Integrand size = 25, antiderivative size = 171 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{2 (a+b)^{7/2} f}-\frac {\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \]
-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)
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 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) \left ((a+b) \left (3 a^2+6 a b-2 b^2+\left (3 a^2+2 b^2\right ) \cos (2 (e+f x))\right ) \csc ^2(e+f x)-3 a (a-4 b) (a+2 b+a \cos (2 (e+f x))) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sec ^5(e+f x)}{24 a (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
-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))
Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4622, 373, 402, 25, 27, 402, 27, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x)^3 \left (a+b \sec (e+f x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4622 |
\(\displaystyle \frac {\int \frac {\sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 373 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a-4 b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {a \left (-10 b \sec ^2(e+f x)+3 a-2 b\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 a (a+b)}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {a \left (-10 b \sec ^2(e+f x)+3 a-2 b\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 a (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {-10 b \sec ^2(e+f x)+3 a-2 b}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\int -\frac {3 a (a-4 b)}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a (a+b)}}{3 (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 (a-4 b) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a+b}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 (a-4 b) \int \frac {1}{1-\frac {(a+b) \sec ^2(e+f x)}{b \sec ^2(e+f x)+a}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}}{a+b}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{(a+b)^{3/2}}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {5 b \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{f}\) |
(Sec[e + f*x]/(2*(a + b)*(1 - Sec[e + f*x]^2)*(a + b*Sec[e + f*x]^2)^(3/2) ) - ((5*b*Sec[e + f*x])/(3*(a + b)*(a + b*Sec[e + f*x]^2)^(3/2)) + ((3*(a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(a + b)^(3/2) + ((13*a - 2*b)*b*Sec[e + f*x])/(a*(a + b)*Sqrt[a + b*Sec[e + f *x]^2]))/(3*(a + b)))/(2*(a + b)))/f
3.2.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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]}, Si mp[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] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4])
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
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 ] \]
[-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)*co s(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*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 + ...
\[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^3\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]