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]

-2/5*x*cos(f*x+e)/f/sin(f*x+e)^(5/2)-4/15/f^2/sin(f*x+e)^(3/2)-6/5*x*cos(f*x+e)/f/sin(f*x+e)^(1/2)+12/5*sin(f*
x+e)^(1/2)/f^2

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Rubi [A]  time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.68, size = 58, normalized size = 0.70 \[ \frac {46 \sin (e+f x)-18 \sin (3 (e+f x))-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))}{30 f^2 \sin ^{\frac {5}{2}}(e+f x)} \]

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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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/sin(f*x+e)^(7/2)+3/5*x*sin(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 \[ \int \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sin \left (f x +e \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\sin }\left (f x +e \right )\right )}{5}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]

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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mupad [B]  time = 4.49, size = 253, normalized size = 3.05 \[ \left (\frac {12}{5\,f^2}+\frac {x\,6{}\mathrm {i}}{5\,f}\right )\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (\frac {x\,3{}\mathrm {i}}{5\,f}-\frac {32+f\,x\,66{}\mathrm {i}}{30\,f^2}\right )}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,12{}\mathrm {i}}{5\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}+\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,16{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*6i)/(5*f) + 12/(5*f^2))*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^(1/2) - (exp(e*2i + f*x*
2i)*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^(1/2)*((x*3i)/(5*f) - (f*x*66i + 32)/(30*f^2)))/
(exp(e*2i + f*x*2i) - 1)^2 - (x*exp(e*2i + f*x*2i)*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^(
1/2)*12i)/(5*f*(exp(e*2i + f*x*2i) - 1)) + (x*exp(e*2i + f*x*2i)*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*
x*1i)*1i)/2)^(1/2)*16i)/(5*f*(exp(e*2i + f*x*2i) - 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {5 x}{\sin ^{\frac {7}{2}}{\left (e + f x \right )}}\, dx + \int 3 x \sqrt {\sin {\left (e + f x \right )}}\, dx}{5} \]

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]

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

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