Integrand size = 23, antiderivative size = 117 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {arctanh}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (12 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b^2 \sec ^5(e+f x) \tan (e+f x)}{6 f} \] Output:
1/16*(8*a^2+12*a*b+5*b^2)*arctanh(sin(f*x+e))/f+1/16*(8*a^2+12*a*b+5*b^2)* sec(f*x+e)*tan(f*x+e)/f+1/24*b*(12*a+5*b)*sec(f*x+e)^3*tan(f*x+e)/f+1/6*b^ 2*sec(f*x+e)^5*tan(f*x+e)/f
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.60 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {a^2 \text {arctanh}(\sin (e+f x))}{2 f}+\frac {3 a b \text {arctanh}(\sin (e+f x))}{4 f}+\frac {5 b^2 \text {arctanh}(\sin (e+f x))}{16 f}+\frac {a^2 \sec (e+f x) \tan (e+f x)}{2 f}+\frac {3 a b \sec (e+f x) \tan (e+f x)}{4 f}+\frac {5 b^2 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a b \sec ^3(e+f x) \tan (e+f x)}{2 f}+\frac {5 b^2 \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b^2 \sec ^5(e+f x) \tan (e+f x)}{6 f} \] Input:
Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]
Output:
(a^2*ArcTanh[Sin[e + f*x]])/(2*f) + (3*a*b*ArcTanh[Sin[e + f*x]])/(4*f) + (5*b^2*ArcTanh[Sin[e + f*x]])/(16*f) + (a^2*Sec[e + f*x]*Tan[e + f*x])/(2* f) + (3*a*b*Sec[e + f*x]*Tan[e + f*x])/(4*f) + (5*b^2*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (a*b*Sec[e + f*x]^3*Tan[e + f*x])/(2*f) + (5*b^2*Sec[e + f *x]^3*Tan[e + f*x])/(24*f) + (b^2*Sec[e + f*x]^5*Tan[e + f*x])/(6*f)
Time = 0.32 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4635, 315, 25, 298, 215, 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.
| \(\displaystyle \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (e+f x)^3 \left (a+b \sec (e+f x)^2\right )^2dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (-a \sin ^2(e+f x)+a+b\right )^2}{\left (1-\sin ^2(e+f x)\right )^4}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\frac {b \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 \left (1-\sin ^2(e+f x)\right )^3}-\frac {1}{6} \int -\frac {(a+b) (6 a+5 b)-3 a (2 a+b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^3}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {(a+b) (6 a+5 b)-3 a (2 a+b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^3}d\sin (e+f x)+\frac {b \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 \left (1-\sin ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \int \frac {1}{\left (1-\sin ^2(e+f x)\right )^2}d\sin (e+f x)+\frac {b (8 a+5 b) \sin (e+f x)}{4 \left (1-\sin ^2(e+f x)\right )^2}\right )+\frac {b \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 \left (1-\sin ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(e+f x)}d\sin (e+f x)+\frac {\sin (e+f x)}{2 \left (1-\sin ^2(e+f x)\right )}\right )+\frac {b (8 a+5 b) \sin (e+f x)}{4 \left (1-\sin ^2(e+f x)\right )^2}\right )+\frac {b \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 \left (1-\sin ^2(e+f x)\right )^3}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (8 a^2+12 a b+5 b^2\right ) \left (\frac {1}{2} \text {arctanh}(\sin (e+f x))+\frac {\sin (e+f x)}{2 \left (1-\sin ^2(e+f x)\right )}\right )+\frac {b (8 a+5 b) \sin (e+f x)}{4 \left (1-\sin ^2(e+f x)\right )^2}\right )+\frac {b \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 \left (1-\sin ^2(e+f x)\right )^3}}{f}\) |
Input:
Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]
Output:
((b*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(6*(1 - Sin[e + f*x]^2)^3) + ((b*(8*a + 5*b)*Sin[e + f*x])/(4*(1 - Sin[e + f*x]^2)^2) + (3*(8*a^2 + 12* a*b + 5*b^2)*(ArcTanh[Sin[e + f*x]]/2 + Sin[e + f*x]/(2*(1 - Sin[e + f*x]^ 2))))/4)/6)/f
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 2.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(147\) |
| default | \(\frac {a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(147\) |
| parts | \(\frac {a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}+\frac {2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(152\) |
