3.3.36 \(\int (d \cos (a+b x))^{3/2} \csc ^2(a+b x) \, dx\) [236]

Optimal. Leaf size=66 \[ -\frac {d \sqrt {d \cos (a+b x)} \csc (a+b x)}{b}-\frac {d^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {d \cos (a+b x)}} \]

[Out]

-d^2*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)/b/
(d*cos(b*x+a))^(1/2)-d*csc(b*x+a)*(d*cos(b*x+a))^(1/2)/b

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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2721, 2720} \begin {gather*} -\frac {d^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {d \cos (a+b x)}}-\frac {d \csc (a+b x) \sqrt {d \cos (a+b x)}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2,x]

[Out]

-((d*Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/b) - (d^2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[
a + b*x]])

Rule 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps

\begin {align*} \int (d \cos (a+b x))^{3/2} \csc ^2(a+b x) \, dx &=-\frac {d \sqrt {d \cos (a+b x)} \csc (a+b x)}{b}-\frac {1}{2} d^2 \int \frac {1}{\sqrt {d \cos (a+b x)}} \, dx\\ &=-\frac {d \sqrt {d \cos (a+b x)} \csc (a+b x)}{b}-\frac {\left (d^2 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{2 \sqrt {d \cos (a+b x)}}\\ &=-\frac {d \sqrt {d \cos (a+b x)} \csc (a+b x)}{b}-\frac {d^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {d \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 56, normalized size = 0.85 \begin {gather*} -\frac {(d \cos (a+b x))^{3/2} \left (\sqrt {\cos (a+b x)} \csc (a+b x)+F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{b \cos ^{\frac {3}{2}}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2,x]

[Out]

-(((d*Cos[a + b*x])^(3/2)*(Sqrt[Cos[a + b*x]]*Csc[a + b*x] + EllipticF[(a + b*x)/2, 2]))/(b*Cos[a + b*x]^(3/2)
))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(86)=172\).
time = 0.72, size = 190, normalized size = 2.88

method	result	size
default	\(-\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{3} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (2 \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{2 \left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \right )^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d^3/(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*
a)^2*d)^(3/2)/cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)*(2*(2*sin(1/2*b*x+1/2*a)^2-1)^(3/2)*(sin(1/2*b*x+1/2*a)^2)
^(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))*cos(1/2*b*x+1/2*a)+4*sin(1/2*b*x+1/2*a)^4-4*sin(1/2*b*x+1/2*a)^2+
1)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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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((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^2, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 91, normalized size = 1.38 \begin {gather*} \frac {i \, \sqrt {2} d^{\frac {3}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, \sqrt {2} d^{\frac {3}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, \sqrt {d \cos \left (b x + a\right )} d}{2 \, b \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(I*sqrt(2)*d^(3/2)*sin(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) - I*sqrt(2)*d^(3
/2)*sin(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) - 2*sqrt(d*cos(b*x + a))*d)/(b*sin(
b*x + a))

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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((d*cos(b*x+a))**(3/2)*csc(b*x+a)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5006 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((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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