\(\int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx\) [284]

Optimal result

Integrand size = 25, antiderivative size = 37 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2643} \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]

[In]

Int[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(9/2),x]

[Out]

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

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*} \text {integral}& = \frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 \cot (a+b x) (c \sin (a+b x))^{9/2}}{7 b c^2 (d \cos (a+b x))^{9/2}} \]

[In]

Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(9/2),x]

[Out]

(2*Cot[a + b*x]*(c*Sin[a + b*x])^(9/2))/(7*b*c^2*(d*Cos[a + b*x])^(9/2))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {2 \sqrt {c \sin \left (b x +a \right )}\, c^{2} \left (\tan ^{3}\left (b x +a \right )\right )}{7 b \,d^{4} \sqrt {d \cos \left (b x +a \right )}}\)	\(40\)

[In]

int((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/7/b*(c*sin(b*x+a))^(1/2)*c^2/d^4/(d*cos(b*x+a))^(1/2)*tan(b*x+a)^3

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=-\frac {2 \, {\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{7 \, b d^{5} \cos \left (b x + a\right )^{4}} \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(9/2),x, algorithm="fricas")

[Out]

-2/7*(c^2*cos(b*x + a)^2 - c^2)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sin(b*x + a)/(b*d^5*cos(b*x + a)^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((c*sin(b*x+a))**(5/2)/(d*cos(b*x+a))**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(9/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(5/2)/(d*cos(b*x + a))^(9/2), x)

Giac [F]

\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(9/2),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(5/2)/(d*cos(b*x + a))^(9/2), x)

Mupad [B] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.41 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2\,c^2\,\sqrt {c\,\sin \left (a+b\,x\right )}\,\left (3\,\sin \left (2\,a+2\,b\,x\right )-\sin \left (6\,a+6\,b\,x\right )\right )}{7\,b\,d^4\,\sqrt {d\,\cos \left (a+b\,x\right )}\,\left (15\,\cos \left (2\,a+2\,b\,x\right )+6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )+10\right )} \]

[In]

int((c*sin(a + b*x))^(5/2)/(d*cos(a + b*x))^(9/2),x)

[Out]

(2*c^2*(c*sin(a + b*x))^(1/2)*(3*sin(2*a + 2*b*x) - sin(6*a + 6*b*x)))/(7*b*d^4*(d*cos(a + b*x))^(1/2)*(15*cos
(2*a + 2*b*x) + 6*cos(4*a + 4*b*x) + cos(6*a + 6*b*x) + 10))