\(\int \frac {\csc ^5(e+f x)}{(a+b \sec ^2(e+f x))^{3/2}} \, dx\) [111]

Optimal result

Integrand size = 25, antiderivative size = 177 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {3 a (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 (a+b)^{7/2} f}-\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \]

[Out]

-3/8*a*(a-4*b)*arctanh(sec(f*x+e)*(a+b)^(1/2)/(a+b*sec(f*x+e)^2)^(1/2))/(a+b)^(7/2)/f-5/8*a*cot(f*x+e)*csc(f*x
+e)/(a+b)^2/f/(a+b*sec(f*x+e)^2)^(1/2)-1/4*cot(f*x+e)^3*csc(f*x+e)/(a+b)/f/(a+b*sec(f*x+e)^2)^(1/2)-1/8*(13*a-
2*b)*b*sec(f*x+e)/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4219, 481, 541, 12, 385, 213} \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {3 a (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 f (a+b)^{7/2}}-\frac {b (13 a-2 b) \sec (e+f x)}{8 f (a+b)^3 \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {5 a \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \sqrt {a+b \sec ^2(e+f x)}} \]

[In]

Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

(-3*a*(a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(7/2)*f) - (5*a*Cot
[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*
f*Sqrt[a + b*Sec[e + f*x]^2]) - ((13*a - 2*b)*b*Sec[e + f*x])/(8*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 4219

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Cos[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^
n)^p/x^(m + 1)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (Gt
Q[m, 0] || EqQ[n, 2] || EqQ[n, 4])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {-a-4 a x^2}{\left (-1+x^2\right )^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{4 (a+b) f} \\ & = -\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {-a (3 a-2 b)+10 a b x^2}{\left (-1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{8 (a+b)^2 f} \\ & = -\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\text {Subst}\left (\int -\frac {3 a^2 (a-4 b)}{\left (-1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a (a+b)^3 f} \\ & = -\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(3 a (a-4 b)) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 (a+b)^3 f} \\ & = -\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(3 a (a-4 b)) \text {Subst}\left (\int \frac {1}{-1-(-a-b) x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 (a+b)^3 f} \\ & = -\frac {3 a (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 (a+b)^{7/2} f}-\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) \left ((a+b)^2 \csc ^4(e+f x)-a (a-4 b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1-\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sec ^3(e+f x)}{8 (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

-1/8*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)^2*Csc[e + f*x]^4 - a*(a - 4*b)*Hypergeometric2F1[-1/2, 2, 1/2, 1
- (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^3)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3611\) vs. \(2(157)=314\).

Time = 1.23 (sec) , antiderivative size = 3612, normalized size of antiderivative = 20.41

method	result	size
default	\(\text {Expression too large to display}\)	\(3612\)

[In]

