3.7.62 \(\int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx\) [662]

Optimal. Leaf size=130 \[ \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]

[Out]

2/7*I*a/d/(e*cos(d*x+c))^(7/2)-6/5*a*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)*Elliptic
E(sin(1/2*d*x+1/2*c),2^(1/2))/d/(e*cos(d*x+c))^(7/2)+2/5*a*cos(d*x+c)*sin(d*x+c)/d/(e*cos(d*x+c))^(7/2)+6/5*a*
cos(d*x+c)^3*sin(d*x+c)/d/(e*cos(d*x+c))^(7/2)

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Rubi [A]
time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567, 3853, 3856, 2719} \begin {gather*} \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]

[Out]

(((2*I)/7)*a)/(d*(e*Cos[c + d*x])^(7/2)) - (6*a*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(5*d*(e*Cos[c +
d*x])^(7/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2)) + (6*a*Cos[c + d*x]^3*Sin[c + d*x]
)/(5*d*(e*Cos[c + d*x])^(7/2))

Rule 2719

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]

Rule 3567

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] + Dist[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])

Rule 3596

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[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]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps

\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {a \int (e \sec (c+d x))^{7/2} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {\left (3 a e^2\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a \cos ^{\frac {7}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 (e \cos (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.23, size = 596, normalized size = 4.58 \begin {gather*} -\frac {i \cos ^{\frac {9}{2}}(c+d x) \left (-\frac {3 i}{5}+\frac {3 \cot (c)}{5}\right ) \left (-\frac {2 i \sqrt {2} e^{i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )}{3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}-\frac {2 \left (\cos \left (\frac {c}{2}\right )+i \sin \left (\frac {c}{2}\right )\right )^2 \left (2 i \cos (c+d x)+F\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right ) (-i \cos (c+d x)-\sin (c+d x)) \sqrt {1-i \cos (c+d x)+\sin (c+d x)} \sqrt {-i \cos (c+d x)+\cos (2 (c+d x))+\sin (c+d x)+i \sin (2 (c+d x))}+E\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right ) \sqrt {1-i \cos (c+d x)+\sin (c+d x)} (i \cos (c+d x)+\sin (c+d x)) \sqrt {-i \cos (c+d x)+\cos (2 (c+d x))+\sin (c+d x)+i \sin (2 (c+d x))}\right )}{\sqrt {\cos (c+d x)}}\right ) (a+i a \tan (c+d x))}{2 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}+\frac {\cos ^5(c+d x) \left (\csc (c) \sec (c) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right )+\sec ^4(c+d x) \left (\frac {2}{7} i \cos (c)+\frac {2 \sin (c)}{7}\right )+\sec (c) \sec ^3(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \sin (d x)+\sec (c) \sec (c+d x) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right ) \sin (d x)+\sec ^2(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \tan (c)\right ) (a+i a \tan (c+d x))}{d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]

[Out]

((-1/2*I)*Cos[c + d*x]^(9/2)*((-3*I)/5 + (3*Cot[c])/5)*((((-2*I)/3)*Sqrt[2]*E^(I*d*x)*Sqrt[1 + E^((2*I)*(c + d
*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]
- (2*(Cos[c/2] + I*Sin[c/2])^2*((2*I)*Cos[c + d*x] + EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]],
-1]*((-I)*Cos[c + d*x] - Sin[c + d*x])*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*Sqrt[(-I)*Cos[c + d*x] + Cos[2
*(c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]] + EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -
1]*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x])*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c +
d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]]))/Sqrt[Cos[c + d*x]])*(a + I*a*Tan[c + d*x]))/(d*(e*Cos[c + d*x])^
(7/2)*(Cos[d*x] + I*Sin[d*x])) + (Cos[c + d*x]^5*(Csc[c]*Sec[c]*((6*Cos[c])/5 - ((6*I)/5)*Sin[c]) + Sec[c + d*
x]^4*(((2*I)/7)*Cos[c] + (2*Sin[c])/7) + Sec[c]*Sec[c + d*x]^3*((2*Cos[c])/5 - ((2*I)/5)*Sin[c])*Sin[d*x] + Se
c[c]*Sec[c + d*x]*((6*Cos[c])/5 - ((6*I)/5)*Sin[c])*Sin[d*x] + Sec[c + d*x]^2*((2*Cos[c])/5 - ((2*I)/5)*Sin[c]
)*Tan[c])*(a + I*a*Tan[c + d*x]))/(d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (137 ) = 274\).
time = 3.67, size = 396, normalized size = 3.05

method	result	size
default	\(\frac {2 \left (336 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+252 \EllipticE \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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+280 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-126 \EllipticE \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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-5 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{35 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \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}\, e^{3} d}\)	\(396\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/35/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*
d*x+1/2*c)^2*e+e)^(1/2)/e^3*(336*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-168*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli
pticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^6-504*cos(1/2*d*x+1/2*c)
*sin(1/2*d*x+1/2*c)^6+252*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/
2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4+280*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-126*EllipticE(cos(1/2*d*x+1/2
*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2-56*sin(1/2*d*x
+1/2*c)^2*cos(1/2*d*x+1/2*c)+21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/
2*d*x+1/2*c),2^(1/2))-5*I*sin(1/2*d*x+1/2*c))*a/d

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

e^(-7/2)*integrate((I*a*tan(d*x + c) + a)/cos(d*x + c)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.08, size = 221, normalized size = 1.70 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (21 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 77 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 23 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {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 )} + 21 \, {\left (i \, \sqrt {2} a e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{\frac {7}{2}} + d e^{\left (8 i \, d x + 8 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-2/35*(2*sqrt(1/2)*(21*I*a*e^(8*I*d*x + 8*I*c) + 77*I*a*e^(6*I*d*x + 6*I*c) + 23*I*a*e^(4*I*d*x + 4*I*c) + 7*I
*a*e^(2*I*d*x + 2*I*c))*sqrt(e^(2*I*d*x + 2*I*c) + 1)*e^(-1/2*I*d*x - 1/2*I*c) + 21*(I*sqrt(2)*a*e^(8*I*d*x +
8*I*c) + 4*I*sqrt(2)*a*e^(6*I*d*x + 6*I*c) + 6*I*sqrt(2)*a*e^(4*I*d*x + 4*I*c) + 4*I*sqrt(2)*a*e^(2*I*d*x + 2*
I*c) + I*sqrt(2)*a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(d*e^(7/2) + d*e^(8*I
*d*x + 8*I*c + 7/2) + 4*d*e^(6*I*d*x + 6*I*c + 7/2) + 6*d*e^(4*I*d*x + 4*I*c + 7/2) + 4*d*e^(2*I*d*x + 2*I*c +
7/2))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6192 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)*e^(-7/2)/cos(d*x + c)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2),x)

[Out]

int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2), x)

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