Integrand size = 41, antiderivative size = 359 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\frac {\left (a^4+9 a^2 b^2+2 b^4\right ) \text {arctanh}(\sin (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}{2 e (b+a \sec (d+e x))^3}+\frac {a^4 b^3 x \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}{\left (a b+a^2 \sec (d+e x)\right )^3}+\frac {a^4 b \left (11 a^2+8 b^2\right ) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \tan (d+e x)}{3 e \left (a b+a^2 \sec (d+e x)\right )^3}+\frac {a^5 \left (3 a^2+5 b^2\right ) \sec (d+e x) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \tan (d+e x)}{6 e \left (a b+a^2 \sec (d+e x)\right )^3}+\frac {b \left (a^2 b+a^3 \sec (d+e x)\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \tan (d+e x)}{3 e \left (a b+a^2 \sec (d+e x)\right )^3} \] Output:
1/2*(a^4+9*a^2*b^2+2*b^4)*arctanh(sin(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*se c(e*x+d)^2)^(3/2)/e/(b+a*sec(e*x+d))^3+a^4*b^3*x*(b^2+2*a*b*sec(e*x+d)+a^2 *sec(e*x+d)^2)^(3/2)/(a*b+a^2*sec(e*x+d))^3+1/3*a^4*b*(11*a^2+8*b^2)*(b^2+ 2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2)*tan(e*x+d)/e/(a*b+a^2*sec(e*x+d)) ^3+1/6*a^5*(3*a^2+5*b^2)*sec(e*x+d)*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2 )^(3/2)*tan(e*x+d)/e/(a*b+a^2*sec(e*x+d))^3+1/3*b*(a^2*b+a^3*sec(e*x+d))^2 *(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2)*tan(e*x+d)/e/(a*b+a^2*sec(e *x+d))^3
Time = 0.82 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.40 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\frac {\cos (d+e x) \sqrt {(b+a \sec (d+e x))^2} \left (6 b^2 \left (3 a^2+b^2\right ) \coth ^{-1}(\sin (d+e x))+a \left (6 b^3 e x+3 a \left (a^2+3 b^2\right ) \text {arctanh}(\sin (d+e x))+3 \left (8 a^2 b+6 b^3+a \left (a^2+3 b^2\right ) \sec (d+e x)\right ) \tan (d+e x)+2 a^2 b \tan ^3(d+e x)\right )\right )}{6 e (a+b \cos (d+e x))} \] Input:
Integrate[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x ]^2)^(3/2),x]
Output:
(Cos[d + e*x]*Sqrt[(b + a*Sec[d + e*x])^2]*(6*b^2*(3*a^2 + b^2)*ArcCoth[Si n[d + e*x]] + a*(6*b^3*e*x + 3*a*(a^2 + 3*b^2)*ArcTanh[Sin[d + e*x]] + 3*( 8*a^2*b + 6*b^3 + a*(a^2 + 3*b^2)*Sec[d + e*x])*Tan[d + e*x] + 2*a^2*b*Tan [d + e*x]^3)))/(6*e*(a + b*Cos[d + e*x]))
Time = 0.77 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.53, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3042, 4662, 27, 3042, 4406, 3042, 4536, 2009}
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.
