3.3.73 \(\int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{7/2}} \, dx\) [273]

Optimal. Leaf size=37 \[ \frac {2 (c \sin (a+b x))^{5/2}}{5 b c d (d \cos (a+b x))^{5/2}} \]

[Out]

2/5*(c*sin(b*x+a))^(5/2)/b/c/d/(d*cos(b*x+a))^(5/2)

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Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2643} \begin {gather*} \frac {2 (c \sin (a+b x))^{5/2}}{5 b c d (d \cos (a+b x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*(d*Cos[a + b*x])^(5/2))

Rule 2643

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[(a*Sin[e +
f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/(a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2,
0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 40, normalized size = 1.08 \begin {gather*} \frac {2 \cot (a+b x) (c \sin (a+b x))^{7/2}}{5 b c^2 (d \cos (a+b x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cot[a + b*x]*(c*Sin[a + b*x])^(7/2))/(5*b*c^2*(d*Cos[a + b*x])^(7/2))

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Maple [A]
time = 0.10, size = 38, normalized size = 1.03

method	result	size
default	\(\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}}}{5 b \left (d \cos \left (b x +a \right )\right )^{\frac {7}{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/5/b*sin(b*x+a)*cos(b*x+a)*(c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(7/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.40, size = 50, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (c \cos \left (b x + a\right )^{2} - c\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(c*cos(b*x + a)^2 - c)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))/(b*d^4*cos(b*x + a)^3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.51, size = 64, normalized size = 1.73 \begin {gather*} -\frac {2\,c\,\left (\cos \left (4\,a+4\,b\,x\right )-1\right )\,\sqrt {c\,\sin \left (a+b\,x\right )}}{5\,b\,d^3\,\sqrt {d\,\cos \left (a+b\,x\right )}\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c*(cos(4*a + 4*b*x) - 1)*(c*sin(a + b*x))^(1/2))/(5*b*d^3*(d*cos(a + b*x))^(1/2)*(4*cos(2*a + 2*b*x) + cos
(4*a + 4*b*x) + 3))

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