Integrand size = 25, antiderivative size = 277 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\frac {a^2 e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a^2 e^2 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 d \sqrt {e \tan (c+d x)}}+\frac {2 a^2 e \sqrt {e \tan (c+d x)}}{d}+\frac {4 a^2 e \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 d}+\frac {2 a^2 (e \tan (c+d x))^{5/2}}{5 d e} \] Output:
1/2*a^2*e^(3/2)*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d-1 /2*a^2*e^(3/2)*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d-1/ 2*a^2*e^(3/2)*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-2/3*a^2*e^2*InverseJacobiAM(c-1/4*Pi+d*x,2^(1/2))*sec(d*x +c)*sin(2*d*x+2*c)^(1/2)/d/(e*tan(d*x+c))^(1/2)+2*a^2*e*(e*tan(d*x+c))^(1/ 2)/d+4/3*a^2*e*sec(d*x+c)*(e*tan(d*x+c))^(1/2)/d+2/5*a^2*(e*tan(d*x+c))^(5 /2)/d/e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 26.01 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.93 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\frac {a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} \arctan (\tan (c+d x))\right ) (e \tan (c+d x))^{3/2} \left (30 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-30 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+15 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-15 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+120 \sqrt {\tan (c+d x)}-80 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(c+d x)\right ) \sqrt {\tan (c+d x)}+80 \sqrt {\sec ^2(c+d x)} \sqrt {\tan (c+d x)}+24 \tan ^{\frac {5}{2}}(c+d x)\right )}{60 d \tan ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(3/2),x]
Output:
(a^2*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(e*Tan[c + d*x])^(3/ 2)*(30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 30*Sqrt[2]*ArcTan[ 1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 15*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 15*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Ta n[c + d*x]] + 120*Sqrt[Tan[c + d*x]] - 80*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 80*Sqrt[Sec[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 24*Tan[c + d*x]^(5/2)))/(60*d*Tan[c + d*x]^(3/2))
Time = 0.68 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.21, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \tan (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^2 (e \tan (c+d x))^{3/2}+a^2 \sec ^2(c+d x) (e \tan (c+d x))^{3/2}+2 a^2 \sec (c+d x) (e \tan (c+d x))^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}+\frac {a^2 e^{3/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}-\frac {a^2 e^{3/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}-\frac {2 a^2 e^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 d \sqrt {e \tan (c+d x)}}+\frac {2 a^2 (e \tan (c+d x))^{5/2}}{5 d e}+\frac {2 a^2 e \sqrt {e \tan (c+d x)}}{d}+\frac {4 a^2 e \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 d}\) |
Input:
Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(3/2),x]
Output:
(a^2*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]* d) - (a^2*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqr t[2]*d) + (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e *Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*a^2*e^2*Ellipti cF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[e*Tan [c + d*x]]) + (2*a^2*e*Sqrt[e*Tan[c + d*x]])/d + (4*a^2*e*Sec[c + d*x]*Sqr t[e*Tan[c + d*x]])/(3*d) + (2*a^2*(e*Tan[c + d*x])^(5/2))/(5*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 = 2.20 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06
| method | result | size |
| parts | \(\frac {2 a^{2} e \left (\sqrt {e \tan \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \tan \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \tan \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d}+\frac {2 a^{2} \left (e \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}-\frac {2 a^{2} \sqrt {e \tan \left (d x +c \right )}\, e \left (\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )-2 \sec \left (d x +c \right )\right )}{3 d}\) | \(295\) |
| 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 \left (48+40 \sec \left (d x +c \right )+12 \sec \left (d x +c \right )^{2}+\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 {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 (-15 \cot \left (d x +c \right )-15 \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 (10 \cot \left (d x +c \right )+10 \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 {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 (-15 \cot \left (d x +c \right )-15 \csc \left (d x +c \right )\right )+i \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 {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 (15 \cot \left (d x +c \right )+15 \csc \left (d x +c \right )\right )+i \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 {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 (-15 \cot \left (d x +c \right )-15 \csc \left (d x +c \right )\right )\right )}{60 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}}}}\) | \(596\) |
Input:
int((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
2*a^2/d*e*((e*tan(d*x+c))^(1/2)-1/8*(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)))+2/5*a^2*(e*tan(d*x+c))^(5/2)/d/e-2/3*a^2/d*(e*tan(d*x+c))^(1/2)* e*((2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*Elli pticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(-cot(d*x+c)+csc(d*x+c )+1)^(1/2)*(csc(d*x+c)+cot(d*x+c))-2*sec(d*x+c))
Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=a^{2} \left (\int \left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int 2 \left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx + \int \left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*sec(d*x+c))**2*(e*tan(d*x+c))**(3/2),x)
Output:
a**2*(Integral((e*tan(c + d*x))**(3/2), x) + Integral(2*(e*tan(c + d*x))** (3/2)*sec(c + d*x), x) + Integral((e*tan(c + d*x))**(3/2)*sec(c + d*x)**2, x))
Exception generated. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),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
\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \tan \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^2*(e*tan(d*x + c))^(3/2), x)
Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \] Input:
int((e*tan(c + d*x))^(3/2)*(a + a/cos(c + d*x))^2,x)
Output:
int((e*tan(c + d*x))^(3/2)*(a + a/cos(c + d*x))^2, x)
\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx=\frac {\sqrt {e}\, a^{2} e \left (6 \sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}+20 \sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right )+30 \sqrt {\tan \left (d x +c \right )}-15 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )}d x \right ) d -3 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\tan \left (d x +c \right )}d x \right ) d -10 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sec \left (d x +c \right )}{\tan \left (d x +c \right )}d x \right ) d \right )}{15 d} \] Input:
int((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),x)
Output:
(sqrt(e)*a**2*e*(6*sqrt(tan(c + d*x))*sec(c + d*x)**2 + 20*sqrt(tan(c + d* x))*sec(c + d*x) + 30*sqrt(tan(c + d*x)) - 15*int(sqrt(tan(c + d*x))/tan(c + d*x),x)*d - 3*int((sqrt(tan(c + d*x))*sec(c + d*x)**2)/tan(c + d*x),x)* d - 10*int((sqrt(tan(c + d*x))*sec(c + d*x))/tan(c + d*x),x)*d))/(15*d)