3.5.78 \(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^n \, dx\) [478]

Optimal. Leaf size=96 \[ \frac {i 2^{\frac {7}{4}+n} a \, _2F_1\left (\frac {3}{4},\frac {1}{4}-n;\frac {7}{4};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{-1+n}}{3 d} \]

[Out]

1/3*I*2^(7/4+n)*a*hypergeom([3/4, 1/4-n],[7/4],1/2-1/2*I*tan(d*x+c))*(e*sec(d*x+c))^(3/2)*(1+I*tan(d*x+c))^(1/
4-n)*(a+I*a*tan(d*x+c))^(-1+n)/d

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Rubi [A]
time = 0.14, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3586, 3604, 72, 71} \begin {gather*} \frac {i a 2^{n+\frac {7}{4}} (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left (\frac {3}{4},\frac {1}{4}-n;\frac {7}{4};\frac {1}{2} (1-i \tan (c+d x))\right )}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/3)*2^(7/4 + n)*a*Hypergeometric2F1[3/4, 1/4 - n, 7/4, (1 - I*Tan[c + d*x])/2]*(e*Sec[c + d*x])^(3/2)*(1 +
I*Tan[c + d*x])^(1/4 - n)*(a + I*a*Tan[c + d*x])^(-1 + n))/d

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 3586

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*S
ec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/2)*(a - b*Tan[e + f*x])^(m/2)), Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a
- b*Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]

Rule 3604

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[a*(c/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f,
m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^n \, dx &=\frac {(e \sec (c+d x))^{3/2} \int (a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{\frac {3}{4}+n} \, dx}{(a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{3/4}}\\ &=\frac {\left (a^2 (e \sec (c+d x))^{3/2}\right ) \text {Subst}\left (\int \frac {(a+i a x)^{-\frac {1}{4}+n}}{\sqrt [4]{a-i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{3/4}}\\ &=\frac {\left (2^{-\frac {1}{4}+n} a^2 (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{-1+n} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {1}{4}-n}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {1}{4}+n}}{\sqrt [4]{a-i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{3/4}}\\ &=\frac {i 2^{\frac {7}{4}+n} a \, _2F_1\left (\frac {3}{4},\frac {1}{4}-n;\frac {7}{4};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{-1+n}}{3 d}\\ \end {align*}

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Mathematica [A]
time = 9.36, size = 170, normalized size = 1.77 \begin {gather*} -\frac {i 2^{\frac {5}{2}+n} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{\frac {3}{2}+n} \left (1+e^{2 i (c+d x)}\right )^{\frac {3}{2}+n} \, _2F_1\left (\frac {3}{4}+n,\frac {3}{2}+n;\frac {7}{4}+n;-e^{2 i (c+d x)}\right ) \sec ^{-\frac {3}{2}-n}(c+d x) (e \sec (c+d x))^{3/2} (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (3+4 n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*2^(5/2 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2 + n)*(1 + E^((2*I)*(c + d*x))
)^(3/2 + n)*Hypergeometric2F1[3/4 + n, 3/2 + n, 7/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-3/2 - n)*(e*Sec[
c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 4*n)*(Cos[d*x] + I*Sin[d*x])^n)

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Maple [F]
time = 0.43, size = 0, normalized size = 0.00 \[\int \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx\]

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))^n,x)

[Out]

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

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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))^n,x, algorithm="maxima")

[Out]

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

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Fricas [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))^n,x, algorithm="fricas")

[Out]

integral(2*sqrt(2)*(2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^n*e^(3/2*I*d*x + 3/2*I*c + 3/2)/(e^(2*I
*d*x + 2*I*c) + 1)^(3/2), x)

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

[Out]

Integral((e*sec(c + d*x))**(3/2)*(I*a*(tan(c + d*x) - I))**n, 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))^n,x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^n*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 )}^n \,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)^n,x)

[Out]

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

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