3.3.38 \(\int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx\) [238]

Optimal. Leaf size=104 \[ \frac {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b} \]

[Out]

-16/21*c*d^3*(d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2)/b-2/7*c*d*(d*csc(b*x+a))^(7/2)*(c*sec(b*x+a))^(1/2)/b+6
4/21*c*d^5*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{21 b}-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(64*c*d^5*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]]) - (16*c*d^3*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a +
b*x]])/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]])/(7*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))^{9/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b}+\frac {1}{7} \left (8 d^2\right ) \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx\\ &=-\frac {16 c d^3 (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b}+\frac {1}{21} \left (32 d^4\right ) \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2} \, dx\\ &=\frac {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 57, normalized size = 0.55 \begin {gather*} -\frac {2 c d^5 \left (-32+8 \csc ^2(a+b x)+3 \csc ^4(a+b x)\right ) \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(-2*c*d^5*(-32 + 8*Csc[a + b*x]^2 + 3*Csc[a + b*x]^4)*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]])

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Maple [A]
time = 60.66, size = 64, normalized size = 0.62

method	result	size
default	\(\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-56 \left (\cos ^{2}\left (b x +a \right )\right )+21\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {9}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{21 b}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/21/b*(32*cos(b*x+a)^4-56*cos(b*x+a)^2+21)*cos(b*x+a)*(d/sin(b*x+a))^(9/2)*(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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)

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Fricas [A]
time = 3.39, size = 85, normalized size = 0.82 \begin {gather*} -\frac {2 \, {\left (32 \, c d^{4} \cos \left (b x + a\right )^{4} - 56 \, c d^{4} \cos \left (b x + a\right )^{2} + 21 \, c d^{4}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{21 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

-2/21*(32*c*d^4*cos(b*x + a)^4 - 56*c*d^4*cos(b*x + a)^2 + 21*c*d^4)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))
/((b*cos(b*x + a)^2 - 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))**(9/2)*(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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)

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Mupad [B]
time = 2.17, size = 110, normalized size = 1.06 \begin {gather*} -\frac {16\,c\,d^4\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (41\,\sin \left (a+b\,x\right )-29\,\sin \left (3\,a+3\,b\,x\right )+12\,\sin \left (5\,a+5\,b\,x\right )-2\,\sin \left (7\,a+7\,b\,x\right )\right )}{21\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \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))^(9/2),x)

[Out]

-(16*c*d^4*(c/cos(a + b*x))^(1/2)*(d/sin(a + b*x))^(1/2)*(41*sin(a + b*x) - 29*sin(3*a + 3*b*x) + 12*sin(5*a +
5*b*x) - 2*sin(7*a + 7*b*x)))/(21*b*(15*cos(2*a + 2*b*x) - 6*cos(4*a + 4*b*x) + cos(6*a + 6*b*x) - 10))

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