| parallelrisch | \(\frac {-360 \left (\frac {\cos \left (6 f x +6 e \right )}{15}+\frac {2 \cos \left (4 f x +4 e \right )}{5}+\cos \left (2 f x +2 e \right )+\frac {2}{3}\right ) \left (a^{2}+\frac {3}{2} a b +\frac {5}{8} b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+360 \left (\frac {\cos \left (6 f x +6 e \right )}{15}+\frac {2 \cos \left (4 f x +4 e \right )}{5}+\cos \left (2 f x +2 e \right )+\frac {2}{3}\right ) \left (a^{2}+\frac {3}{2} a b +\frac {5}{8} b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (144 a^{2}+408 a b +170 b^{2}\right ) \sin \left (3 f x +3 e \right )+\left (48 a^{2}+72 a b +30 b^{2}\right ) \sin \left (5 f x +5 e \right )+96 \left (a^{2}+\frac {7}{2} a b +\frac {33}{8} b^{2}\right ) \sin \left (f x +e \right )}{48 f \left (\cos \left (6 f x +6 e \right )+6 \cos \left (4 f x +4 e \right )+15 \cos \left (2 f x +2 e \right )+10\right )}\) | \(233\) |
| norman | \(\frac {\frac {\left (8 a^{2}+4 a b +15 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}+\frac {\left (8 a^{2}+4 a b +15 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {\left (8 a^{2}+20 a b +11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {\left (8 a^{2}+20 a b +11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}-\frac {\left (72 a^{2}+84 a b -5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {\left (72 a^{2}+84 a b -5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}-\frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(267\) |
| risch | \(-\frac {i {\mathrm e}^{i \left (f x +e \right )} \left (24 \,{\mathrm e}^{10 i \left (f x +e \right )} a^{2}+36 \,{\mathrm e}^{10 i \left (f x +e \right )} a b +15 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+72 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+204 \,{\mathrm e}^{8 i \left (f x +e \right )} a b +85 \,{\mathrm e}^{8 i \left (f x +e \right )} b^{2}+48 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+168 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+198 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-48 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-168 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-198 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-72 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-204 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-85 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-24 a^{2}-36 a b -15 b^{2}\right )}{24 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2}}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a b}{4 f}-\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b^{2}}{16 f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a^{2}}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a b}{4 f}+\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b^{2}}{16 f}\) | \(406\) |
Input:
int(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/f*(a^2*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+tan(f*x+e)))+2*a*b*( -(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+3/8*ln(sec(f*x+e)+tan(f*x+e )))+b^2*(-(-1/6*sec(f*x+e)^5-5/24*sec(f*x+e)^3-5/16*sec(f*x+e))*tan(f*x+e) +5/16*ln(sec(f*x+e)+tan(f*x+e))))
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.22 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )} \sin \left (f x + e\right )}{96 \, f \cos \left (f x + e\right )^{6}} \] Input:
integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")
Output:
1/96*(3*(8*a^2 + 12*a*b + 5*b^2)*cos(f*x + e)^6*log(sin(f*x + e) + 1) - 3* (8*a^2 + 12*a*b + 5*b^2)*cos(f*x + e)^6*log(-sin(f*x + e) + 1) + 2*(3*(8*a ^2 + 12*a*b + 5*b^2)*cos(f*x + e)^4 + 2*(12*a*b + 5*b^2)*cos(f*x + e)^2 + 8*b^2)*sin(f*x + e))/(f*cos(f*x + e)^6)
\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{3}{\left (e + f x \right )}\, dx \] Input:
integrate(sec(f*x+e)**3*(a+b*sec(f*x+e)**2)**2,x)
Output:
Integral((a + b*sec(e + f*x)**2)**2*sec(e + f*x)**3, x)
Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.42 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{5} - 8 \, {\left (6 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} + 20 \, a b + 11 \, b^{2}\right )} \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1}}{96 \, f} \] Input:
integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")
