3.264 \(\int \frac {(d \csc (a+b x))^{7/2}}{(c \sec (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{5/2}} \]

[Out]

-2/5*c*d*(d*csc(b*x+a))^(5/2)/b/(c*sec(b*x+a))^(5/2)

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Rubi [A]  time = 0.05, antiderivative size = 33, 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 = {2619} \[ -\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*(c*Sec[a + b*x])^(5/2))

Rule 2619

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 45, normalized size = 1.36 \[ -\frac {2 d^3 \cot ^2(a+b x) \sqrt {d \csc (a+b x)}}{5 b c \sqrt {c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*Cot[a + b*x]^2*Sqrt[d*Csc[a + b*x]])/(5*b*c*Sqrt[c*Sec[a + b*x]])

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fricas [B]  time = 0.86, size = 59, normalized size = 1.79 \[ \frac {2 \, d^{3} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{3}}{5 \, {\left (b c^{2} \cos \left (b x + a\right )^{2} - b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.10, size = 42, normalized size = 1.27 \[ -\frac {2 \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {7}{2}} \cos \left (b x +a \right ) \sin \left (b x +a \right )}{5 b \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.29, size = 70, normalized size = 2.12 \[ \frac {2\,d^3\,\left (\cos \left (4\,a+4\,b\,x\right )-1\right )\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{5\,b\,c\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\left (\cos \left (4\,a+4\,b\,x\right )-4\,\cos \left (2\,a+2\,b\,x\right )+3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*d^3*(cos(4*a + 4*b*x) - 1)*(d/sin(a + b*x))^(1/2))/(5*b*c*(c/cos(a + b*x))^(1/2)*(cos(4*a + 4*b*x) - 4*cos(
2*a + 2*b*x) + 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((d*csc(b*x+a))**(7/2)/(c*sec(b*x+a))**(3/2),x)

[Out]

Timed out

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