\(\int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\) [443]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f} \]

[Out]

3/32*arctan((b*sec(f*x+e))^(1/2)/b^(1/2))/b^(5/2)/f+3/32*arctanh((b*sec(f*x+e))^(1/2)/b^(1/2))/b^(5/2)/f-1/16*
cot(f*x+e)^2*(b*sec(f*x+e))^(1/2)/b^3/f-1/4*cot(f*x+e)^4*(b*sec(f*x+e))^(1/2)/b^3/f

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2702, 294, 296, 335, 218, 212, 209} \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f} \]

[In]

Int[Csc[e + f*x]^5/(b*Sec[e + f*x])^(5/2),x]

[Out]

(3*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(5/2)*f) + (3*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(5/2
)*f) - (Cot[e + f*x]^2*Sqrt[b*Sec[e + f*x]])/(16*b^3*f) - (Cot[e + f*x]^4*Sqrt[b*Sec[e + f*x]])/(4*b^3*f)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^{3/2}}{\left (-1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \sec (e+f x)\right )}{b^5 f} \\ & = -\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{8 b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )} \, dx,x,b \sec (e+f x)\right )}{32 b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {3 \text {Subst}\left (\int \frac {1}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{16 b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}+\frac {3 \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 b^2 f}+\frac {3 \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 b^2 f} \\ & = \frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {\left (6 \arctan \left (\sqrt {\sec (e+f x)}\right )-3 \log \left (1-\sqrt {\sec (e+f x)}\right )+3 \log \left (1+\sqrt {\sec (e+f x)}\right )-\frac {2 (5+3 \cos (2 (e+f x))) \csc ^4(e+f x)}{\sec ^{\frac {3}{2}}(e+f x)}\right ) \sqrt {\sec (e+f x)}}{64 b^2 f \sqrt {b \sec (e+f x)}} \]

[In]

Integrate[Csc[e + f*x]^5/(b*Sec[e + f*x])^(5/2),x]

[Out]

((6*ArcTan[Sqrt[Sec[e + f*x]]] - 3*Log[1 - Sqrt[Sec[e + f*x]]] + 3*Log[1 + Sqrt[Sec[e + f*x]]] - (2*(5 + 3*Cos
[2*(e + f*x)])*Csc[e + f*x]^4)/Sec[e + f*x]^(3/2))*Sqrt[Sec[e + f*x]])/(64*b^2*f*Sqrt[b*Sec[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(99)=198\).

Time = 0.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.83

method	result	size
default	\(-\frac {\left (12 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-3 \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )-3 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right ) \left (\sin ^{2}\left (f x +e \right )\right )+3 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \left (\sin ^{2}\left (f x +e \right )\right )+4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\right ) \left (\csc ^{4}\left (f x +e \right )\right )}{64 f \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b \sec \left (f x +e \right )}\, b^{2}}\)	\(348\)

[In]

int(csc(f*x+e)^5/(b*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/64/f*(12*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-3*sin(f*x+e)^2*cos(f*x+e)*arctan(1/2/(-cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2))-3*ln((2*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x+e)+
1)^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)+1))*sin(f*x+e)^2*cos(f*x+e)+3*arctan(1/2/(-cos(f*x+e)/(cos(f*x+e)+1)^2)^
(1/2))*sin(f*x+e)^2+3*ln((2*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
1/2)-cos(f*x+e)+1)/(cos(f*x+e)+1))*sin(f*x+e)^2+4*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))/(-cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)/(b*sec(f*x+e))^(1/2)/b^2*csc(f*x+e)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (99) = 198\).

Time = 0.39 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.72 \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\left [-\frac {6 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + 3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (3 \, \cos \left (f x + e\right )^{4} + \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (b^{3} f \cos \left (f x + e\right )^{4} - 2 \, b^{3} f \cos \left (f x + e\right )^{2} + b^{3} f\right )}}, -\frac {6 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - 3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (3 \, \cos \left (f x + e\right )^{4} + \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (b^{3} f \cos \left (f x + e\right )^{4} - 2 \, b^{3} f \cos \left (f x + e\right )^{2} + b^{3} f\right )}}\right ] \]

[In]

integrate(csc(f*x+e)^5/(b*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[-1/128*(6*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b)*arctan(1/2*sqrt(-b)*sqrt(b/cos(f*x + e))*(cos(f*x
+ e) + 1)/b) + 3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b)*log((b*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 -
cos(f*x + e))*sqrt(-b)*sqrt(b/cos(f*x + e)) - 6*b*cos(f*x + e) + b)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 8
*(3*cos(f*x + e)^4 + cos(f*x + e)^2)*sqrt(b/cos(f*x + e)))/(b^3*f*cos(f*x + e)^4 - 2*b^3*f*cos(f*x + e)^2 + b^
3*f), -1/128*(6*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(b)*arctan(1/2*sqrt(b/cos(f*x + e))*(cos(f*x + e)
- 1)/sqrt(b)) - 3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(b)*log((b*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 +
cos(f*x + e))*sqrt(b)*sqrt(b/cos(f*x + e)) + 6*b*cos(f*x + e) + b)/(cos(f*x + e)^2 - 2*cos(f*x + e) + 1)) + 8*
(3*cos(f*x + e)^4 + cos(f*x + e)^2)*sqrt(b/cos(f*x + e)))/(b^3*f*cos(f*x + e)^4 - 2*b^3*f*cos(f*x + e)^2 + b^3
*f)]

Sympy [F]

\[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(csc(f*x+e)**5/(b*sec(f*x+e))**(5/2),x)

[Out]

Integral(csc(e + f*x)**5/(b*sec(e + f*x))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=-\frac {b {\left (\frac {4 \, {\left (3 \, b^{2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {5}{2}}\right )}}{b^{6} - \frac {2 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {b^{6}}{\cos \left (f x + e\right )^{4}}} - \frac {6 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{b^{\frac {7}{2}}}\right )}}{64 \, f} \]

[In]

integrate(csc(f*x+e)^5/(b*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

-1/64*b*(4*(3*b^2*sqrt(b/cos(f*x + e)) + (b/cos(f*x + e))^(5/2))/(b^6 - 2*b^6/cos(f*x + e)^2 + b^6/cos(f*x + e
)^4) - 6*arctan(sqrt(b/cos(f*x + e))/sqrt(b))/b^(7/2) + 3*log(-(sqrt(b) - sqrt(b/cos(f*x + e)))/(sqrt(b) + sqr
t(b/cos(f*x + e))))/b^(7/2))/f

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=-\frac {\frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, \sqrt {b \cos \left (f x + e\right )} b^{3} \cos \left (f x + e\right )^{3} + \sqrt {b \cos \left (f x + e\right )} b^{3} \cos \left (f x + e\right )\right )}}{{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )}^{2} b^{2}}}{32 \, f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]

[In]

integrate(csc(f*x+e)^5/(b*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-1/32*(3*arctan(sqrt(b*cos(f*x + e))/sqrt(-b))/(sqrt(-b)*b^2) + 3*arctan(sqrt(b*cos(f*x + e))/sqrt(b))/b^(5/2)
+ 2*(3*sqrt(b*cos(f*x + e))*b^3*cos(f*x + e)^3 + sqrt(b*cos(f*x + e))*b^3*cos(f*x + e))/((b^2*cos(f*x + e)^2
- b^2)^2*b^2))/(f*sgn(cos(f*x + e)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int(1/(sin(e + f*x)^5*(b/cos(e + f*x))^(5/2)),x)

[Out]

int(1/(sin(e + f*x)^5*(b/cos(e + f*x))^(5/2)), x)