Output:
1/96*(3*(8*a^2 + 12*a*b + 5*b^2)*log(sin(f*x + e) + 1) - 3*(8*a^2 + 12*a*b + 5*b^2)*log(sin(f*x + e) - 1) - 2*(3*(8*a^2 + 12*a*b + 5*b^2)*sin(f*x + e)^5 - 8*(6*a^2 + 12*a*b + 5*b^2)*sin(f*x + e)^3 + 3*(8*a^2 + 20*a*b + 11* b^2)*sin(f*x + e))/(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1))/f
Time = 0.16 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.56 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a^{2} \sin \left (f x + e\right )^{5} + 36 \, a b \sin \left (f x + e\right )^{5} + 15 \, b^{2} \sin \left (f x + e\right )^{5} - 48 \, a^{2} \sin \left (f x + e\right )^{3} - 96 \, a b \sin \left (f x + e\right )^{3} - 40 \, b^{2} \sin \left (f x + e\right )^{3} + 24 \, a^{2} \sin \left (f x + e\right ) + 60 \, a b \sin \left (f x + e\right ) + 33 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3}}}{96 \, f} \] Input:
integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Output:
1/96*(3*(8*a^2 + 12*a*b + 5*b^2)*log(abs(sin(f*x + e) + 1)) - 3*(8*a^2 + 1 2*a*b + 5*b^2)*log(abs(sin(f*x + e) - 1)) - 2*(24*a^2*sin(f*x + e)^5 + 36* a*b*sin(f*x + e)^5 + 15*b^2*sin(f*x + e)^5 - 48*a^2*sin(f*x + e)^3 - 96*a* b*sin(f*x + e)^3 - 40*b^2*sin(f*x + e)^3 + 24*a^2*sin(f*x + e) + 60*a*b*si n(f*x + e) + 33*b^2*sin(f*x + e))/(sin(f*x + e)^2 - 1)^3)/f
Time = 16.02 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.15 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\mathrm {atanh}\left (\sin \left (e+f\,x\right )\right )\,\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )}{f}-\frac {\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )\,{\sin \left (e+f\,x\right )}^5+\left (-a^2-2\,a\,b-\frac {5\,b^2}{6}\right )\,{\sin \left (e+f\,x\right )}^3+\left (\frac {a^2}{2}+\frac {5\,a\,b}{4}+\frac {11\,b^2}{16}\right )\,\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^6-3\,{\sin \left (e+f\,x\right )}^4+3\,{\sin \left (e+f\,x\right )}^2-1\right )} \] Input:
int((a + b/cos(e + f*x)^2)^2/cos(e + f*x)^3,x)
Output:
(atanh(sin(e + f*x))*((3*a*b)/4 + a^2/2 + (5*b^2)/16))/f - (sin(e + f*x)*( (5*a*b)/4 + a^2/2 + (11*b^2)/16) - sin(e + f*x)^3*(2*a*b + a^2 + (5*b^2)/6 ) + sin(e + f*x)^5*((3*a*b)/4 + a^2/2 + (5*b^2)/16))/(f*(3*sin(e + f*x)^2 - 3*sin(e + f*x)^4 + sin(e + f*x)^6 - 1))
Time = 0.16 (sec) , antiderivative size = 690, normalized size of antiderivative = 5.90 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx =\text {Too large to display} \] Input:
int(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x)
Output:
( - 24*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**6*a**2 - 36*log(tan((e + f* x)/2) - 1)*sin(e + f*x)**6*a*b - 15*log(tan((e + f*x)/2) - 1)*sin(e + f*x) **6*b**2 + 72*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**4*a**2 + 108*log(tan ((e + f*x)/2) - 1)*sin(e + f*x)**4*a*b + 45*log(tan((e + f*x)/2) - 1)*sin( e + f*x)**4*b**2 - 72*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**2*a**2 - 108 *log(tan((e + f*x)/2) - 1)*sin(e + f*x)**2*a*b - 45*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**2*b**2 + 24*log(tan((e + f*x)/2) - 1)*a**2 + 36*log(tan( (e + f*x)/2) - 1)*a*b + 15*log(tan((e + f*x)/2) - 1)*b**2 + 24*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**6*a**2 + 36*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**6*a*b + 15*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**6*b**2 - 72*log( tan((e + f*x)/2) + 1)*sin(e + f*x)**4*a**2 - 108*log(tan((e + f*x)/2) + 1) *sin(e + f*x)**4*a*b - 45*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**4*b**2 + 72*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**2*a**2 + 108*log(tan((e + f*x) /2) + 1)*sin(e + f*x)**2*a*b + 45*log(tan((e + f*x)/2) + 1)*sin(e + f*x)** 2*b**2 - 24*log(tan((e + f*x)/2) + 1)*a**2 - 36*log(tan((e + f*x)/2) + 1)* a*b - 15*log(tan((e + f*x)/2) + 1)*b**2 - 24*sin(e + f*x)**5*a**2 - 36*sin (e + f*x)**5*a*b - 15*sin(e + f*x)**5*b**2 + 48*sin(e + f*x)**3*a**2 + 96* sin(e + f*x)**3*a*b + 40*sin(e + f*x)**3*b**2 - 24*sin(e + f*x)*a**2 - 60* sin(e + f*x)*a*b - 33*sin(e + f*x)*b**2)/(48*f*(sin(e + f*x)**6 - 3*sin(e + f*x)**4 + 3*sin(e + f*x)**2 - 1))