3.70 \(\int (\frac{x}{\sin ^{\frac{7}{2}}(e+f x)}+\frac{3}{5} x \sqrt{\sin (e+f x)}) \, dx\)

Optimal. Leaf size=83 \[ -\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}} \]

[Out]

(-2*x*Cos[e + f*x])/(5*f*Sin[e + f*x]^(5/2)) - 4/(15*f^2*Sin[e + f*x]^(3/2)) - (6*x*Cos[e + f*x])/(5*f*Sqrt[Si
n[e + f*x]]) + (12*Sqrt[Sin[e + f*x]])/(5*f^2)

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Rubi [A]  time = 0.0858376, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3315} \[ -\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sin[e + f*x]^(7/2) + (3*x*Sqrt[Sin[e + f*x]])/5,x]

[Out]

(-2*x*Cos[e + f*x])/(5*f*Sin[e + f*x]^(5/2)) - 4/(15*f^2*Sin[e + f*x]^(3/2)) - (6*x*Cos[e + f*x])/(5*f*Sqrt[Si
n[e + f*x]]) + (12*Sqrt[Sin[e + f*x]])/(5*f^2)

Rule 3315

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

Rubi steps

\begin{align*} \int \left (\frac{x}{\sin ^{\frac{7}{2}}(e+f x)}+\frac{3}{5} x \sqrt{\sin (e+f x)}\right ) \, dx &=\frac{3}{5} \int x \sqrt{\sin (e+f x)} \, dx+\int \frac{x}{\sin ^{\frac{7}{2}}(e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{3}{5} \int \frac{x}{\sin ^{\frac{3}{2}}(e+f x)} \, dx+\frac{3}{5} \int x \sqrt{\sin (e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}\\ \end{align*}

Mathematica [A]  time = 0.585283, size = 58, normalized size = 0.7 \[ \frac{46 \sin (e+f x)-18 \sin (3 (e+f x))-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))}{30 f^2 \sin ^{\frac{5}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sin[e + f*x]^(7/2) + (3*x*Sqrt[Sin[e + f*x]])/5,x]

[Out]

(-21*f*x*Cos[e + f*x] + 9*f*x*Cos[3*(e + f*x)] + 46*Sin[e + f*x] - 18*Sin[3*(e + f*x)])/(30*f^2*Sin[e + f*x]^(
5/2))

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Maple [F]  time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,x}{5}\sqrt{\sin \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x)

[Out]

int(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\sin \left (f x + e\right )} + \frac{x}{\sin \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/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(3/5*x*sqrt(sin(f*x + e)) + x/sin(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/sin(f*x+e)^(7/2)+3/5*x*sin(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/sin(f*x+e)**(7/2)+3/5*x*sin(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{3}{5} \, x \sqrt{\sin \left (f x + e\right )} + \frac{x}{\sin \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/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(3/5*x*sqrt(sin(f*x + e)) + x/sin(f*x + e)^(7/2), x)