Integrand size = 25, antiderivative size = 197 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {(5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{16 a^{3/2} f}-\frac {(5 a-b) (a+b) \sqrt {a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{16 a f}-\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f} \]
-1/16*(5*a-b)*(a+b)^2*arctanh(cos(f*x+e)*a^(1/2)/(a+b-b*cos(f*x+e)^2)^(1/2 ))/a^(3/2)/f-1/24*(5*a-b)*(a+b-b*cos(f*x+e)^2)^(3/2)*cot(f*x+e)*csc(f*x+e) ^3/a/f-1/6*(a+b-b*cos(f*x+e)^2)^(5/2)*cot(f*x+e)*csc(f*x+e)^5/a/f-1/16*(5* a-b)*(a+b)*cot(f*x+e)*csc(f*x+e)*(a+b-b*cos(f*x+e)^2)^(1/2)/a/f
Time = 1.60 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-6 (5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cos (e+f x)}{\sqrt {2 a+b-b \cos (2 (e+f x))}}\right )-\sqrt {2} \sqrt {a} \sqrt {2 a+b-b \cos (2 (e+f x))} \csc ^2(e+f x) \left (\left (15 a^2+22 a b+3 b^2\right ) \cos (e+f x)+2 a \cot (e+f x) \csc (e+f x) \left (5 a+7 b+4 a \csc ^2(e+f x)\right )\right )}{96 a^{3/2} f} \]
(-6*(5*a - b)*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a + b - b*Cos[2*(e + f*x )]]*Csc[e + f*x]^2*((15*a^2 + 22*a*b + 3*b^2)*Cos[e + f*x] + 2*a*Cot[e + f *x]*Csc[e + f*x]*(5*a + 7*b + 4*a*Csc[e + f*x]^2)))/(96*a^(3/2)*f)
Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3665, 296, 292, 292, 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.
| \(\displaystyle \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin (e+f x)^2\right )^{3/2}}{\sin (e+f x)^7}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {\left (-b \cos ^2(e+f x)+a+b\right )^{3/2}}{\left (1-\cos ^2(e+f x)\right )^4}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle -\frac {\frac {(5 a-b) \int \frac {\left (-b \cos ^2(e+f x)+a+b\right )^{3/2}}{\left (1-\cos ^2(e+f x)\right )^3}d\cos (e+f x)}{6 a}+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a \left (1-\cos ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle -\frac {\frac {(5 a-b) \left (\frac {3}{4} (a+b) \int \frac {\sqrt {-b \cos ^2(e+f x)+a+b}}{\left (1-\cos ^2(e+f x)\right )^2}d\cos (e+f x)+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )}{6 a}+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a \left (1-\cos ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle -\frac {\frac {(5 a-b) \left (\frac {3}{4} (a+b) \left (\frac {1}{2} (a+b) \int \frac {1}{\left (1-\cos ^2(e+f x)\right ) \sqrt {-b \cos ^2(e+f x)+a+b}}d\cos (e+f x)+\frac {\cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )}{6 a}+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a \left (1-\cos ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {(5 a-b) \left (\frac {3}{4} (a+b) \left (\frac {1}{2} (a+b) \int \frac {1}{1-\frac {a \cos ^2(e+f x)}{-b \cos ^2(e+f x)+a+b}}d\frac {\cos (e+f x)}{\sqrt {-b \cos ^2(e+f x)+a+b}}+\frac {\cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )}{6 a}+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a \left (1-\cos ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {(5 a-b) \left (\frac {3}{4} (a+b) \left (\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{2 \sqrt {a}}+\frac {\cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )}{6 a}+\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a \left (1-\cos ^2(e+f x)\right )^3}}{f}\) |
-(((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(5/2))/(6*a*(1 - Cos[e + f*x]^ 2)^3) + ((5*a - b)*((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(4*(1 - Cos[e + f*x]^2)^2) + (3*(a + b)*(((a + b)*ArcTanh[(Sqrt[a]*Cos[e + f*x]) /Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*Sqrt[a]) + (Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*(1 - Cos[e + f*x]^2))))/4))/(6*a))/f)
3.2.37.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(177)=354\).