| \(\displaystyle \int (a+b \sec (d+e x)) \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sec (d+e x)) \left (a^2 \sec (d+e x)^2+2 a b \sec (d+e x)+b^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4662 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \int 8 \left (\sec (d+e x) a^2+b a\right )^3 (a+b \sec (d+e x))dx}{8 \left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \int \left (\sec (d+e x) a^2+b a\right )^3 (a+b \sec (d+e x))dx}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \int \left (\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a\right )^3 \left (a+b \csc \left (d+e x+\frac {\pi }{2}\right )\right )dx}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 4406 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \left (\frac {1}{3} \int \left (\sec (d+e x) a^2+b a\right ) \left (3 b^2 a^3+\left (3 a^2+5 b^2\right ) \sec ^2(d+e x) a^3+b \left (8 a^2+3 b^2\right ) \sec (d+e x) a^2\right )dx+\frac {b \tan (d+e x) \left (a^3 \sec (d+e x)+a^2 b\right )^2}{3 e}\right )}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \left (\frac {1}{3} \int \left (\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a\right ) \left (3 b^2 a^3+\left (3 a^2+5 b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right )^2 a^3+b \left (8 a^2+3 b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {b \tan (d+e x) \left (a^3 \sec (d+e x)+a^2 b\right )^2}{3 e}\right )}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (6 b^3 a^4+2 b \left (11 a^2+8 b^2\right ) \sec ^2(d+e x) a^4+3 \left (a^4+9 b^2 a^2+2 b^4\right ) \sec (d+e x) a^3\right )dx+\frac {a^5 \left (3 a^2+5 b^2\right ) \tan (d+e x) \sec (d+e x)}{2 e}\right )+\frac {b \tan (d+e x) \left (a^3 \sec (d+e x)+a^2 b\right )^2}{3 e}\right )}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2} \left (\frac {b \tan (d+e x) \left (a^3 \sec (d+e x)+a^2 b\right )^2}{3 e}+\frac {1}{3} \left (\frac {a^5 \left (3 a^2+5 b^2\right ) \tan (d+e x) \sec (d+e x)}{2 e}+\frac {1}{2} \left (6 a^4 b^3 x+\frac {2 a^4 b \left (11 a^2+8 b^2\right ) \tan (d+e x)}{e}+\frac {3 a^3 \left (a^4+9 a^2 b^2+2 b^4\right ) \text {arctanh}(\sin (d+e x))}{e}\right )\right )\right )}{\left (a^2 \sec (d+e x)+a b\right )^3}\) |
Input:
Int[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^( 3/2),x]
Output:
((b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*((b*(a^2*b + a^3*Se c[d + e*x])^2*Tan[d + e*x])/(3*e) + ((a^5*(3*a^2 + 5*b^2)*Sec[d + e*x]*Tan [d + e*x])/(2*e) + (6*a^4*b^3*x + (3*a^3*(a^4 + 9*a^2*b^2 + 2*b^4)*ArcTanh [Sin[d + e*x]])/e + (2*a^4*b*(11*a^2 + 8*b^2)*Tan[d + e*x])/e)/2)/3))/(a*b + a^2*Sec[d + e*x])^3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b* c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2 Int[Simp[2*A*a + (2*B*a + b*(2* A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, e, f, A, B, C}, x]
Int[((A_) + (B_.)*sec[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sec[(d_.) + (e_.)* (x_)] + (c_.)*sec[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[(a + b*Sec [d + e*x] + c*Sec[d + e*x]^2)^n/(b + 2*c*Sec[d + e*x])^(2*n) Int[(A + B*S ec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
Time = 3.07 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {\sqrt {\left (b \cos \left (e x +d \right )+a \right )^{2} \sec \left (e x +d \right )^{2}}\, \left (-3 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) \cos \left (e x +d \right ) a^{4}-27 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) \cos \left (e x +d \right ) a^{2} b^{2}-6 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) \cos \left (e x +d \right ) b^{4}+3 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) \cos \left (e x +d \right ) a^{4}+27 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) \cos \left (e x +d \right ) a^{2} b^{2}+6 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) \cos \left (e x +d \right ) b^{4}+6 \cos \left (e x +d \right ) a \,b^{3} \left (e x +d \right )+3 a^{4} \tan \left (e x +d \right )+a^{3} b \left (22 \sin \left (e x +d \right )+2 \sec \left (e x +d \right ) \tan \left (e x +d \right )\right )+9 a^{2} b^{2} \tan \left (e x +d \right )+18 a \,b^{3} \sin \left (e x +d \right )\right )}{6 e \left (b \cos \left (e x +d \right )+a \right )}\) | \(298\) |
