Integrand size = 28, antiderivative size = 124 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}} \] Output:
2/15*a^3*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/e^4/cos(d*x+c)^(1/2)/(e*s ec(d*x+c))^(1/2)-2/9*I*(a+I*a*tan(d*x+c))^3/d/(e*sec(d*x+c))^(9/2)-4/15*I* (a^3+I*a^3*tan(d*x+c))/d/e^2/(e*sec(d*x+c))^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.87 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=-\frac {a^3 e^{-2 i (c+d x)} \left (11+16 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+4 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (-i+\tan (c+d x))^3}{90 d e^2 (e \sec (c+d x))^{5/2}} \] Input:
Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(9/2),x]
Output:
-1/90*(a^3*(11 + 16*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 4*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d* x))])*(-I + Tan[c + d*x])^3)/(d*e^2*E^((2*I)*(c + d*x))*(e*Sec[c + d*x])^( 5/2))
Time = 0.64 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3978, 3042, 3977, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {a \int \frac {(i \tan (c+d x) a+a)^2}{(e \sec (c+d x))^{5/2}}dx}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {(i \tan (c+d x) a+a)^2}{(e \sec (c+d x))^{5/2}}dx}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle \frac {a \left (\frac {a^2 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx}{5 e^2}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {a^2 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 e^2}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a \left (\frac {a^2 \int \sqrt {\cos (c+d x)}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {a^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a \left (\frac {2 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(9/2),x]
Output:
(((-2*I)/9)*(a + I*a*Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(9/2)) + (a*((2* a^2*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((4*I)/5)*(a^2 + I*a^2*Tan[c + d*x]))/(d*(e*Sec[c + d*x])^(5/2)) ))/(3*e^2)
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[2*b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^( n - 1)/(f*m)), x] - Simp[b^2*((m + 2*n - 2)/(d^2*m)) Int[(d*Sec[e + f*x]) ^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILtQ[m, 0] & & LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/( a*f*m)), x] + Simp[a*((m + n)/(m*d^2)) Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b ^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (108 ) = 216\).
Time = 18.48 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.08
| method | result | size |
| default | \(\frac {2 a^{3} \left (\sin \left (d x +c \right ) \left (20 \cos \left (d x +c \right )^{4}+20 \cos \left (d x +c \right )^{3}+\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+3\right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )+3 i \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )+i \left (-20 \cos \left (d x +c \right )^{5}-20 \cos \left (d x +c \right )^{4}+9 \cos \left (d x +c \right )^{3}+9 \cos \left (d x +c \right )^{2}\right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{4}}\) | \(258\) |
| risch | \(-\frac {i \left (5 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+12\right ) a^{3} \sqrt {2}}{90 d \,e^{4} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a^{3} \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{15 d \,e^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(339\) |
| parts | \(\frac {2 a^{3} \left (\sin \left (d x +c \right ) \left (5 \cos \left (d x +c \right )^{4}+5 \cos \left (d x +c \right )^{3}+7 \cos \left (d x +c \right )^{2}+7 \cos \left (d x +c \right )+21\right )-21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{4}}-\frac {i a^{3} \left (\frac {2 \cos \left (d x +c \right )^{4}}{9}-\frac {2 \cos \left (d x +c \right )^{2}}{5}\right )}{d \,e^{4} \sqrt {e \sec \left (d x +c \right )}}-\frac {2 i a^{3}}{3 d \left (e \sec \left (d x +c \right )\right )^{\frac {9}{2}}}-\frac {2 a^{3} \left (\sin \left (d x +c \right ) \left (-5 \cos \left (d x +c \right )^{4}-5 \cos \left (d x +c \right )^{3}+2 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+6\right )-6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{4}}\) | \(498\) |
Input:
int((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(9/2),x,method=_RETURNVERBOSE)
Output:
2/45*a^3/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)/e^4*(sin(d*x+c)*(20*cos(d*x +c)^4+20*cos(d*x+c)^3+cos(d*x+c)^2+cos(d*x+c)+3)-3*I*(1/(cos(d*x+c)+1))^(1 /2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+2+sec(d*x+c))*EllipticE( I*(cot(d*x+c)-csc(d*x+c)),I)+3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos (d*x+c)+1))^(1/2)*(cos(d*x+c)+2+sec(d*x+c))*EllipticF(I*(cot(d*x+c)-csc(d* x+c)),I)+I*(-20*cos(d*x+c)^5-20*cos(d*x+c)^4+9*cos(d*x+c)^3+9*cos(d*x+c)^2 ))
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {12 i \, \sqrt {2} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-5 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 16 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 11 i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{90 \, d e^{5}} \] Input:
integrate((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(9/2),x, algorithm="fricas")
Output:
1/90*(12*I*sqrt(2)*a^3*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse( -4, 0, e^(I*d*x + I*c))) + sqrt(2)*(-5*I*a^3*e^(5*I*d*x + 5*I*c) - 16*I*a^ 3*e^(3*I*d*x + 3*I*c) - 11*I*a^3*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I *c) + 1))*e^(1/2*I*d*x + 1/2*I*c))/(d*e^5)
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(d*x+c))**3/(e*sec(d*x+c))**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(9/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)^3/(e*sec(d*x + c))^(9/2), x)
\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(9/2),x, algorithm="giac")
Output:
integrate((I*a*tan(d*x + c) + a)^3/(e*sec(d*x + c))^(9/2), x)
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)^3/(e/cos(c + d*x))^(9/2),x)
Output:
int((a + a*tan(c + d*x)*1i)^3/(e/cos(c + d*x))^(9/2), x)
\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {\sqrt {e}\, a^{3} \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x -\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{5}}d x \right ) i -3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{5}}d x \right )+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right )^{5}}d x \right ) i \right )}{e^{5}} \] Input:
int((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(9/2),x)
Output:
(sqrt(e)*a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x) - int((sqrt(sec(c + d*x))*tan(c + d*x)**3)/sec(c + d*x)**5,x)*i - 3*int((sqrt(sec(c + d*x)) *tan(c + d*x)**2)/sec(c + d*x)**5,x) + 3*int((sqrt(sec(c + d*x))*tan(c + d *x))/sec(c + d*x)**5,x)*i))/e**5