3.2.85 \(\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx\) [185]

Optimal. Leaf size=123 \[ -\frac {6 a e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {6 a e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \]

[Out]

2/7*I*a*(e*sec(d*x+c))^(7/2)/d+2/5*a*e*(e*sec(d*x+c))^(5/2)*sin(d*x+c)/d-6/5*a*e^4*(cos(1/2*d*x+1/2*c)^2)^(1/2
)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+6/5*a*e^3*s
in(d*x+c)*(e*sec(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3567, 3853, 3856, 2719} \begin {gather*} -\frac {6 a e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {6 a e^3 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {2 a e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a*e^4*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((2*I)/7)*a*(e*Sec[c + d
*x])^(7/2))/d + (6*a*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*e*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x]
)/(5*d)

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 2.16, size = 156, normalized size = 1.27 \begin {gather*} \frac {a e e^{-i d x} (e \sec (c+d x))^{5/2} (\cos (d x)-i \sin (d x)) (\cos (c+3 d x)+i \sin (c+3 d x)) \left (-36 i-28 i \cos (2 (c+d x))+7 i e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+7 \sec (c+d x) \sin (3 (c+d x))+27 \tan (c+d x)\right )}{70 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e*(e*Sec[c + d*x])^(5/2)*(Cos[d*x] - I*Sin[d*x])*(Cos[c + 3*d*x] + I*Sin[c + 3*d*x])*(-36*I - (28*I)*Cos[2*
(c + d*x)] + ((7*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^
((2*I)*(c + d*x)) + 7*Sec[c + d*x]*Sin[3*(c + d*x)] + 27*Tan[c + d*x]))/(70*d*E^(I*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. 364 vs. \(2 (130 ) = 260\).
time = 6.70, size = 365, normalized size = 2.97

method	result	size
default	\(-\frac {2 a \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (21 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-21 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+21 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{3}\left (d x +c \right )\right )-5 i \sin \left (d x +c \right )-7 \cos \left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}}}{35 d \sin \left (d x +c \right )^{5}}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*a/d*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^2*(21*I*cos(d*x+c)^4*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c
)/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)-21*I*cos(d*x+c)^4*sin(d*x+c)*(1/(1+cos(d*x+c
)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)+21*I*cos(d*x+c)^3*sin(d*
x+c)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)-21*I
*cos(d*x+c)^3*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(I*(-1+cos(d*x+c)
)/sin(d*x+c),I)+21*cos(d*x+c)^4-14*cos(d*x+c)^3-5*I*sin(d*x+c)-7*cos(d*x+c))*(e/cos(d*x+c))^(7/2)/sin(d*x+c)^5

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

[Out]

e^(7/2)*integrate((I*a*tan(d*x + c) + a)*sec(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 = 0.11, size = 195, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (21 i \, a e^{\left (7 i \, d x + 7 i \, c + \frac {7}{2}\right )} + 77 i \, a e^{\left (5 i \, d x + 5 i \, c + \frac {7}{2}\right )} + 23 i \, a e^{\left (3 i \, d x + 3 i \, c + \frac {7}{2}\right )} + 7 i \, a e^{\left (i \, d x + i \, c + \frac {7}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 21 \, {\left (i \, \sqrt {2} a e^{\frac {7}{2}} + i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\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^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(sqrt(2)*(21*I*a*e^(7*I*d*x + 7*I*c + 7/2) + 77*I*a*e^(5*I*d*x + 5*I*c + 7/2) + 23*I*a*e^(3*I*d*x + 3*I*
c + 7/2) + 7*I*a*e^(I*d*x + I*c + 7/2))*e^(1/2*I*d*x + 1/2*I*c)/sqrt(e^(2*I*d*x + 2*I*c) + 1) + 21*(I*sqrt(2)*
a*e^(7/2) + I*sqrt(2)*a*e^(6*I*d*x + 6*I*c + 7/2) + 3*I*sqrt(2)*a*e^(4*I*d*x + 4*I*c + 7/2) + 3*I*sqrt(2)*a*e^
(2*I*d*x + 2*I*c + 7/2))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(d*e^(6*I*d*x +
6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 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((e*sec(d*x+c))^(7/2)*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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