| parts | \(\frac {a \sqrt {\left (b \cos \left (e x +d \right )+a \right )^{2} \sec \left (e x +d \right )^{2}}\, \left (-\ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) \cos \left (e x +d \right ) a^{3}-6 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) \cos \left (e x +d \right ) a \,b^{2}+\ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) \cos \left (e x +d \right ) a^{3}+6 \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) \cos \left (e x +d \right ) a \,b^{2}+2 \cos \left (e x +d \right ) \left (e x +d \right ) b^{3}+6 \sin \left (e x +d \right ) a^{2} b +\tan \left (e x +d \right ) a^{3}\right )}{2 e \left (b \cos \left (e x +d \right )+a \right )}+\frac {b \sqrt {\left (b \cos \left (e x +d \right )+a \right )^{2} \sec \left (e x +d \right )^{2}}\, \left (-9 \cos \left (e x +d \right ) \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) a^{2} b -6 \cos \left (e x +d \right ) \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )-1\right ) b^{3}+9 \cos \left (e x +d \right ) \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) a^{2} b +6 \cos \left (e x +d \right ) \ln \left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )+1\right ) b^{3}+a^{3} \left (4 \sin \left (e x +d \right )+2 \sec \left (e x +d \right ) \tan \left (e x +d \right )\right )+9 \tan \left (e x +d \right ) a^{2} b +18 \sin \left (e x +d \right ) a \,b^{2}\right )}{6 e \left (b \cos \left (e x +d \right )+a \right )}\) | \(402\) |
| risch | \(\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right ) \sqrt {\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right )^{2}}{\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right )^{2}}}\, a \,b^{3} x}{{\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b}-\frac {i \sqrt {\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right )^{2}}{\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right )^{2}}}\, a \left (3 a^{3} {\mathrm e}^{5 i \left (e x +d \right )}+9 a \,b^{2} {\mathrm e}^{5 i \left (e x +d \right )}-18 a^{2} b \,{\mathrm e}^{4 i \left (e x +d \right )}-18 b^{3} {\mathrm e}^{4 i \left (e x +d \right )}-48 a^{2} b \,{\mathrm e}^{2 i \left (e x +d \right )}-36 b^{3} {\mathrm e}^{2 i \left (e x +d \right )}-3 a^{3} {\mathrm e}^{i \left (e x +d \right )}-9 a \,b^{2} {\mathrm e}^{i \left (e x +d \right )}-22 a^{2} b -18 b^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right ) \left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right )^{2} e}+\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right ) \sqrt {\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right )^{2}}{\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right )^{2}}}\, \left (a^{4}+9 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{2 \left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right ) e}-\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right ) \sqrt {\frac {\left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right )^{2}}{\left ({\mathrm e}^{2 i \left (e x +d \right )}+1\right )^{2}}}\, \left (a^{4}+9 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}-i\right )}{2 \left ({\mathrm e}^{2 i \left (e x +d \right )} b +2 \,{\mathrm e}^{i \left (e x +d \right )} a +b \right ) e}\) | \(539\) |
Input:
int((a+b*sec(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x,metho d=_RETURNVERBOSE)
Output:
1/6/e*((b*cos(e*x+d)+a)^2*sec(e*x+d)^2)^(1/2)/(b*cos(e*x+d)+a)*(-3*ln(-cot (e*x+d)+csc(e*x+d)-1)*cos(e*x+d)*a^4-27*ln(-cot(e*x+d)+csc(e*x+d)-1)*cos(e *x+d)*a^2*b^2-6*ln(-cot(e*x+d)+csc(e*x+d)-1)*cos(e*x+d)*b^4+3*ln(-cot(e*x+ d)+csc(e*x+d)+1)*cos(e*x+d)*a^4+27*ln(-cot(e*x+d)+csc(e*x+d)+1)*cos(e*x+d) *a^2*b^2+6*ln(-cot(e*x+d)+csc(e*x+d)+1)*cos(e*x+d)*b^4+6*cos(e*x+d)*a*b^3* (e*x+d)+3*a^4*tan(e*x+d)+a^3*b*(22*sin(e*x+d)+2*sec(e*x+d)*tan(e*x+d))+9*a ^2*b^2*tan(e*x+d)+18*a*b^3*sin(e*x+d))
