3.4.24 \(\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx\) [324]

Optimal. Leaf size=88 \[ \frac {F_1(1+n;1-m,1;2+n;-i \tan (e+f x),i \tan (e+f x)) (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^{1+n} (a+i a \tan (e+f x))^m}{d f (1+n)} \]

[Out]

AppellF1(1+n,1-m,1,2+n,-I*tan(f*x+e),I*tan(f*x+e))*(d*tan(f*x+e))^(1+n)*(a+I*a*tan(f*x+e))^m/d/f/(1+n)/((1+I*t
an(f*x+e))^m)

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Rubi [A]
time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3645, 140, 138} \begin {gather*} \frac {(1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \tan (e+f x))^{n+1} F_1(n+1;1-m,1;n+2;-i \tan (e+f x),i \tan (e+f x))}{d f (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^m,x]

[Out]

(AppellF1[1 + n, 1 - m, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + I*a*Tan[e +
f*x])^m)/(d*f*(1 + n)*(1 + I*Tan[e + f*x])^m)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 140

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[c^IntPart[n]*((c +
d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 3645

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

Rubi steps

\begin {align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {\left (i a (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n \left (1+\frac {x}{a}\right )^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {F_1(1+n;1-m,1;2+n;-i \tan (e+f x),i \tan (e+f x)) (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^{1+n} (a+i a \tan (e+f x))^m}{d f (1+n)}\\ \end {align*}

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Mathematica [F]
time = 5.48, size = 0, normalized size = 0.00 \begin {gather*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^m,x]

[Out]

Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^m, x]

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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^m,x)

[Out]

int((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^m,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((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((I*a*tan(f*x + e) + a)^m*(d*tan(f*x + e))^n, 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((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((2*a*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1))^m*((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x +
2*I*e) + 1))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))**n*(a+I*a*tan(f*x+e))**m,x)

[Out]

Integral((d*tan(e + f*x))**n*(I*a*(tan(e + f*x) - I))**m, 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((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^m*(d*tan(f*x + e))^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(e + f*x))^n*(a + a*tan(e + f*x)*1i)^m,x)

[Out]

int((d*tan(e + f*x))^n*(a + a*tan(e + f*x)*1i)^m, x)

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