\(\int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) [889]

Optimal result

Integrand size = 23, antiderivative size = 159 \[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]

[Out]

1/5*AppellF1(5/2,1,-n,7/2,-I*tan(d*x+c),-b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^n/d/cot(d*x+c)^(5/2)/((1+b*tan(d*x+c
)/a)^n)+1/5*AppellF1(5/2,1,-n,7/2,I*tan(d*x+c),-b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^n/d/cot(d*x+c)^(5/2)/((1+b*ta
n(d*x+c)/a)^n)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4326, 3656, 926, 129, 525, 524} \[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {(a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]

[In]

Int[(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2),x]

[Out]

(AppellF1[5/2, 1, -n, 7/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/(5*d*Cot[c + d*x]
^(5/2)*(1 + (b*Tan[c + d*x])/a)^n) + (AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*
Tan[c + d*x])^n)/(5*d*Cot[c + d*x]^(5/2)*(1 + (b*Tan[c + d*x])/a)^n)

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 3656

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

Rule 4326

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx \\ & = \frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {i x^{3/2} (a+b x)^n}{2 (i-x)}+\frac {i x^{3/2} (a+b x)^n}{2 (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{i-x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{i+x} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^n}{i-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^n}{i+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {x^4 \left (1+\frac {b x^2}{a}\right )^n}{i-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {x^4 \left (1+\frac {b x^2}{a}\right )^n}{i+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \]

[In]

Integrate[(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2),x]

[Out]

Integrate[(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x]

Maple [F]

\[\int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n}}{\cot \left (d x +c \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*tan(d*x+c))^n/cot(d*x+c)^(3/2),x)

[Out]

int((a+b*tan(d*x+c))^n/cot(d*x+c)^(3/2),x)

Fricas [F]

\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*tan(d*x+c))^n/cot(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

integral((b*tan(d*x + c) + a)^n/cot(d*x + c)^(3/2), x)

Sympy [F]

\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

[In]

integrate((a+b*tan(d*x+c))**n/cot(d*x+c)**(3/2),x)

[Out]

Integral((a + b*tan(c + d*x))**n/cot(c + d*x)**(3/2), x)

Maxima [F]

\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*tan(d*x+c))^n/cot(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n/cot(d*x + c)^(3/2), x)

Giac [F]

\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*tan(d*x+c))^n/cot(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^n/cot(d*x + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*tan(c + d*x))^n/cot(c + d*x)^(3/2),x)

[Out]

int((a + b*tan(c + d*x))^n/cot(c + d*x)^(3/2), x)