3.3.33 \(\int (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)} \, dx\) [233]

Optimal. Leaf size=31 \[ -\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 31, 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 \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 31.08, size = 42, normalized size = 1.35

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

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

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Fricas [A]
time = 1.78, size = 36, normalized size = 1.16 \begin {gather*} -\frac {2 \, d \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))**(3/2)*(c*sec(b*x+a))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3433 deep

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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))^(3/2)*(c*sec(b*x+a))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.37, size = 36, normalized size = 1.16 \begin {gather*} -\frac {2\,d\,\cos \left (a+b\,x\right )\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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