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.45 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\frac {12 \, a b^{3} e x \cos \left (e x + d\right )^{3} + 3 \, {\left (a^{4} + 9 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (e x + d\right )^{3} \log \left (\sin \left (e x + d\right ) + 1\right ) - 3 \, {\left (a^{4} + 9 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (e x + d\right )^{3} \log \left (-\sin \left (e x + d\right ) + 1\right ) + 2 \, {\left (2 \, a^{3} b + 2 \, {\left (11 \, a^{3} b + 9 \, a b^{3}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{12 \, e \cos \left (e x + d\right )^{3}} \] Input:
integrate((a+b*sec(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x , algorithm="fricas")
Output:
1/12*(12*a*b^3*e*x*cos(e*x + d)^3 + 3*(a^4 + 9*a^2*b^2 + 2*b^4)*cos(e*x + d)^3*log(sin(e*x + d) + 1) - 3*(a^4 + 9*a^2*b^2 + 2*b^4)*cos(e*x + d)^3*lo g(-sin(e*x + d) + 1) + 2*(2*a^3*b + 2*(11*a^3*b + 9*a*b^3)*cos(e*x + d)^2 + 3*(a^4 + 3*a^2*b^2)*cos(e*x + d))*sin(e*x + d))/(e*cos(e*x + d)^3)
\[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\int \left (a + b \sec {\left (d + e x \right )}\right ) \left (\left (a \sec {\left (d + e x \right )} + b\right )^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+b*sec(e*x+d))*(b**2+2*a*b*sec(e*x+d)+a**2*sec(e*x+d)**2)**(3/ 2),x)
Output:
Integral((a + b*sec(d + e*x))*((a*sec(d + e*x) + b)**2)**(3/2), x)
Time = 0.14 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.23 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\frac {3 \, {\left (4 \, b^{3} \arctan \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right ) + {\left (a^{3} + 6 \, a b^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right ) - {\left (a^{3} + 6 \, a b^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - 1\right ) - \frac {2 \, {\left (\frac {{\left (a^{3} + 6 \, a^{2} b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {{\left (a^{3} - 6 \, a^{2} b\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}\right )}}{\frac {2 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {\sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} - 1}\right )} a + {\left (3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right ) - 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - 1\right ) - \frac {2 \, {\left (\frac {3 \, {\left (2 \, a^{3} + 3 \, a^{2} b + 6 \, a b^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {4 \, {\left (a^{3} + 9 \, a b^{2}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, {\left (2 \, a^{3} - 3 \, a^{2} b + 6 \, a b^{2}\right )} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}}\right )}}{\frac {3 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {3 \, \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {\sin \left (e x + d\right )^{6}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{6}} - 1}\right )} b}{6 \, e} \] Input:
integrate((a+b*sec(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x , algorithm="maxima")
Output:
1/6*(3*(4*b^3*arctan(sin(e*x + d)/(cos(e*x + d) + 1)) + (a^3 + 6*a*b^2)*lo g(sin(e*x + d)/(cos(e*x + d) + 1) + 1) - (a^3 + 6*a*b^2)*log(sin(e*x + d)/ (cos(e*x + d) + 1) - 1) - 2*((a^3 + 6*a^2*b)*sin(e*x + d)/(cos(e*x + d) + 1) + (a^3 - 6*a^2*b)*sin(e*x + d)^3/(cos(e*x + d) + 1)^3)/(2*sin(e*x + d)^ 2/(cos(e*x + d) + 1)^2 - sin(e*x + d)^4/(cos(e*x + d) + 1)^4 - 1))*a + (3* (3*a^2*b + 2*b^3)*log(sin(e*x + d)/(cos(e*x + d) + 1) + 1) - 3*(3*a^2*b + 2*b^3)*log(sin(e*x + d)/(cos(e*x + d) + 1) - 1) - 2*(3*(2*a^3 + 3*a^2*b + 6*a*b^2)*sin(e*x + d)/(cos(e*x + d) + 1) - 4*(a^3 + 9*a*b^2)*sin(e*x + d)^ 3/(cos(e*x + d) + 1)^3 + 3*(2*a^3 - 3*a^2*b + 6*a*b^2)*sin(e*x + d)^5/(cos (e*x + d) + 1)^5)/(3*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 - 3*sin(e*x + d)^ 4/(cos(e*x + d) + 1)^4 + sin(e*x + d)^6/(cos(e*x + d) + 1)^6 - 1))*b)/e
Time = 0.40 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.69 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*sec(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x , algorithm="giac")
Output:
