3.255 \(\int \frac{(d \csc (a+b x))^{7/2}}{\sqrt{c \sec (a+b x)}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{4 c d^3 \sqrt{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{4 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}} \]

[Out]

(-4*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*(c*Sec[a + b*x])^(3/2)) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*(c*Sec[a +
b*x])^(3/2)) - (4*d^4*EllipticE[a - Pi/4 + b*x, 2])/(5*b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*
a + 2*b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.192731, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2625, 2630, 2572, 2639} \[ -\frac{4 c d^3 \sqrt{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{4 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(7/2)/Sqrt[c*Sec[a + b*x]],x]

[Out]

(-4*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*(c*Sec[a + b*x])^(3/2)) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*(c*Sec[a +
b*x])^(3/2)) - (4*d^4*EllipticE[a - Pi/4 + b*x, 2])/(5*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 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 \frac{(d \csc (a+b x))^{7/2}}{\sqrt{c \sec (a+b x)}} \, dx &=-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}+\frac{1}{5} \left (2 d^2\right ) \int \frac{(d \csc (a+b x))^{3/2}}{\sqrt{c \sec (a+b x)}} \, dx\\ &=-\frac{4 c d^3 \sqrt{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac{1}{5} \left (4 d^4\right ) \int \frac{1}{\sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}} \, dx\\ &=-\frac{4 c d^3 \sqrt{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac{\left (4 d^4\right ) \int \sqrt{c \cos (a+b x)} \sqrt{d \sin (a+b x)} \, dx}{5 \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{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac{\left (4 d^4\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{5 \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{d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac{4 d^4 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{5 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 = 1.04791, size = 104, normalized size = 0.81 \[ -\frac{2 d^2 \tan ^2(a+b x) (d \csc (a+b x))^{3/2} \left (\sin (2 (a+b x)) \sqrt [4]{-\cot ^2(a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4},\frac{1}{2},\csc ^2(a+b x)\right )-(\cos (2 (a+b x))-2) \cot ^3(a+b x)\right )}{5 b \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(7/2)/Sqrt[c*Sec[a + b*x]],x]

[Out]

(-2*d^2*(d*Csc[a + b*x])^(3/2)*(-((-2 + Cos[2*(a + b*x)])*Cot[a + b*x]^3) + (-Cot[a + b*x]^2)^(1/4)*Hypergeome
tric2F1[-1/2, 1/4, 1/2, Csc[a + b*x]^2]*Sin[2*(a + b*x)])*Tan[a + b*x]^2)/(5*b*Sqrt[c*Sec[a + b*x]])

________________________________________________________________________________________

Maple [B]  time = 0.207, size = 992, normalized size = 7.8 \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))^(7/2)/(c*sec(b*x+a))^(1/2),x)

[Out]

-1/5/b*2^(1/2)*(4*cos(b*x+a)^3*(-(-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)^3*(-(-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*cos(b*x+a)^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)*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)^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+co
s(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*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+co
s(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))-2*cos(b*x+a)^3*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)*Ellipti
cF((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+cos(b*x+a)^2*2^(1/2)+2*cos(b*x+a)*2^(1/2))*(d/s
in(b*x+a))^(7/2)*sin(b*x+a)/(c/cos(b*x+a))^(1/2)/cos(b*x+a)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{7}{2}}}{\sqrt{c \sec \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(7/2)/sqrt(c*sec(b*x + a)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \left (b x + a\right )} d^{3} \csc \left (b x + a\right )^{3}}{c \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))^(7/2)/(c*sec(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(7/2)/(c*sec(b*x+a))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{7}{2}}}{\sqrt{c \sec \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(7/2)/sqrt(c*sec(b*x + a)), x)