3.91 \(\int (\frac {x}{\csc ^{\frac {3}{2}}(e+f x)}-\frac {1}{3} x \sqrt {\csc (e+f x)}) \, dx\)

Optimal. Leaf size=42 \[ \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]

[Out]

4/9/f^2/csc(f*x+e)^(3/2)-2/3*x*cos(f*x+e)/f/csc(f*x+e)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Csc[e + f*x]^(3/2) - (x*Sqrt[Csc[e + f*x]])/3,x]

[Out]

4/(9*f^2*Csc[e + f*x]^(3/2)) - (2*x*Cos[e + f*x])/(3*f*Sqrt[Csc[e + f*x]])

Rule 4187

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(d*(b*Csc[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x], x] + Simp[((c + d*x)*Cos[e + f*x]
*(b*Csc[e + f*x])^(n + 1))/(b*f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1]

Rule 4189

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \left (\frac {x}{\csc ^{\frac {3}{2}}(e+f x)}-\frac {1}{3} x \sqrt {\csc (e+f x)}\right ) \, dx &=-\left (\frac {1}{3} \int x \sqrt {\csc (e+f x)} \, dx\right )+\int \frac {x}{\csc ^{\frac {3}{2}}(e+f x)} \, dx\\ &=\frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}+\frac {1}{3} \int x \sqrt {\csc (e+f x)} \, dx-\frac {1}{3} \left (\sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {x}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 29, normalized size = 0.69 \[ -\frac {2 (3 f x \cot (e+f x)-2)}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Csc[e + f*x]^(3/2) - (x*Sqrt[Csc[e + f*x]])/3,x]

[Out]

(-2*(-2 + 3*f*x*Cot[e + f*x]))/(9*f^2*Csc[e + f*x]^(3/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{3} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/3*x*sqrt(csc(f*x + e)) + x/csc(f*x + e)^(3/2), x)

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {x}{\csc \left (f x +e \right )^{\frac {3}{2}}}-\frac {x \left (\sqrt {\csc }\left (f x +e \right )\right )}{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x)

[Out]

int(x/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{3} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(-1/3*x*sqrt(csc(f*x + e)) + x/csc(f*x + e)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{3/2}}-\frac {x\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}}{3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1/sin(e + f*x))^(3/2) - (x*(1/sin(e + f*x))^(1/2))/3,x)

[Out]

int(x/(1/sin(e + f*x))^(3/2) - (x*(1/sin(e + f*x))^(1/2))/3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {3 x}{\csc ^{\frac {3}{2}}{\left (e + f x \right )}}\right )\, dx + \int x \sqrt {\csc {\left (e + f x \right )}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)**(3/2)-1/3*x*csc(f*x+e)**(1/2),x)

[Out]

-(Integral(-3*x/csc(e + f*x)**(3/2), x) + Integral(x*sqrt(csc(e + f*x)), x))/3

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