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]

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]])

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Rubi [A]  time = 0.0486463, antiderivative size = 55, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0961792, 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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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{7}{2}}}\,{d x} \end{align*}

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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Fricas [A]  time = 0.503384, size = 147, normalized size = 2.67 \begin{align*} \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}} \end{align*}

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

[Out]

Timed out

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

[Out]

Timed out