\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f} \]

[Out]

-cot(f*x+e)*(a+b*sec(f*x+e))^(3/2)/f-3*(a-b)*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a
-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/f+3*(a-b)*cot(f*x+e)*El
lipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b
*(1+sec(f*x+e))/(a-b))^(1/2)/f

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Rubi [A]
time = 0.17, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3960, 3914, 3917, 4089} \begin {gather*} \frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f}-\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*
Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (3*(a - b)*Sqrt[a + b]*Cot[e
+ f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(
a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*(a + b*Sec[e + f*x])^(3/2))/f

Rule 3914

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[a - b, Int[Csc[e + f
*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[b, Int[Csc[e + f*x]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]),
x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3960

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[Tan[e + f*x]*((a
+ b*Csc[e + f*x])^m/f), x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e
, f, m}, x]

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin {align*} \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx &=-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac {1}{2} (3 b) \int \sec (e+f x) \sqrt {a+b \sec (e+f x)} \, dx\\ &=-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac {1}{2} (3 (a-b) b) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx+\frac {1}{2} \left (3 b^2\right ) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx\\ &=-\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}\\ \end {align*}

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Mathematica [A]
time = 9.01, size = 276, normalized size = 1.21 \begin {gather*} \frac {\cos (e+f x) (a+b \sec (e+f x))^{3/2} ((-b-a \cos (e+f x)) \csc (e+f x)+3 b \sin (e+f x))}{f (b+a \cos (e+f x))}+\frac {3 b (a+b \sec (e+f x))^{3/2} \left (-\frac {(a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}-(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b+a \cos (e+f x))^2 \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((-b - a*Cos[e + f*x])*Csc[e + f*x] + 3*b*Sin[e + f*x]))/(f*(b + a*Co
s[e + f*x])) + (3*b*(a + b*Sec[e + f*x])^(3/2)*(-(((a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x
]))]*(EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a +
b)]))/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) - (b + a*Cos[e + f*x])*Tan[(e + f*x)/2]))/(f*(b + a*Cos[e + f*x])
^2*Sqrt[Sec[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(848\) vs. \(2(208)=416\).
time = 0.24, size = 849, normalized size = 3.72

method	result	size
default	\(\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (3 \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \cos \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) a b +3 \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \cos \left (f x +e \right ) b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right )-3 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \cos \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) a b -3 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \cos \left (f x +e \right ) b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right )+3 \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) a b +3 \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right )-3 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) a b \sin \left (f x +e \right )-3 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b^{2} \sin \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right ) a^{2}-3 \left (\cos ^{2}\left (f x +e \right )\right ) a b +\cos \left (f x +e \right ) a b -3 \cos \left (f x +e \right ) b^{2}+2 b^{2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}}{f \left (a \cos \left (f x +e \right )+b \right ) \sin \left (f x +e \right )^{5}}\)	\(849\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(-1+cos(f*x+e))^2*(3*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*(cos(f*x+e)/(cos
(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*sin(f*x+e)*a*b+3*EllipticE((-1+cos(f*x+e))/sin
(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*b^2*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)
/(a+b))^(1/2)*sin(f*x+e)-3*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*(cos(f*x+e)/(c
os(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*sin(f*x+e)*a*b-3*EllipticF((-1+cos(f*x+e))/s
in(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*b^2*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+
1)/(a+b))^(1/2)*sin(f*x+e)+3*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)
+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*sin(f*x+e)*a*b+3*EllipticE((-1+cos(f*x+e))/sin(f*x+e)
,((a-b)/(a+b))^(1/2))*b^2*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*sin(
f*x+e)-3*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*EllipticF((-1+cos(f*x
+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a*b*sin(f*x+e)-3*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos
(f*x+e)+1)/(a+b))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^2*sin(f*x+e)-cos(f*x+e)^2*
a^2-3*cos(f*x+e)^2*a*b+cos(f*x+e)*a*b-3*cos(f*x+e)*b^2+2*b^2)*(cos(f*x+e)+1)^2*((a*cos(f*x+e)+b)/cos(f*x+e))^(
1/2)/(a*cos(f*x+e)+b)/sin(f*x+e)^5

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

[Out]

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

[Out]

integral((b*csc(f*x + e)^2*sec(f*x + e) + a*csc(f*x + e)^2)*sqrt(b*sec(f*x + e) + a), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2*(a+b*sec(f*x+e))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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