3.93 \(\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\)
Optimal. Leaf size=28 \[ \frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
[Out]
Sin[a + b*x]^3/(3*b*Sin[2*a + 2*b*x]^(3/2))
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Rubi [A] time = 0.0280523, antiderivative size = 28, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{4292} \[ \frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
[In]
Int[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]
[Out]
Sin[a + b*x]^3/(3*b*Sin[2*a + 2*b*x]^(3/2))
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{5}{2}}(2 a+2 b x)} \, dx &=\frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}\\ \end{align*}
Mathematica [A] time = 0.0542326, size = 27, normalized size = 0.96 \[ \frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 (a+b x))} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]
[Out]
Sin[a + b*x]^3/(3*b*Sin[2*(a + b*x)]^(3/2))
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Maple [C] time = 44.19, size = 727, normalized size = 26. \begin{align*} \text{result too large to display} \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)^(5/2),x)
[Out]
-1/48*(-tan(1/2*b*x+1/2*a)/(tan(1/2*b*x+1/2*a)^2-1))^(1/2)*(tan(1/2*b*x+1/2*a)^2-1)*(6*(tan(1/2*b*x+1/2*a)+1)^
(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticE((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2
^(1/2))*tan(1/2*b*x+1/2*a)^6-3*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*
a))^(1/2)*EllipticF((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))*tan(1/2*b*x+1/2*a)^6+18*(tan(1/2*b*x+1/2*a)+1)^(
1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticE((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^
(1/2))*tan(1/2*b*x+1/2*a)^4-9*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*a
))^(1/2)*EllipticF((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))*tan(1/2*b*x+1/2*a)^4+6*tan(1/2*b*x+1/2*a)^8+18*(t
an(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticE((tan(1/2*b*x+
1/2*a)+1)^(1/2),1/2*2^(1/2))*tan(1/2*b*x+1/2*a)^2-9*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/
2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticF((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))*tan(1/2*b*x+1/2*a)^2-2*tan(
1/2*b*x+1/2*a)^6+6*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)*(-tan(1/2*b*x+1/2*a))^(1/2)*El
lipticE((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))-3*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/
2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticF((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))+10*tan(1/2*b*x+1/2*a)^4-14*
tan(1/2*b*x+1/2*a)^2)/(tan(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)^2-1))^(1/2)/(tan(1/2*b*x+1/2*a)^2+1)^3/(tan(1/2*
b*x+1/2*a)^3-tan(1/2*b*x+1/2*a))^(1/2)/b
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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{5}{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)^(5/2),x, algorithm="maxima")
[Out]
integrate(sin(b*x + a)^3/sin(2*b*x + 2*a)^(5/2), x)
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Fricas [B] time = 0.493171, size = 132, normalized size = 4.71 \begin{align*} -\frac{\cos \left (b x + a\right )^{2} - \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right )}{12 \, b \cos \left (b x + a\right )^{2}} \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)^(5/2),x, algorithm="fricas")
[Out]
-1/12*(cos(b*x + a)^2 - sqrt(2)*sqrt(cos(b*x + a)*sin(b*x + a))*sin(b*x + a))/(b*cos(b*x + a)^2)
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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)**(5/2),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)^(5/2),x, algorithm="giac")
[Out]
Timed out