3.1.63 \(\int \csc ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx\) [63]

Optimal. Leaf size=105 \[ -\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {4 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{7 b} \]

[Out]

-4/7*d*csc(b*x+a)/b/(d*tan(b*x+a))^(1/2)-2/7*d*csc(b*x+a)^3/b/(d*tan(b*x+a))^(1/2)-4/7*csc(b*x+a)*(sin(a+1/4*P
i+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^(1/2)*(d*tan(b*x+a))^(1/
2)/b

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Rubi [A]
time = 0.10, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2679, 2681, 2653, 2720} \begin {gather*} -\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {4 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^5*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-4*d*Csc[a + b*x])/(7*b*Sqrt[d*Tan[a + b*x]]) - (2*d*Csc[a + b*x]^3)/(7*b*Sqrt[d*Tan[a + b*x]]) + (4*Csc[a +
b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(7*b)

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2679

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[b*(a*Sin[e +
f*x])^(m + 2)*((b*Tan[e + f*x])^(n - 1)/(a^2*f*(m + n + 1))), x] + Dist[(m + 2)/(a^2*(m + n + 1)), Int[(a*Sin
[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n + 1, 0]
&& IntegersQ[2*m, 2*n]

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[Cos[e + f*x]
^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

Rule 2720

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

Rubi steps

\begin {align*} \int \csc ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx &=-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {6}{7} \int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx\\ &=-\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {4}{7} \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx\\ &=-\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {\left (4 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{7 \sqrt {\sin (a+b x)}}\\ &=-\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {1}{7} \left (4 \csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {4 d \csc (a+b x)}{7 b \sqrt {d \tan (a+b x)}}-\frac {2 d \csc ^3(a+b x)}{7 b \sqrt {d \tan (a+b x)}}+\frac {4 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{7 b}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.51, size = 124, normalized size = 1.18 \begin {gather*} -\frac {2 d \cos (2 (a+b x)) \csc ^3(a+b x) \left ((-2+\cos (2 (a+b x))) \sec ^2(a+b x)^{3/2}-4 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right ) \tan ^{\frac {7}{2}}(a+b x)\right )}{7 b \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)} \left (-1+\tan ^2(a+b x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^5*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*d*Cos[2*(a + b*x)]*Csc[a + b*x]^3*((-2 + Cos[2*(a + b*x)])*(Sec[a + b*x]^2)^(3/2) - 4*(-1)^(1/4)*EllipticF
[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(7/2)))/(7*b*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b
*x]]*(-1 + Tan[a + b*x]^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(549\) vs. \(2(116)=232\).
time = 0.40, size = 550, normalized size = 5.24

method	result	size
default	\(-\frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (4 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+4 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (b x +a \right )\right )-4 \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \cos \left (b x +a \right )-4 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+3 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}\, \sqrt {2}}{7 b \sin \left (b x +a \right )^{8}}\)	\(550\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^5*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/7/b*(-1+cos(b*x+a))^2*(4*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((
cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*cos(b*x+a)^3*sin(b*x+a)*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))^(1/2)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((c
os(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*sin(b*x+a)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/
2*2^(1/2))*cos(b*x+a)^2-4*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((-1+cos(b*x+a))
/sin(b*x+a))^(1/2)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*s
in(b*x+a)*cos(b*x+a)-4*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b
*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*sin(b*x+a)*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)+3*cos(b*x+a)*2^(1/2))*(cos(b*x+a)+1)^2*(d*sin(b*x+a)/cos(b*x+a))^(1/2)/sin(b*x+a
)^8*2^(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(csc(b*x+a)^5*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a)^5, x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.11, size = 155, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (2 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 2 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {-i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - {\left (2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{7 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^5*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-2/7*(2*(cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*sqrt(I*d)*ellipticF(cos(b*x + a) + I*sin(b*x + a), -1) + 2*(co
s(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*sqrt(-I*d)*ellipticF(cos(b*x + a) - I*sin(b*x + a), -1) - (2*cos(b*x + a)
^3 - 3*cos(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)))/(b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d \tan {\left (a + b x \right )}} \csc ^{5}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**5*(d*tan(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(d*tan(a + b*x))*csc(a + b*x)**5, x)

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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(csc(b*x+a)^5*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a)^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{{\sin \left (a+b\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(a + b*x))^(1/2)/sin(a + b*x)^5,x)

[Out]

int((d*tan(a + b*x))^(1/2)/sin(a + b*x)^5, x)

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