3.241 \(\int (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=125 \[ -\frac{4 c^2 d^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{4 c d^3 \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{2 c d \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]

[Out]

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

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Rubi [A]  time = 0.204733, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2625, 2626, 2630, 2572, 2639} \[ -\frac{4 c^2 d^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{4 c d^3 \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{2 c d \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2625

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*(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]

Rule 2626

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] + Dist[(b^2*(m + n - 2))/(n - 1), Int[(a*Csc[e + f
*x])^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]

Rule 2630

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

Rule 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

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

Mathematica [C]  time = 0.511562, size = 99, normalized size = 0.79 \[ -\frac{2 c d \tan ^2(a+b x) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} \left (2 \cos ^2(a+b x) \sqrt [4]{-\cot ^2(a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4},\frac{1}{2},\csc ^2(a+b x)\right )+\cos (2 (a+b x)) \cot ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*d*Sqrt[d*Csc[a + b*x]]*(Cos[2*(a + b*x)]*Cot[a + b*x]^2 + 2*Cos[a + b*x]^2*(-Cot[a + b*x]^2)^(1/4)*Hyper
geometric2F1[-1/2, 1/4, 1/2, Csc[a + b*x]^2])*Sqrt[c*Sec[a + b*x]]*Tan[a + b*x]^2)/b

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Maple [B]  time = 0.187, size = 503, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*2^(1/2)*(4*cos(b*x+a)*(-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a
))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticE((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/
2))-2*cos(b*x+a)*(-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*
((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+4*(-(
-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin
(b*x+a))^(1/2)*EllipticE((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-2*(-(-1+cos(b*x+a)-sin(b*
x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*Ellip
ticF((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-2*cos(b*x+a)*2^(1/2)+2^(1/2))*cos(b*x+a)*(d/s
in(b*x+a))^(3/2)*(c/cos(b*x+a))^(3/2)*sin(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \left (b x + a\right )} c d \csc \left (b x + a\right ) \sec \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))*c*d*csc(b*x + a)*sec(b*x + a), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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