3.3.51 \(\int \frac {(c \sec (a+b x))^{5/2}}{\sqrt {d \csc (a+b x)}} \, dx\) [251]

Optimal. Leaf size=33 \[ \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2699} \begin {gather*} \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 58.52, size = 42, normalized size = 1.27

method	result	size
default	\(\frac {2 \cos \left (b x +a \right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{3 b \sqrt {\frac {d}{\sin \left (b x +a \right )}}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sec(b*x+a))^(5/2)/(d*csc(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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*sec(b*x+a))^(5/2)/(d*csc(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
time = 2.51, size = 58, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{3 \, b d \cos \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sec(b*x+a))^(5/2)/(d*csc(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-2/3*(c^2*cos(b*x + a)^2 - c^2)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))/(b*d*cos(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((c*sec(b*x+a))**(5/2)/(d*csc(b*x+a))**(1/2),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*sec(b*x+a))^(5/2)/(d*csc(b*x+a))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.86, size = 66, normalized size = 2.00 \begin {gather*} \frac {c^2\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (\cos \left (a+b\,x\right )-\cos \left (3\,a+3\,b\,x\right )\right )}{3\,b\,d\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/cos(a + b*x))^(5/2)/(d/sin(a + b*x))^(1/2),x)

[Out]

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

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