3.94 \(\int (\frac{x}{\csc ^{\frac{7}{2}}(e+f x)}-\frac{5}{21} x \sqrt{\csc (e+f x)}) \, dx\)

Optimal. Leaf size=83 \[ \frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}} \]

[Out]

4/(49*f^2*Csc[e + f*x]^(7/2)) - (2*x*Cos[e + f*x])/(7*f*Csc[e + f*x]^(5/2)) + 20/(63*f^2*Csc[e + f*x]^(3/2)) -
 (10*x*Cos[e + f*x])/(21*f*Sqrt[Csc[e + f*x]])

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Rubi [A]  time = 0.133373, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Csc[e + f*x]^(7/2) - (5*x*Sqrt[Csc[e + f*x]])/21,x]

[Out]

4/(49*f^2*Csc[e + f*x]^(7/2)) - (2*x*Cos[e + f*x])/(7*f*Csc[e + f*x]^(5/2)) + 20/(63*f^2*Csc[e + f*x]^(3/2)) -
 (10*x*Cos[e + f*x])/(21*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{7}{2}}(e+f x)}-\frac{5}{21} x \sqrt{\csc (e+f x)}\right ) \, dx &=-\left (\frac{5}{21} \int x \sqrt{\csc (e+f x)} \, dx\right )+\int \frac{x}{\csc ^{\frac{7}{2}}(e+f x)} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{5}{7} \int \frac{x}{\csc ^{\frac{3}{2}}(e+f x)} \, dx-\frac{1}{21} \left (5 \sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}}+\frac{5}{21} \int x \sqrt{\csc (e+f x)} \, dx-\frac{1}{21} \left (5 \sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.22757, size = 57, normalized size = 0.69 \[ \frac{-36 \cos (2 (e+f x))-483 f x \cot (e+f x)+63 f x \cos (3 (e+f x)) \csc (e+f x)+316}{882 f^2 \csc ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Csc[e + f*x]^(7/2) - (5*x*Sqrt[Csc[e + f*x]])/21,x]

[Out]

(316 - 36*Cos[2*(e + f*x)] - 483*f*x*Cot[e + f*x] + 63*f*x*Cos[3*(e + f*x)]*Csc[e + f*x])/(882*f^2*Csc[e + f*x
]^(3/2))

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \csc \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,x}{21}\sqrt{\csc \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/csc(f*x+e)^(7/2)-5/21*x*csc(f*x+e)^(1/2),x)

[Out]

int(x/csc(f*x+e)^(7/2)-5/21*x*csc(f*x+e)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \left (f x + e\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(7/2)-5/21*x*csc(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(-5/21*x*sqrt(csc(f*x + e)) + x/csc(f*x + e)^(7/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(7/2)-5/21*x*csc(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)**(7/2)-5/21*x*csc(f*x+e)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \left (f x + e\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/csc(f*x+e)^(7/2)-5/21*x*csc(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(-5/21*x*sqrt(csc(f*x + e)) + x/csc(f*x + e)^(7/2), x)