\(\int \frac {\csc ^3(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [123]

Optimal result

Integrand size = 25, antiderivative size = 171 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{2 (a+b)^{7/2} f}-\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \]

[Out]

-1/2*(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-1/2*cot(f*x+e)*csc(f*x+e)/
(a+b)/f/(a+b*sec(f*x+e)^2)^(3/2)-5/6*b*sec(f*x+e)/(a+b)^2/f/(a+b*sec(f*x+e)^2)^(3/2)-1/6*(13*a-2*b)*b*sec(f*x+
e)/a/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 171, 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, 482, 541, 12, 385, 213} \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{2 f (a+b)^{7/2}}-\frac {b (13 a-2 b) \sec (e+f x)}{6 a f (a+b)^3 \sqrt {a+b \sec ^2(e+f x)}}-\frac {5 b \sec (e+f x)}{6 f (a+b)^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]

[In]

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

[Out]

-1/2*((a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/((a + b)^(7/2)*f) - (Cot[e + f
*x]*Csc[e + f*x])/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (5*b*Sec[e + f*x])/(6*(a + b)^2*f*(a + b*Sec[e
+ f*x]^2)^(3/2)) - ((13*a - 2*b)*b*Sec[e + f*x])/(6*a*(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 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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^2}{\left (-1+x^2\right )^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {a-4 b x^2}{\left (-1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{2 (a+b) f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\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 )}{6 a (a+b)^2 f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (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 )}{6 a^2 (a+b)^3 f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(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 )}{2 (a+b)^3 f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(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 )}{2 (a+b)^3 f} \\ & = -\frac {(a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{2 (a+b)^{7/2} f}-\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (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 = 1.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) \left ((a+b) \left (3 a^2+6 a b-2 b^2+\left (3 a^2+2 b^2\right ) \cos (2 (e+f x))\right ) \csc ^2(e+f x)-3 a (a-4 b) (a+2 b+a \cos (2 (e+f x))) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sec ^5(e+f x)}{24 a (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]

[In]

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

[Out]

-1/24*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)*(3*a^2 + 6*a*b - 2*b^2 + (3*a^2 + 2*b^2)*Cos[2*(e + f*x)])*Csc[
e + f*x]^2 - 3*a*(a - 4*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (a*Sin[e + f*x]^
2)/(a + b)])*Sec[e + f*x]^5)/(a*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(8688\) vs. \(2(151)=302\).

Time = 5.54 (sec) , antiderivative size = 8689, normalized size of antiderivative = 50.81

method	result	size
default	\(\text {Expression too large to display}\)	\(8689\)

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (151) = 302\).

Time = 0.43 (sec) , antiderivative size = 941, normalized size of antiderivative = 5.50 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (a^{4} - 4 \, a^{3} b\right )} \cos \left (f x + e\right )^{6} - {\left (a^{4} - 6 \, a^{3} b + 8 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} - a^{2} b^{2} + 4 \, a b^{3} - {\left (2 \, a^{3} b - 9 \, a^{2} b^{2} + 4 \, a b^{3}\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^{4} - 3 \, a^{3} b - 4 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (9 \, a^{3} b + 4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (13 \, a^{2} b^{2} + 11 \, a b^{3} - 2 \, b^{4}\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}}}}{12 \, {\left ({\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right )} f\right )}}, \frac {3 \, {\left ({\left (a^{4} - 4 \, a^{3} b\right )} \cos \left (f x + e\right )^{6} - {\left (a^{4} - 6 \, a^{3} b + 8 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} - a^{2} b^{2} + 4 \, a b^{3} - {\left (2 \, a^{3} b - 9 \, a^{2} b^{2} + 4 \, a b^{3}\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^{4} - 3 \, a^{3} b - 4 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (9 \, a^{3} b + 4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (13 \, a^{2} b^{2} + 11 \, a b^{3} - 2 \, b^{4}\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}}}}{6 \, {\left ({\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right )} f\right )}}\right ] \]

[In]

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

[Out]

[-1/12*(3*((a^4 - 4*a^3*b)*cos(f*x + e)^6 - (a^4 - 6*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 4*a*b^3 - (
2*a^3*b - 9*a^2*b^2 + 4*a*b^3)*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^4 - 3*a^3*b - 4*a^2*b
^2)*cos(f*x + e)^5 + 2*(9*a^3*b + 4*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^3 + (13*a^2*b^2 + 11*a*b^3 - 2*b^4)*
cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*
f*cos(f*x + e)^6 - (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b
+ 7*a^5*b^2 + 8*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4
+ 4*a^2*b^5 + a*b^6)*f), 1/6*(3*((a^4 - 4*a^3*b)*cos(f*x + e)^6 - (a^4 - 6*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 -
a^2*b^2 + 4*a*b^3 - (2*a^3*b - 9*a^2*b^2 + 4*a*b^3)*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^4 - 3*a^3*b - 4*a^2*b^2)*cos(f*x + e)^5 + 2*
(9*a^3*b + 4*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^3 + (13*a^2*b^2 + 11*a*b^3 - 2*b^4)*cos(f*x + e))*sqrt((a*c
os(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (a^7
+ 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b + 7*a^5*b^2 + 8*a^4*b^
3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b^6)*f)
]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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