\(\int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx\) [83]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]

[Out]

-40/21*d^3*csc(b*x+a)/b/(d*tan(b*x+a))^(1/2)-20/21*d^3*csc(b*x+a)^3/b/(d*tan(b*x+a))^(1/2)-40/21*d^2*csc(b*x+a
)*(sin(a+1/4*Pi+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*t
an(b*x+a))^(1/2)/b+2/3*d*csc(b*x+a)^5*(d*tan(b*x+a))^(3/2)/b

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2673, 2679, 2681, 2653, 2720} \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 d^2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]

[In]

Int[Csc[a + b*x]^7*(d*Tan[a + b*x])^(5/2),x]

[Out]

(-40*d^3*Csc[a + b*x])/(21*b*Sqrt[d*Tan[a + b*x]]) - (20*d^3*Csc[a + b*x]^3)/(21*b*Sqrt[d*Tan[a + b*x]]) + (40
*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(21*b) + (2*d*Csc[
a + b*x]^5*(d*Tan[a + b*x])^(3/2))/(3*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 2673

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*(n - 1))), x] - Dist[b^2*((m + 2)/(a^2*(n - 1))), Int[(a*Sin[e
+ f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || (EqQ
[m, -1] && EqQ[n, 3/2])) && IntegersQ[2*m, 2*n]

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*} \text {integral}& = \frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (10 d^2\right ) \int \csc ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{7} \left (20 d^2\right ) \int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{21} \left (40 d^2\right ) \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {\left (40 d^2 \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}{21 \sqrt {\sin (a+b x)}} \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{21} \left (40 d^2 \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 {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {d^2 \csc (a+b x) \left ((1+10 \cos (2 (a+b x))-5 \cos (4 (a+b x))) \csc ^3(a+b x) \sec (a+b x) \sqrt {\sec ^2(a+b x)}+80 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sqrt {\tan (a+b x)}\right ) \sqrt {d \tan (a+b x)}}{21 b \sqrt {\sec ^2(a+b x)}} \]

[In]

Integrate[Csc[a + b*x]^7*(d*Tan[a + b*x])^(5/2),x]

[Out]

-1/21*(d^2*Csc[a + b*x]*((1 + 10*Cos[2*(a + b*x)] - 5*Cos[4*(a + b*x)])*Csc[a + b*x]^3*Sec[a + b*x]*Sqrt[Sec[a
+ b*x]^2] + 80*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Tan[a + b*x]])*Sqrt[d*
Tan[a + b*x]])/(b*Sqrt[Sec[a + b*x]^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(147)=294\).

Time = 151.87 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.26

method	result	size
default	\(\frac {\sin \left (b x +a \right ) \tan \left (b x +a \right ) \left (40 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{4}\left (b x +a \right )\right ) \sin \left (b x +a \right )+40 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )-40 \sin \left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-40 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-20 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+30 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-7 \sqrt {2}\right ) \sqrt {d \tan \left (b x +a \right )}\, d^{2} \sqrt {2}}{21 b \left (-1+\cos \left (b x +a \right )\right )^{3} \left (\cos \left (b x +a \right )+1\right )^{3}}\)	\(456\)

[In]

int(csc(b*x+a)^7*(d*tan(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/21/b*sin(b*x+a)*tan(b*x+a)*(40*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-
csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)^4*sin(b*x+a)+40*(1+csc(b*x
+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-c
ot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)^3*sin(b*x+a)-40*sin(b*x+a)*cos(b*x+a)^2*(1+csc(b*x+a)-cot(b*x+a))^(1/
2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/
2*2^(1/2))-40*sin(b*x+a)*cos(b*x+a)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)
+1+cot(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-20*2^(1/2)*cos(b*x+a)^4+30*cos(b*x
+a)^2*2^(1/2)-7*2^(1/2))*(d*tan(b*x+a))^(1/2)*d^2/(-1+cos(b*x+a))^3/(cos(b*x+a)+1)^3*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {2 \, {\left (20 \, {\left (d^{2} \cos \left (b x + a\right )^{5} - 2 \, d^{2} \cos \left (b x + a\right )^{3} + d^{2} \cos \left (b x + a\right )\right )} \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + 20 \, {\left (d^{2} \cos \left (b x + a\right )^{5} - 2 \, d^{2} \cos \left (b x + a\right )^{3} + d^{2} \cos \left (b x + a\right )\right )} \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - {\left (20 \, d^{2} \cos \left (b x + a\right )^{4} - 30 \, d^{2} \cos \left (b x + a\right )^{2} + 7 \, d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{21 \, {\left (b \cos \left (b x + a\right )^{5} - 2 \, b \cos \left (b x + a\right )^{3} + b \cos \left (b x + a\right )\right )}} \]

[In]

integrate(csc(b*x+a)^7*(d*tan(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/21*(20*(d^2*cos(b*x + a)^5 - 2*d^2*cos(b*x + a)^3 + d^2*cos(b*x + a))*sqrt(I*d)*elliptic_f(arcsin(cos(b*x +
a) + I*sin(b*x + a)), -1) + 20*(d^2*cos(b*x + a)^5 - 2*d^2*cos(b*x + a)^3 + d^2*cos(b*x + a))*sqrt(-I*d)*elli
ptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - (20*d^2*cos(b*x + a)^4 - 30*d^2*cos(b*x + a)^2 + 7*d^2)*sq
rt(d*sin(b*x + a)/cos(b*x + a)))/(b*cos(b*x + a)^5 - 2*b*cos(b*x + a)^3 + b*cos(b*x + a))

Sympy [F(-1)]

Timed out. \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+a)**7*(d*tan(b*x+a))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{7} \,d x } \]

[In]

integrate(csc(b*x+a)^7*(d*tan(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*tan(b*x + a))^(5/2)*csc(b*x + a)^7, x)

Giac [F]

\[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{7} \,d x } \]

[In]

integrate(csc(b*x+a)^7*(d*tan(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((d*tan(b*x + a))^(5/2)*csc(b*x + a)^7, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^7} \,d x \]

[In]

int((d*tan(a + b*x))^(5/2)/sin(a + b*x)^7,x)

[Out]

int((d*tan(a + b*x))^(5/2)/sin(a + b*x)^7, x)