Integrand size = 15, antiderivative size = 109 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^3}+\frac {(a+5 b) \text {arctanh}(\sin (x))}{2 (a+b)^3}-\frac {(a-b) b \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )} \]
1/2*b^(3/2)*(5*a+b)*arctan(sin(x)*b^(1/2)/a^(1/2))/a^(3/2)/(a+b)^3+1/2*(a+ 5*b)*arctanh(sin(x))/(a+b)^3-1/2*(a-b)*b*sin(x)/a/(a+b)^2/(a+b*sin(x)^2)+1 /2*sec(x)*tan(x)/(a+b)/(a+b*sin(x)^2)
Time = 1.00 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {-\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{a^{3/2}}+\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{a^{3/2}}-2 (a+5 b) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+2 (a+5 b) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {4 b^2 (a+b) \sin (x)}{a (2 a+b-b \cos (2 x))}}{4 (a+b)^3} \]
(-((b^(3/2)*(5*a + b)*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]])/a^(3/2)) + (b^(3/2 )*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/a^(3/2) - 2*(a + 5*b)*Log[Co s[x/2] - Sin[x/2]] + 2*(a + 5*b)*Log[Cos[x/2] + Sin[x/2]] + (a + b)/(Cos[x /2] - Sin[x/2])^2 - (a + b)/(Cos[x/2] + Sin[x/2])^2 + (4*b^2*(a + b)*Sin[x ])/(a*(2*a + b - b*Cos[2*x])))/(4*(a + b)^3)
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3669, 316, 402, 27, 397, 218, 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.
| \(\displaystyle \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (x)^3 \left (a+b \sin (x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (a+b \sin ^2(x)\right )^2}d\sin (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {3 b \sin ^2(x)+a+2 b}{\left (1-\sin ^2(x)\right ) \left (b \sin ^2(x)+a\right )^2}d\sin (x)}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (a^2+4 b a+b^2+(a-b) b \sin ^2(x)\right )}{\left (1-\sin ^2(x)\right ) \left (b \sin ^2(x)+a\right )}d\sin (x)}{2 a (a+b)}-\frac {b (a-b) \sin (x)}{a (a+b) \left (a+b \sin ^2(x)\right )}}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a^2+4 b a+b^2+(a-b) b \sin ^2(x)}{\left (1-\sin ^2(x)\right ) \left (b \sin ^2(x)+a\right )}d\sin (x)}{a (a+b)}-\frac {b (a-b) \sin (x)}{a (a+b) \left (a+b \sin ^2(x)\right )}}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {b^2 (5 a+b) \int \frac {1}{b \sin ^2(x)+a}d\sin (x)}{a+b}+\frac {a (a+5 b) \int \frac {1}{1-\sin ^2(x)}d\sin (x)}{a+b}}{a (a+b)}-\frac {b (a-b) \sin (x)}{a (a+b) \left (a+b \sin ^2(x)\right )}}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {a (a+5 b) \int \frac {1}{1-\sin ^2(x)}d\sin (x)}{a+b}+\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{a (a+b)}-\frac {b (a-b) \sin (x)}{a (a+b) \left (a+b \sin ^2(x)\right )}}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {a (a+5 b) \text {arctanh}(\sin (x))}{a+b}}{a (a+b)}-\frac {b (a-b) \sin (x)}{a (a+b) \left (a+b \sin ^2(x)\right )}}{2 (a+b)}+\frac {\sin (x)}{2 (a+b) \left (1-\sin ^2(x)\right ) \left (a+b \sin ^2(x)\right )}\) |
Sin[x]/(2*(a + b)*(1 - Sin[x]^2)*(a + b*Sin[x]^2)) + (((b^(3/2)*(5*a + b)* ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + (a*(a + 5*b)*ArcTanh [Sin[x]])/(a + b))/(a*(a + b)) - ((a - b)*b*Sin[x])/(a*(a + b)*(a + b*Sin[ x]^2)))/(2*(a + b))
3.4.22.3.1 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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((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[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, 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[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 1.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {b^{2} \left (\frac {\left (a +b \right ) \sin \left (x \right )}{2 a \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}+\frac {\left (5 a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (x \right )-1\right )}+\frac {\left (-a -5 b \right ) \ln \left (\sin \left (x \right )-1\right )}{4 \left (a +b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (1+\sin \left (x \right )\right )}+\frac {\left (a +5 b \right ) \ln \left (1+\sin \left (x \right )\right )}{4 \left (a +b \right )^{3}}\) | \(119\) |
| risch | \(-\frac {i \left (a b \,{\mathrm e}^{7 i x}-b^{2} {\mathrm e}^{7 i x}-4 a^{2} {\mathrm e}^{5 i x}-3 a b \,{\mathrm e}^{5 i x}-b^{2} {\mathrm e}^{5 i x}+4 \,{\mathrm e}^{3 i x} a^{2}+3 a b \,{\mathrm e}^{3 i x}+b^{2} {\mathrm e}^{3 i x}-{\mathrm e}^{i x} a b +{\mathrm e}^{i x} b^{2}\right )}{\left (a +b \right )^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} a \left (b \,{\mathrm e}^{4 i x}-4 a \,{\mathrm e}^{2 i x}-2 b \,{\mathrm e}^{2 i x}+b \right )}+\frac {\ln \left ({\mathrm e}^{i x}+i\right ) a}{2 a^{3}+6 a^{2} b +6 a \,b^{2}+2 b^{3}}+\frac {5 \ln \left ({\mathrm e}^{i x}+i\right ) b}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{i x}-i\right ) a}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {5 \ln \left ({\mathrm e}^{i x}-i\right ) b}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{3}}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a^{2} \left (a +b \right )^{3}}-\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{3}}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a^{2} \left (a +b \right )^{3}}\) | \(447\) |
1/(a+b)^3*b^2*(1/2*(a+b)/a*sin(x)/(a+b*sin(x)^2)+1/2*(5*a+b)/a/(a*b)^(1/2) *arctan(b*sin(x)/(a*b)^(1/2)))-1/4/(a+b)^2/(sin(x)-1)+1/4/(a+b)^3*(-a-5*b) *ln(sin(x)-1)-1/4/(a+b)^2/(1+sin(x))+1/4*(a+5*b)/(a+b)^3*ln(1+sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (93) = 186\).