int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/64/f/(a+b)^(13/2)*(2*a*b*(a+b)^(7/2)+(a+b)^(7/2)*a^2+(a+b)^(7/2)*b^2+7*a^2*(1-cos(f*x+e))^2*(a+b)^(7/2)*csc(
f*x+e)^2-3*b^2*(1-cos(f*x+e))^2*(a+b)^(7/2)*csc(f*x+e)^2+(a+b)^(7/2)*a^2*(1-cos(f*x+e))^10*csc(f*x+e)^10+(a+b)
^(7/2)*b^2*(1-cos(f*x+e))^10*csc(f*x+e)^10-12*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+b*(1-cos(f*x+e))^2+(a
*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*
x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*sin(f*x+e)^2))*(a*(1-cos(f*x+e))^4*c
sc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^
2+a+b)^(1/2)*a^4*b*(1-cos(f*x+e))^4*csc(f*x+e)^4-108*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+b*(1-cos(f*x+e
))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1
-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*sin(f*x+e)^2))*(a*(1-cos(f*x+
e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(
f*x+e)^2+a+b)^(1/2)*a^3*b^2*(1-cos(f*x+e))^4*csc(f*x+e)^4-132*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+b*(1-
cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)
^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*sin(f*x+e)^2))*(a*(1
-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e
))^2*csc(f*x+e)^2+a+b)^(1/2)*a^2*b^3*(1-cos(f*x+e))^4*csc(f*x+e)^4-48*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))
^2+b*(1-cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*cs
c(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*sin(f*x+e)^2
))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-c
os(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*a*b^4*(1-cos(f*x+e))^4*csc(f*x+e)^4-12*ln((a*(1-cos(f*x+e))^2*csc(f*x+e)^
2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(
f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)-a+b)/(a+b)^(1/2))*(a*(1-cos(f*
x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*cs
c(f*x+e)^2+a+b)^(1/2)*a^4*b*(1-cos(f*x+e))^4*csc(f*x+e)^4-108*ln((a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x
+e))^2*csc(f*x+e)^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(
f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)-a+b)/(a+b)^(1/2))*(a*(1-cos(f*x+e))^4*csc(f*
x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b
)^(1/2)*a^3*b^2*(1-cos(f*x+e))^4*csc(f*x+e)^4-132*ln((a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f
*x+e)^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b
*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)-a+b)/(a+b)^(1/2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-
cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*a^2*
b^3*(1-cos(f*x+e))^4*csc(f*x+e)^4-48*ln((a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+(a*(1
-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e
))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)-a+b)/(a+b)^(1/2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4
*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*a*b^4*(1-cos(f*x+
e))^4*csc(f*x+e)^4+7*(a+b)^(7/2)*a^2*(1-cos(f*x+e))^8*csc(f*x+e)^8-3*(a+b)^(7/2)*b^2*(1-cos(f*x+e))^8*csc(f*x+
e)^8-8*a^2*(1-cos(f*x+e))^6*(a+b)^(7/2)*csc(f*x+e)^6-14*b^2*(1-cos(f*x+e))^6*(a+b)^(7/2)*csc(f*x+e)^6-8*a^2*(1
-cos(f*x+e))^4*(a+b)^(7/2)*csc(f*x+e)^4-14*b^2*(1-cos(f*x+e))^4*(a+b)^(7/2)*csc(f*x+e)^4+12*ln(2/(1-cos(f*x+e)
)^2*(-a*(1-cos(f*x+e))^2+b*(1-cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2
*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f
*x+e)^2+b*sin(f*x+e)^2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2
*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*a^5*(1-cos(f*x+e))^4*csc(f*x+e)^4+12*ln((a*(1-cos(f
*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(
f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)-a+b)/(a+b)
^(1/2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b
*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*a^5*(1-cos(f*x+e))^4*csc(f*x+e)^4+4*a*b*(1-cos(f*x+e))^2*(a+b)^(7/2)
*csc(f*x+e)^2+2*(1-cos(f*x+e))^10*a*b*(a+b)^(7/2)*csc(f*x+e)^10+4*(1-cos(f*x+e))^8*a*b*(a+b)^(7/2)*csc(f*x+e)^
8+98*(1-cos(f*x+e))^6*a*b*(a+b)^(7/2)*csc(f*x+e)^6+98*a*b*(1-cos(f*x+e))^4*(a+b)^(7/2)*csc(f*x+e)^4)*(a*(1-cos
(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2
*csc(f*x+e)^2+a+b)/(1-cos(f*x+e))^4*sin(f*x+e)^4/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^3/((a*(1-cos(f*x+e))^4*csc(
f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a
+b)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^2)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (157) = 314\).

Time = 0.42 (sec) , antiderivative size = 873, normalized size of antiderivative = 4.93 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (a^{3} - 4 \, a^{2} b\right )} \cos \left (f x + e\right )^{6} - {\left (2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b - 4 \, a b^{2} + {\left (a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a + b} \log \left (\frac {2 \, {\left (a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a + b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + 2 \, b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (3 \, {\left (a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{16 \, {\left ({\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f\right )}}, \frac {3 \, {\left ({\left (a^{3} - 4 \, a^{2} b\right )} \cos \left (f x + e\right )^{6} - {\left (2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b - 4 \, a b^{2} + {\left (a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {-a - b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a + b}\right ) + {\left (3 \, {\left (a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \, {\left ({\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f\right )}}\right ] \]

[In]

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(3*((a^3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^
3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 + 2*sqrt(a + b)*sqrt((a*cos(f*x + e
)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^3 - 3*a^2*b - 4*a*b^2)*cos(f*
x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x + e)^3 - (13*a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*
sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6
- (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8
*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f), 1/8*(3*((a^
3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^3 - 6*a^2*b +
8*a*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x
+ e)/(a + b)) + (3*(a^3 - 3*a^2*b - 4*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x +
e)^3 - (13*a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 + 4*a^4*
b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5
)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a
^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)]

Sympy [F]

\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(csc(f*x+e)**5/(a+b*sec(f*x+e)**2)**(3/2),x)

[Out]

Integral(csc(e + f*x)**5/(a + b*sec(e + f*x)**2)**(3/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^5\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]

[In]

int(1/(sin(e + f*x)^5*(a + b/cos(e + f*x)^2)^(3/2)),x)

[Out]

int(1/(sin(e + f*x)^5*(a + b/cos(e + f*x)^2)^(3/2)), x)