Integrand size = 25, antiderivative size = 307 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {a^2 e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e} \] Output:
1/2*a^2*e^(5/2)*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d-1 /2*a^2*e^(5/2)*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d+1/ 2*a^2*e^(5/2)*arctanh(2^(1/2)*(e*tan(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*tan(d* x+c)))*2^(1/2)/d-12/5*a^2*e^2*cos(d*x+c)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/ 2))*(e*tan(d*x+c))^(1/2)/d/sin(2*d*x+2*c)^(1/2)+2/3*a^2*e*(e*tan(d*x+c))^( 3/2)/d-12/5*a^2*e*cos(d*x+c)*(e*tan(d*x+c))^(3/2)/d+4/5*a^2*e*sec(d*x+c)*( e*tan(d*x+c))^(3/2)/d+2/7*a^2*(e*tan(d*x+c))^(7/2)/d/e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.38 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\frac {2 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} \arctan (\tan (c+d x))\right ) (e \tan (c+d x))^{3/2} \left (35-42 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-35 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )+42 \sqrt {\sec ^2(c+d x)}+15 \tan ^2(c+d x)\right )}{105 d} \] Input:
Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]
Output:
(2*a^2*e*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(e*Tan[c + d*x]) ^(3/2)*(35 - 42*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 35*Hyp ergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2] + 42*Sqrt[Sec[c + d*x]^2] + 1 5*Tan[c + d*x]^2))/(105*d)
Time = 0.77 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3042, 4374, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \tan (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^2 (e \tan (c+d x))^{5/2}+a^2 \sec ^2(c+d x) (e \tan (c+d x))^{5/2}+2 a^2 \sec (c+d x) (e \tan (c+d x))^{5/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}\) |
Input:
Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]
Output:
(a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]* d) - (a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqr t[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e *Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (12*a^2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Sin[2* c + 2*d*x]]) + (2*a^2*e*(e*Tan[c + d*x])^(3/2))/(3*d) - (12*a^2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (4*a^2*e*Sec[c + d*x]*(e*Tan[c + d*x] )^(3/2))/(5*d) + (2*a^2*(e*Tan[c + d*x])^(7/2))/(7*d*e)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Time = 2.46 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.57
| method | result | size |
| parts | \(\frac {2 a^{2} e \left (\frac {\left (e \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \tan \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \tan \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}+\frac {2 a^{2} \left (e \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}-\frac {a^{2} \sqrt {2}\, \sqrt {-\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sqrt {e \tan \left (d x +c \right )}\, e^{2} \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (6 \cot \left (d x +c \right )+6 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-3 \cot \left (d x +c \right )-3 \csc \left (d x +c \right )\right )-6 \cot \left (d x +c \right )+8 \csc \left (d x +c \right )-2 \sec \left (d x +c \right )^{2} \csc \left (d x +c \right )\right )}{5 d \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\) | \(482\) |
| default | \(-\frac {a^{2} \sqrt {2}\, \sqrt {-\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sqrt {e \tan \left (d x +c \right )}\, e^{2} \left (i \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (105 \cot \left (d x +c \right )+105 \csc \left (d x +c \right )\right )+i \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-105 \cot \left (d x +c \right )-105 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \left (504 \cot \left (d x +c \right )+504 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-252 \cot \left (d x +c \right )-252 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-105 \cot \left (d x +c \right )-105 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-105 \cot \left (d x +c \right )-105 \csc \left (d x +c \right )\right )-424 \cot \left (d x +c \right )+672 \csc \left (d x +c \right )-20 \sec \left (d x +c \right ) \csc \left (d x +c \right )-168 \sec \left (d x +c \right )^{2} \csc \left (d x +c \right )-60 \sec \left (d x +c \right )^{3} \csc \left (d x +c \right )\right )}{420 d \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\) | \(738\) |
Input:
int((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
2*a^2/d*e*(1/3*(e*tan(d*x+c))^(3/2)-1/8*e^2/(e^2)^(1/4)*2^(1/2)*(ln((e*tan (d*x+c)-(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*tan(d*x+c )+(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/ (e^2)^(1/4)*(e*tan(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*tan(d *x+c))^(1/2)+1)))+2/7*a^2*(e*tan(d*x+c))^(7/2)/d/e-1/5*a^2/d*2^(1/2)*(-2*s in(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(e*tan(d*x+c))^(1/2)*e^2/(-si n(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*((-cot(d*x+c)+csc(d*x+c)+1)^(1 /2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*Ell ipticE((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(6*cot(d*x+c)+6*csc(d *x+c))+(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2 )*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2 ),1/2*2^(1/2))*(-3*cot(d*x+c)-3*csc(d*x+c))-6*cot(d*x+c)+8*csc(d*x+c)-2*se c(d*x+c)^2*csc(d*x+c))
Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**2*(e*tan(d*x+c))**(5/2),x)
Output:
Timed out
Exception generated. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^2*(e*tan(d*x + c))^(5/2), x)
Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \] Input:
int((e*tan(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2,x)
Output:
int((e*tan(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2, x)
\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\sqrt {e}\, a^{2} e^{2} \left (\int \sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}d x +\int \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}d x +2 \left (\int \sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}d x \right )\right ) \] Input:
int((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x)
Output:
sqrt(e)*a**2*e**2*(int(sqrt(tan(c + d*x))*sec(c + d*x)**2*tan(c + d*x)**2, x) + int(sqrt(tan(c + d*x))*tan(c + d*x)**2,x) + 2*int(sqrt(tan(c + d*x))* sec(c + d*x)*tan(c + d*x)**2,x))