3.2.93 \(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx\) [193]

Optimal. Leaf size=138 \[ -\frac {14 a^2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {14 a^2 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d} \]

[Out]

14/15*I*a^2*(e*sec(d*x+c))^(3/2)/d-14/5*a^2*e^2*(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)+14/5*a^2*e*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/d+2
/5*I*(e*sec(d*x+c))^(3/2)*(a^2+I*a^2*tan(d*x+c))/d

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Rubi [A]
time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3579, 3567, 3853, 3856, 2719} \begin {gather*} -\frac {14 a^2 e^2 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 {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {14 a^2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2,x]

[Out]

(-14*a^2*e^2*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((14*I)/15)*a^2*(e*Se
c[c + d*x])^(3/2))/d + (14*a^2*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (((2*I)/5)*(e*Sec[c + d*x])^(3/2)*
(a^2 + I*a^2*Tan[c + d*x]))/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 3579

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 - 1)/(f*(m + n - 1))), x] + Dist[a*((m + 2*n - 2)/(m + n - 1)), Int[(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] &&
GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

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))^{3/2} (a+i a \tan (c+d x))^2 \, dx &=\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {1}{5} (7 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx\\ &=\frac {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {1}{5} \left (7 a^2\right ) \int (e \sec (c+d x))^{3/2} \, dx\\ &=\frac {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {14 a^2 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \left (7 a^2 e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx\\ &=\frac {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {14 a^2 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}-\frac {\left (7 a^2 e^2\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {14 a^2 e^2 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 {14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac {14 a^2 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{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 = 2.57, size = 267, normalized size = 1.93 \begin {gather*} \frac {(e \sec (c+d x))^{3/2} \left (-\frac {14 i \sqrt {2} \left (3 \sqrt {1+e^{2 i (c+d x)}}-e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}}}+\frac {1}{2} \csc (c) \sec ^{\frac {5}{2}}(c+d x) (\cos (2 c)-i \sin (2 c)) (36 \cos (d x)+27 \cos (2 c+d x)+21 \cos (2 c+3 d x)-20 i \sin (d x)+20 i \sin (2 c+d x))\right ) (a+i a \tan (c+d x))^2}{15 d \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2,x]

[Out]

((e*Sec[c + d*x])^(3/2)*(((-14*I)*Sqrt[2]*(3*Sqrt[1 + E^((2*I)*(c + d*x))] - E^((2*I)*d*x)*(-1 + E^((2*I)*c))*
Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/((-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I
)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (Csc[c]*Sec[c + d*x]^(5/2)*(Cos[2*c] - I*Sin[2*c])*(36*Cos[d*x
] + 27*Cos[2*c + d*x] + 21*Cos[2*c + 3*d*x] - (20*I)*Sin[d*x] + (20*I)*Sin[2*c + d*x]))/2)*(a + I*a*Tan[c + d*
x])^2)/(15*d*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^2)

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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. 373 vs. \(2 (143 ) = 286\).
time = 0.49, size = 374, normalized size = 2.71

method	result	size
default	\(\frac {2 a^{2} \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+21 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 \left (\cos ^{3}\left (d x +c \right )\right )+10 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+24 \left (\cos ^{2}\left (d x +c \right )\right )-3\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{15 d \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

2/15*a^2/d*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^2*(21*I*cos(d*x+c)^3*(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)*sin(d*x+c)-21*I*cos(d*x+c)^3*(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)*sin(d*x+c)+21*I*cos(d*x+c)^2*(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)*sin(d*x+c)-21*
I*cos(d*x+c)^2*(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)*sin(d*x+c)-21*cos(d*x+c)^3+10*I*cos(d*x+c)*sin(d*x+c)+24*cos(d*x+c)^2-3)*(e/cos(d*x+c))^(3/2)/sin(d*x+c)
^5/cos(d*x+c)

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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))^(3/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

e^(3/2)*integrate((I*a*tan(d*x + c) + a)^2*sec(d*x + c)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 166, normalized size = 1.20 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (21 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c + \frac {3}{2}\right )} + 16 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c + \frac {3}{2}\right )} + 7 i \, a^{2} e^{\left (i \, d x + i \, c + \frac {3}{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^{2} e^{\frac {3}{2}} + i \, \sqrt {2} a^{2} e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 2 i \, \sqrt {2} a^{2} e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, 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))^(3/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-2/15*(sqrt(2)*(21*I*a^2*e^(5*I*d*x + 5*I*c + 3/2) + 16*I*a^2*e^(3*I*d*x + 3*I*c + 3/2) + 7*I*a^2*e^(I*d*x + I
*c + 3/2))*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^2*e^(3/2) + I*sqrt(2)*a^2*e
^(4*I*d*x + 4*I*c + 3/2) + 2*I*sqrt(2)*a^2*e^(2*I*d*x + 2*I*c + 3/2))*weierstrassZeta(-4, 0, weierstrassPInver
se(-4, 0, e^(I*d*x + I*c))))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{2} \left (\int \left (- \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))**(3/2)*(a+I*a*tan(d*x+c))**2,x)

[Out]

-a**2*(Integral(-(e*sec(c + d*x))**(3/2), x) + Integral((e*sec(c + d*x))**(3/2)*tan(c + d*x)**2, x) + Integral
(-2*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x), x))

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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))^(3/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^2*e^(3/2)*sec(d*x + c)^(3/2), 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 )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^2,x)

[Out]

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^2, x)

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