3.3.62 \(\int \frac {(d \csc (a+b x))^{11/2}}{(c \sec (a+b x))^{3/2}} \, dx\) [262]

Optimal. Leaf size=110 \[ \frac {8 d^5 \sqrt {d \csc (a+b x)}}{45 b c \sqrt {c \sec (a+b x)}}+\frac {2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt {c \sec (a+b x)}} \]

[Out]

2/45*d^3*(d*csc(b*x+a))^(5/2)/b/c/(c*sec(b*x+a))^(1/2)-2/9*d*(d*csc(b*x+a))^(9/2)/b/c/(c*sec(b*x+a))^(1/2)+8/4
5*d^5*(d*csc(b*x+a))^(1/2)/b/c/(c*sec(b*x+a))^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2703, 2705, 2699} \begin {gather*} \frac {8 d^5 \sqrt {d \csc (a+b x)}}{45 b c \sqrt {c \sec (a+b x)}}+\frac {2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt {c \sec (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(11/2)/(c*Sec[a + b*x])^(3/2),x]

[Out]

(8*d^5*Sqrt[d*Csc[a + b*x]])/(45*b*c*Sqrt[c*Sec[a + b*x]]) + (2*d^3*(d*Csc[a + b*x])^(5/2))/(45*b*c*Sqrt[c*Sec
[a + b*x]]) - (2*d*(d*Csc[a + b*x])^(9/2))/(9*b*c*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]

Rule 2703

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

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 \frac {(d \csc (a+b x))^{11/2}}{(c \sec (a+b x))^{3/2}} \, dx &=-\frac {2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt {c \sec (a+b x)}}-\frac {d^2 \int (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)} \, dx}{9 c^2}\\ &=\frac {2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt {c \sec (a+b x)}}-\frac {\left (4 d^4\right ) \int (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)} \, dx}{45 c^2}\\ &=\frac {8 d^5 \sqrt {d \csc (a+b x)}}{45 b c \sqrt {c \sec (a+b x)}}+\frac {2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt {c \sec (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 57, normalized size = 0.52 \begin {gather*} \frac {2 d^3 (-7+2 \cos (2 (a+b x))) \cot ^2(a+b x) (d \csc (a+b x))^{5/2}}{45 b c \sqrt {c \sec (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(11/2)/(c*Sec[a + b*x])^(3/2),x]

[Out]

(2*d^3*(-7 + 2*Cos[2*(a + b*x)])*Cot[a + b*x]^2*(d*Csc[a + b*x])^(5/2))/(45*b*c*Sqrt[c*Sec[a + b*x]])

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Maple [A]
time = 34.36, size = 54, normalized size = 0.49

method	result	size
default	\(\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-9\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {11}{2}} \cos \left (b x +a \right ) \sin \left (b x +a \right )}{45 b \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(11/2)/(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/45/b*(4*cos(b*x+a)^2-9)*(d/sin(b*x+a))^(11/2)*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 \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))^(11/2)/(c*sec(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(11/2)/(c*sec(b*x + a))^(3/2), x)

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Fricas [A]
time = 4.24, size = 88, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (4 \, d^{5} \cos \left (b x + a\right )^{5} - 9 \, d^{5} \cos \left (b x + a\right )^{3}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{45 \, {\left (b c^{2} \cos \left (b x + a\right )^{4} - 2 \, b c^{2} \cos \left (b x + a\right )^{2} + b c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(11/2)/(c*sec(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

2/45*(4*d^5*cos(b*x + a)^5 - 9*d^5*cos(b*x + a)^3)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))/(b*c^2*cos(b*x +
a)^4 - 2*b*c^2*cos(b*x + a)^2 + b*c^2)

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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))**(11/2)/(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))^(11/2)/(c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(11/2)/(c*sec(b*x + a))^(3/2), x)

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Mupad [B]
time = 3.69, size = 125, normalized size = 1.14 \begin {gather*} \frac {8\,d^5\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (9\,\cos \left (2\,a+2\,b\,x\right )+14\,\cos \left (4\,a+4\,b\,x\right )-9\,\cos \left (6\,a+6\,b\,x\right )+\cos \left (8\,a+8\,b\,x\right )-15\right )}{45\,b\,c\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\left (28\,\cos \left (4\,a+4\,b\,x\right )-56\,\cos \left (2\,a+2\,b\,x\right )-8\,\cos \left (6\,a+6\,b\,x\right )+\cos \left (8\,a+8\,b\,x\right )+35\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d/sin(a + b*x))^(11/2)/(c/cos(a + b*x))^(3/2),x)

[Out]

(8*d^5*(d/sin(a + b*x))^(1/2)*(9*cos(2*a + 2*b*x) + 14*cos(4*a + 4*b*x) - 9*cos(6*a + 6*b*x) + cos(8*a + 8*b*x
) - 15))/(45*b*c*(c/cos(a + b*x))^(1/2)*(28*cos(4*a + 4*b*x) - 56*cos(2*a + 2*b*x) - 8*cos(6*a + 6*b*x) + cos(
8*a + 8*b*x) + 35))

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