Time = 0.41 (sec) , antiderivative size = 560, normalized size of antiderivative = 5.14 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\left [\frac {{\left ({\left (5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} - {\left (5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) - {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )}}, \frac {2 \, {\left ({\left (5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} - {\left (5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \left (x\right )\right ) + {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) - {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )}}\right ] \]
[1/4*(((5*a*b^2 + b^3)*cos(x)^4 - (5*a^2*b + 6*a*b^2 + b^3)*cos(x)^2)*sqrt (-b/a)*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*l og(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*c os(x)^2)*log(-sin(x) + 1) - 2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - b^3)*cos(x )^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4 - (a^5 + 4* a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2), 1/4*(2*((5*a*b^2 + b^3)* cos(x)^4 - (5*a^2*b + 6*a*b^2 + b^3)*cos(x)^2)*sqrt(b/a)*arctan(sqrt(b/a)* sin(x)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2 )*log(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2 )*cos(x)^2)*log(-sin(x) + 1) - 2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - b^3)*co s(x)^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2)]
\[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\int \frac {\sec ^{3}{\left (x \right )}}{\left (a + b \sin ^{2}{\left (x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (93) = 186\).
Time = 0.48 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.02 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) - 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} - \frac {{\left (a b - b^{2}\right )} \sin \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sin \left (x\right )}{2 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \sin \left (x\right )^{4} - a^{4} - 2 \, a^{3} b - a^{2} b^{2} + {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} \sin \left (x\right )^{2}\right )}} \]
1/4*(a + 5*b)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(a + 5 *b)*log(sin(x) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/2*(5*a*b^2 + b^3)* arctan(b*sin(x)/sqrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)) - 1/2*((a*b - b^2)*sin(x)^3 + (a^2 + b^2)*sin(x))/((a^3*b + 2*a^2*b^2 + a *b^3)*sin(x)^4 - a^4 - 2*a^3*b - a^2*b^2 + (a^4 + a^3*b - a^2*b^2 - a*b^3) *sin(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (93) = 186\).
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.78 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (a + 5 \, b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} - \frac {a b \sin \left (x\right )^{3} - b^{2} \sin \left (x\right )^{3} + a^{2} \sin \left (x\right ) + b^{2} \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{4} + a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} - a\right )} {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )}} \]
1/4*(a + 5*b)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(a + 5 *b)*log(-sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/2*(5*a*b^2 + b^3) *arctan(b*sin(x)/sqrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b) ) - 1/2*(a*b*sin(x)^3 - b^2*sin(x)^3 + a^2*sin(x) + b^2*sin(x))/((b*sin(x) ^4 + a*sin(x)^2 - b*sin(x)^2 - a)*(a^3 + 2*a^2*b + a*b^2))
Time = 14.64 (sec) , antiderivative size = 2009, normalized size of antiderivative = 18.43 \[ \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\text {Too large to display} \]
(log(sin(x) + 1)*(a + 5*b))/(4*(a + b)^3) - log(sin(x) - 1)*(b/(a + b)^3 + 1/(4*(a + b)^2)) - ((sin(x)*(a^2 + b^2))/(2*a*(2*a*b + a^2 + b^2)) + (b*s in(x)^3*(a - b))/(2*a*(2*a*b + a^2 + b^2)))/(b*sin(x)^4 - a + sin(x)^2*(a - b)) - (atan(((((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 + a^4*b ^3))/(2*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2)) + ((5*a + b)*(- a^3*b^3)^(1/2)*((2*a*b^10 + 20*a^2*b^9 + 80*a^3*b^8 + 172*a^4*b^7 + 220*a^ 5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8*b^3 + 2*a^9*b^2)/(6*a^7*b + a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4 + 20*a^5*b^3 + 15*a^6*b^2) - (sin(x)*(5 *a + b)*(-a^3*b^3)^(1/2)*(16*a^2*b^9 + 80*a^3*b^8 + 144*a^4*b^7 + 80*a^5*b ^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^3 - 16*a^9*b^2))/(8*(3*a^5*b + a^ 6 + a^3*b^3 + 3*a^4*b^2)*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2) )))/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)))*(5*a + b)*(-a^3*b^3)^(1/2)* 1i)/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)) + (((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 + a^4*b^3))/(2*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3* b^3 + 6*a^4*b^2)) - ((5*a + b)*(-a^3*b^3)^(1/2)*((2*a*b^10 + 20*a^2*b^9 + 80*a^3*b^8 + 172*a^4*b^7 + 220*a^5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8 *b^3 + 2*a^9*b^2)/(6*a^7*b + a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4 + 20*a ^5*b^3 + 15*a^6*b^2) + (sin(x)*(5*a + b)*(-a^3*b^3)^(1/2)*(16*a^2*b^9 + 80 *a^3*b^8 + 144*a^4*b^7 + 80*a^5*b^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^ 3 - 16*a^9*b^2))/(8*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)*(4*a^5*b + a^...