Time = 1.39 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.87
| method | result | size |
| default | \(-\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, \left (15 a^{4} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right )+27 a^{3} b \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right )+9 b^{2} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right ) a^{2}-3 b^{3} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right ) a +30 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {7}{2}}+44 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {5}{2}} b +6 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {3}{2}} b^{2}+20 \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {7}{2}}+28 \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {5}{2}} b +16 a^{\frac {7}{2}} \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\right )}{96 \sin \left (f x +e \right )^{6} a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(565\) |
-1/96*(cos(f*x+e)^2*(a+b*sin(f*x+e)^2))^(1/2)*(15*a^4*ln(((a-b)*cos(f*x+e) ^2+2*a^(1/2)*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)+a+b)/sin(f*x+e)^2) *sin(f*x+e)^6+27*a^3*b*ln(((a-b)*cos(f*x+e)^2+2*a^(1/2)*(-b*cos(f*x+e)^4+( a+b)*cos(f*x+e)^2)^(1/2)+a+b)/sin(f*x+e)^2)*sin(f*x+e)^6+9*b^2*ln(((a-b)*c os(f*x+e)^2+2*a^(1/2)*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)+a+b)/sin( f*x+e)^2)*sin(f*x+e)^6*a^2-3*b^3*ln(((a-b)*cos(f*x+e)^2+2*a^(1/2)*(-b*cos( f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)+a+b)/sin(f*x+e)^2)*sin(f*x+e)^6*a+30*si n(f*x+e)^4*(cos(f*x+e)^2*(a+b*sin(f*x+e)^2))^(1/2)*a^(7/2)+44*sin(f*x+e)^4 *(cos(f*x+e)^2*(a+b*sin(f*x+e)^2))^(1/2)*a^(5/2)*b+6*sin(f*x+e)^4*(cos(f*x +e)^2*(a+b*sin(f*x+e)^2))^(1/2)*a^(3/2)*b^2+20*sin(f*x+e)^2*(cos(f*x+e)^2* (a+b*sin(f*x+e)^2))^(1/2)*a^(7/2)+28*sin(f*x+e)^2*(cos(f*x+e)^2*(a+b*sin(f *x+e)^2))^(1/2)*a^(5/2)*b+16*a^(7/2)*(cos(f*x+e)^2*(a+b*sin(f*x+e)^2))^(1/ 2))/sin(f*x+e)^6/a^(5/2)/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
Time = 1.13 (sec) , antiderivative size = 752, normalized size of antiderivative = 3.82 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left ({\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + {\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} + a^{2} + 2 \, a b + b^{2}\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left ({\left (15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{192 \, {\left (a^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )}}, \frac {3 \, {\left ({\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (-\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{2 \, {\left (a b \cos \left (f x + e\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )}}\right ) + 2 \, {\left ({\left (15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{96 \, {\left (a^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )}}\right ] \]
[-1/192*(3*((5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^6 - 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - 5*a^3 - 9*a^2*b - 3*a*b^2 + b^3 + 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 4 *((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1 )) - 4*((15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e)^5 - 2*(20*a^3 + 29*a^2* b + 3*a*b^2)*cos(f*x + e)^3 + 3*(11*a^3 + 12*a^2*b + a*b^2)*cos(f*x + e))* sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^6 - 3*a^2*f*cos(f*x + e)^4 + 3*a^2*f*cos(f*x + e)^2 - a^2*f), 1/96*(3*((5*a^3 + 9*a^2*b + 3*a*b ^2 - b^3)*cos(f*x + e)^6 - 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e )^4 - 5*a^3 - 9*a^2*b - 3*a*b^2 + b^3 + 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3 )*cos(f*x + e)^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sq rt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*c os(f*x + e))) + 2*((15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e)^5 - 2*(20*a^ 3 + 29*a^2*b + 3*a*b^2)*cos(f*x + e)^3 + 3*(11*a^3 + 12*a^2*b + a*b^2)*cos (f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^6 - 3*a^2* f*cos(f*x + e)^4 + 3*a^2*f*cos(f*x + e)^2 - a^2*f)]
Timed out. \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{7} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1623 vs. \(2 (177) = 354\).
Time = 0.94 (sec) , antiderivative size = 1623, normalized size of antiderivative = 8.24 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
1/384*(sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*ta n(1/2*f*x + 1/2*e)^2 + a)*((a*tan(1/2*f*x + 1/2*e)^2 + (8*a^3 + 7*a^2*b)/a ^2)*tan(1/2*f*x + 1/2*e)^2 + (37*a^3 + 51*a^2*b + 6*a*b^2)/a^2) + 24*(5*a^ 3 + 9*a^2*b + 3*a*b^2 - b^3)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqr t(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a) - 12*(5*a^(7/2) + 9*a^(5/2)*b + 3* a^(3/2)*b^2 - sqrt(a)*b^3)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt (a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^2 + 2*(45*(sqrt(a)*tan(1/2*f *x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 + 132*(sqrt(a)*tan(1/2*f*x + 1/2 *e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*t an(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b + 108*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2 *f*x + 1/2*e)^2 + a))^5*a*b^2 + 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt( a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 63*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2 *f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) + 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 +...
Timed out. \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^7} \,d x \]