3.3.40 \(\int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx\) [240]

Optimal. Leaf size=69 \[ \frac {8 c d^3 \sqrt {c \sec (a+b x)}}{3 b \sqrt {d \csc (a+b x)}}-\frac {2 c d (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{3 b} \]

[Out]

-2/3*c*d*(d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2)/b+8/3*c*d^3*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2705, 2699} \begin {gather*} \frac {8 c d^3 \sqrt {c \sec (a+b x)}}{3 b \sqrt {d \csc (a+b x)}}-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

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

Rule 2699

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]

Rule 2705

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

Rubi steps

\begin {align*} \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac {2 c d (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{3 b}+\frac {1}{3} \left (4 d^2\right ) \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2} \, dx\\ &=\frac {8 c d^3 \sqrt {c \sec (a+b x)}}{3 b \sqrt {d \csc (a+b x)}}-\frac {2 c d (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 45, normalized size = 0.65 \begin {gather*} -\frac {2 c d^3 \left (-4+\csc ^2(a+b x)\right ) \sqrt {c \sec (a+b x)}}{3 b \sqrt {d \csc (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(-2*c*d^3*(-4 + Csc[a + b*x]^2)*Sqrt[c*Sec[a + b*x]])/(3*b*Sqrt[d*Csc[a + b*x]])

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Maple [A]
time = 58.42, size = 54, normalized size = 0.78

method	result	size
default	\(-\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-3\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{3 b}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(5/2)*(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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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((d*csc(b*x+a))^(5/2)*(c*sec(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2), x)

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Fricas [A]
time = 2.27, size = 58, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (4 \, c d^{2} \cos \left (b x + a\right )^{2} - 3 \, c d^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{3 \, b \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(5/2)*(c*sec(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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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((d*csc(b*x+a))**(5/2)*(c*sec(b*x+a))**(3/2),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((d*csc(b*x+a))^(5/2)*(c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2), x)

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Mupad [B]
time = 0.76, size = 61, normalized size = 0.88 \begin {gather*} \frac {2\,c\,d^2\,\left (2\,\sin \left (a+b\,x\right )-\sin \left (3\,a+3\,b\,x\right )\right )\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{3\,b\,{\sin \left (a+b\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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