Integrand size = 26, antiderivative size = 124 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d} \]
-2/7*I*a*(e*cos(d*x+c))^(7/2)/d+10/21*a*(e*cos(d*x+c))^(7/2)*(cos(1/2*d*x+ 1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d /cos(d*x+c)^(7/2)+2/7*a*(e*cos(d*x+c))^(7/2)*tan(d*x+c)/d+10/21*a*(e*cos(d *x+c))^(7/2)*sec(d*x+c)^2*tan(d*x+c)/d
Time = 1.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\frac {a e^3 \sqrt {e \cos (c+d x)} (\cos (d x)-i \sin (d x)) \left (10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (c+d x)-i \sin (c+d x))+\sqrt {\cos (c+d x)} (-8 i+2 i \cos (2 (c+d x))+5 \sin (2 (c+d x)))\right ) (\cos (c+2 d x)+i \sin (c+2 d x))}{21 d \sqrt {\cos (c+d x)}} \]
(a*e^3*Sqrt[e*Cos[c + d*x]]*(Cos[d*x] - I*Sin[d*x])*(10*EllipticF[(c + d*x )/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]) + Sqrt[Cos[c + d*x]]*(-8*I + (2*I) *Cos[2*(c + d*x)] + 5*Sin[2*(c + d*x)]))*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x ]))/(21*d*Sqrt[Cos[c + d*x]])
Time = 0.86 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3998, 3042, 3967, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int (a+i a \tan (c+d x)) (e \cos (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x)) (e \cos (c+d x))^{7/2}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \int \frac {i \tan (c+d x) a+a}{(e \sec (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \int \frac {i \tan (c+d x) a+a}{(e \sec (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \int \frac {1}{(e \sec (c+d x))^{7/2}}dx-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \left (\frac {\int \sqrt {e \sec (c+d x)}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \left (\frac {\int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (a \left (\frac {5 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )-\frac {2 i a}{7 d (e \sec (c+d x))^{7/2}}\right )\) |
(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*((((-2*I)/7)*a)/(d*(e*Sec[c + d*x])^(7/2)) + a*((2*Sin[c + d*x])/(7*d*e*(e*Sec[c + d*x])^(5/2)) + (5*( (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d *e^2) + (2*Sin[c + d*x])/(3*d*e*Sqrt[e*Sec[c + d*x]])))/(7*e^2)))
3.7.55.3.1 Defintions of rubi rules used
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_.), x_Symbol] :> Simp[(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m Int[( a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m , n}, x] && !IntegerQ[m]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[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]
Time = 9.94 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85
| method | result | size |
| parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 i a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d}\) | \(229\) |
| default | \(-\frac {2 a \,e^{4} \left (48 i \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+72 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}+3 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(241\) |
-2/21*a*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^4*(48* cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-72* cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c) ^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*cos(1/2*d*x+1/2*c))/( -e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c) /(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d-2/7*I*a*(e*cos(d*x+c))^(7/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\frac {{\left (-20 i \, \sqrt {2} a e^{\frac {7}{2}} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {\frac {1}{2}} {\left (-3 i \, a e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, a e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{42 \, d} \]
1/42*(-20*I*sqrt(2)*a*e^(7/2)*e^(I*d*x + I*c)*weierstrassPInverse(-4, 0, e ^(I*d*x + I*c)) + sqrt(1/2)*(-3*I*a*e^3*e^(4*I*d*x + 4*I*c) - 16*I*a*e^3*e ^(2*I*d*x + 2*I*c) + 7*I*a*e^3)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I* d*x - 1/2*I*c))*e^(-I*d*x - I*c)/d
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
\[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]