Integrand size = 28, antiderivative size = 125 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=-\frac {2 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}} \] Output:
-2/15*a^4*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/e^4/cos(d*x+c)^(1/2)/(e* sec(d*x+c))^(1/2)-4/9*I*a*(a+I*a*tan(d*x+c))^3/d/(e*sec(d*x+c))^(9/2)+4/15 *I*(a^4+I*a^4*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 = 3.98 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=-\frac {i a^4 e^{i (c+d x)} \left (2+7 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}-2 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sqrt {e \sec (c+d x)}}{45 d e^5} \] Input:
Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2),x]
Output:
((-1/45*I)*a^4*E^(I*(c + d*x))*(2 + 7*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) - 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(d*e^5)
Time = 0.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3977, 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.
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -\frac {a^2 \int \frac {(i \tan (c+d x) a+a)^2}{(e \sec (c+d x))^{5/2}}dx}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 \int \frac {(i \tan (c+d x) a+a)^2}{(e \sec (c+d x))^{5/2}}dx}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -\frac {a^2 \left (\frac {a^2 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx}{5 e^2}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 \left (\frac {a^2 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 e^2}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle -\frac {a^2 \left (\frac {a^2 \int \sqrt {\cos (c+d x)}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 \left (\frac {a^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {a^2 \left (\frac {2 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{5 d (e \sec (c+d x))^{5/2}}\right )}{3 e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2),x]
Output:
(((-4*I)/9)*a*(a + I*a*Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(9/2)) - (a^2* ((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[(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. 269 vs. \(2 (109 ) = 218\).
Time = 26.88 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(-\frac {2 a^{4} \left (\sin \left (d x +c \right ) \left (-40 \cos \left (d x +c \right )^{4}-40 \cos \left (d x +c \right )^{3}+16 \cos \left (d x +c \right )^{2}+16 \cos \left (d x +c \right )+3\right )+i \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (3 \cos \left (d x +c \right )+6+3 \sec \left (d x +c \right )\right )+i \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (-3 \cos \left (d x +c \right )-6-3 \sec \left (d x +c \right )\right )+i \left (40 \cos \left (d x +c \right )^{5}+40 \cos \left (d x +c \right )^{4}-36 \cos \left (d x +c \right )^{3}-36 \cos \left (d x +c \right )^{2}\right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{4}}\) | \(270\) |
| risch | \(-\frac {i \left (5 \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}-6\right ) a^{4} \sqrt {2}}{45 d \,e^{4} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}+\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a^{4} \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{15 d \,e^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(339\) |
| parts | \(\frac {2 a^{4} \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 {2 a^{4} \left (\sin \left (d x +c \right ) \left (5 \cos \left (d x +c \right )^{4}+5 \cos \left (d x +c \right )^{3}-11 \cos \left (d x +c \right )^{2}-11 \cos \left (d x +c \right )+12\right )+12 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 )-12 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 )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{4}}-\frac {4 i a^{4} \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 {8 i a^{4}}{9 d \left (e \sec \left (d x +c \right )\right )^{\frac {9}{2}}}-\frac {4 a^{4} \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}}\) | \(715\) |
Input:
int((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x,method=_RETURNVERBOSE)
Output:
-2/45*a^4/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)/e^4*(sin(d*x+c)*(-40*cos(d *x+c)^4-40*cos(d*x+c)^3+16*cos(d*x+c)^2+16*cos(d*x+c)+3)+I*EllipticF(I*(co t(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1 ))^(1/2)*(3*cos(d*x+c)+6+3*sec(d*x+c))+I*EllipticE(I*(cot(d*x+c)-csc(d*x+c )),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-3*cos(d *x+c)-6-3*sec(d*x+c))+I*(40*cos(d*x+c)^5+40*cos(d*x+c)^4-36*cos(d*x+c)^3-3 6*cos(d*x+c)^2))
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\frac {-6 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-5 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 7 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 i \, a^{4} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{45 \, d e^{5}} \] Input:
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="fricas")
Output:
1/45*(-6*I*sqrt(2)*a^4*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse( -4, 0, e^(I*d*x + I*c))) + sqrt(2)*(-5*I*a^4*e^(5*I*d*x + 5*I*c) - 7*I*a^4 *e^(3*I*d*x + 3*I*c) - 2*I*a^4*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. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(d*x+c))**4/(e*sec(d*x+c))**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(9/2), x)
\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="giac")
Output:
integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(9/2), x)
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)^4/(e/cos(c + d*x))^(9/2),x)
Output:
int((a + a*tan(c + d*x)*1i)^4/(e/cos(c + d*x))^(9/2), x)
\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx=\frac {\sqrt {e}\, a^{4} \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x +\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{5}}d x -4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{5}}d x \right ) i -6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{5}}d x \right )+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right )^{5}}d x \right ) i \right )}{e^{5}} \] Input:
int((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x)
Output:
(sqrt(e)*a**4*(int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x) + int((sqrt(sec(c + d*x))*tan(c + d*x)**4)/sec(c + d*x)**5,x) - 4*int((sqrt(sec(c + d*x))*t an(c + d*x)**3)/sec(c + d*x)**5,x)*i - 6*int((sqrt(sec(c + d*x))*tan(c + d *x)**2)/sec(c + d*x)**5,x) + 4*int((sqrt(sec(c + d*x))*tan(c + d*x))/sec(c + d*x)**5,x)*i))/e**5