3.94 \(\int \frac {\sin ^3(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx\)

Optimal. Leaf size=55 \[ \frac {\sin ^3(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{5 b \sqrt {\sin (2 a+2 b x)}} \]

[Out]

1/5*sin(b*x+a)^3/b/sin(2*b*x+2*a)^(5/2)+1/5*sin(b*x+a)/b/sin(2*b*x+2*a)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4296, 4292} \[ \frac {\sin ^3(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{5 b \sqrt {\sin (2 a+2 b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2),x]

[Out]

Sin[a + b*x]^3/(5*b*Sin[2*a + 2*b*x]^(5/2)) + Sin[a + b*x]/(5*b*Sqrt[Sin[2*a + 2*b*x]])

Rule 4292

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[((e*Sin[a +
b*x])^m*(g*Sin[c + d*x])^(p + 1))/(b*g*m), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 4296

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> -Simp[((e*Sin[a +
b*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^2*(m + 2*p + 2))/(4*g^2*(p + 1)), Int[(e*Sin[a
+ b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[
d/b, 2] &&  !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && Int
egersQ[2*m, 2*p]

Rubi steps

\begin {align*} \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^3(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {1}{5} \int \frac {\sin (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=\frac {\sin ^3(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{5 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 35, normalized size = 0.64 \[ \frac {\sqrt {\sin (2 (a+b x))} \sec (a+b x) \left (\sec ^2(a+b x)+4\right )}{40 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2),x]

[Out]

(Sec[a + b*x]*(4 + Sec[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/(40*b)

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fricas [A]  time = 0.49, size = 55, normalized size = 1.00 \[ \frac {4 \, \cos \left (b x + a\right )^{3} + \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{40 \, b \cos \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(7/2),x, algorithm="fricas")

[Out]

1/40*(4*cos(b*x + a)^3 + sqrt(2)*(4*cos(b*x + a)^2 + 1)*sqrt(cos(b*x + a)*sin(b*x + a)))/(b*cos(b*x + a)^3)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(7/2),x, algorithm="giac")

[Out]

Timed out

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (b x +a \right )}{\sin \left (2 b x +2 a \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^3/sin(2*b*x+2*a)^(7/2),x)

[Out]

int(sin(b*x+a)^3/sin(2*b*x+2*a)^(7/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(7/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x + a)^3/sin(2*b*x + 2*a)^(7/2), x)

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mupad [B]  time = 3.26, size = 88, normalized size = 1.60 \[ \frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}{5\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^3/sin(2*a + 2*b*x)^(7/2),x)

[Out]

(exp(a*1i + b*x*1i)*((exp(- a*2i - b*x*2i)*1i)/2 - (exp(a*2i + b*x*2i)*1i)/2)^(1/2)*(3*exp(a*2i + b*x*2i) + ex
p(a*4i + b*x*4i) + 1))/(5*b*(exp(a*2i + b*x*2i) + 1)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**3/sin(2*b*x+2*a)**(7/2),x)

[Out]

Timed out

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