1/6*(6*(e*x + d)*a*b^3*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)) + 3*(a^4*sgn (b*cos(e*x + d)^2 + a*cos(e*x + d)) + 9*a^2*b^2*sgn(b*cos(e*x + d)^2 + a*c os(e*x + d)) + 2*b^4*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)))*log(abs(tan(1 /2*e*x + 1/2*d) + 1)) - 3*(a^4*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)) + 9* a^2*b^2*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)) + 2*b^4*sgn(b*cos(e*x + d)^ 2 + a*cos(e*x + d)))*log(abs(tan(1/2*e*x + 1/2*d) - 1)) + 2*(3*a^4*sgn(b*c os(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^5 - 24*a^3*b*sgn(b*co s(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^5 + 9*a^2*b^2*sgn(b*co s(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^5 - 18*a*b^3*sgn(b*cos (e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^5 + 40*a^3*b*sgn(b*cos( e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^3 + 36*a*b^3*sgn(b*cos(e *x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d)^3 - 3*a^4*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d) - 24*a^3*b*sgn(b*cos(e*x + d) ^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d) - 9*a^2*b^2*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d) - 18*a*b^3*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d))*tan(1/2*e*x + 1/2*d))/(tan(1/2*e*x + 1/2*d)^2 - 1)^3)/e
Timed out. \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\int \left (a+\frac {b}{\cos \left (d+e\,x\right )}\right )\,{\left (b^2+\frac {a^2}{{\cos \left (d+e\,x\right )}^2}+\frac {2\,a\,b}{\cos \left (d+e\,x\right )}\right )}^{3/2} \,d x \] Input:
int((a + b/cos(d + e*x))*(b^2 + a^2/cos(d + e*x)^2 + (2*a*b)/cos(d + e*x)) ^(3/2),x)
Output:
int((a + b/cos(d + e*x))*(b^2 + a^2/cos(d + e*x)^2 + (2*a*b)/cos(d + e*x)) ^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.36 \[ \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2} \, dx=\frac {-3 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) \sin \left (e x +d \right )^{2} a^{4}-27 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) \sin \left (e x +d \right )^{2} a^{2} b^{2}-6 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) \sin \left (e x +d \right )^{2} b^{4}+3 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) a^{4}+27 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) a^{2} b^{2}+6 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right ) b^{4}+3 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) \sin \left (e x +d \right )^{2} a^{4}+27 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) \sin \left (e x +d \right )^{2} a^{2} b^{2}+6 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) \sin \left (e x +d \right )^{2} b^{4}-3 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) a^{4}-27 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) a^{2} b^{2}-6 \cos \left (e x +d \right ) \mathrm {log}\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right ) b^{4}+6 \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} a \,b^{3} e x -3 \cos \left (e x +d \right ) \sin \left (e x +d \right ) a^{4}-9 \cos \left (e x +d \right ) \sin \left (e x +d \right ) a^{2} b^{2}-6 \cos \left (e x +d \right ) a \,b^{3} e x +22 \sin \left (e x +d \right )^{3} a^{3} b +18 \sin \left (e x +d \right )^{3} a \,b^{3}-24 \sin \left (e x +d \right ) a^{3} b -18 \sin \left (e x +d \right ) a \,b^{3}}{6 \cos \left (e x +d \right ) e \left (\sin \left (e x +d \right )^{2}-1\right )} \] Input:
int((a+b*sec(e*x+d))*(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x)
Output:
( - 3*cos(d + e*x)*log(tan((d + e*x)/2) - 1)*sin(d + e*x)**2*a**4 - 27*cos (d + e*x)*log(tan((d + e*x)/2) - 1)*sin(d + e*x)**2*a**2*b**2 - 6*cos(d + e*x)*log(tan((d + e*x)/2) - 1)*sin(d + e*x)**2*b**4 + 3*cos(d + e*x)*log(t an((d + e*x)/2) - 1)*a**4 + 27*cos(d + e*x)*log(tan((d + e*x)/2) - 1)*a**2 *b**2 + 6*cos(d + e*x)*log(tan((d + e*x)/2) - 1)*b**4 + 3*cos(d + e*x)*log (tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**4 + 27*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**2*b**2 + 6*cos(d + e*x)*log(tan((d + e*x) /2) + 1)*sin(d + e*x)**2*b**4 - 3*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*a **4 - 27*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*a**2*b**2 - 6*cos(d + e*x) *log(tan((d + e*x)/2) + 1)*b**4 + 6*cos(d + e*x)*sin(d + e*x)**2*a*b**3*e* x - 3*cos(d + e*x)*sin(d + e*x)*a**4 - 9*cos(d + e*x)*sin(d + e*x)*a**2*b* *2 - 6*cos(d + e*x)*a*b**3*e*x + 22*sin(d + e*x)**3*a**3*b + 18*sin(d + e* x)**3*a*b**3 - 24*sin(d + e*x)*a**3*b - 18*sin(d + e*x)*a*b**3)/(6*cos(d + e*x)*e*(sin(d + e*x)**2 - 1))