\(\int \tan ^m(c+d x) (b \tan (c+d x))^n (A+B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [46]

Optimal result

Integrand size = 39, antiderivative size = 154 \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x) (b \tan (c+d x))^n}{d (2+m+n)} \]

[Out]

C*tan(d*x+c)^(1+m)*(b*tan(d*x+c))^n/d/(1+m+n)+(A-C)*hypergeom([1, 1/2+1/2*m+1/2*n],[3/2+1/2*m+1/2*n],-tan(d*x+
c)^2)*tan(d*x+c)^(1+m)*(b*tan(d*x+c))^n/d/(1+m+n)+B*hypergeom([1, 1+1/2*m+1/2*n],[2+1/2*m+1/2*n],-tan(d*x+c)^2
)*tan(d*x+c)^(2+m)*(b*tan(d*x+c))^n/d/(2+m+n)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {20, 3711, 3619, 3557, 371} \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {(A-C) \tan ^{m+1}(c+d x) (b \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),-\tan ^2(c+d x)\right )}{d (m+n+1)}+\frac {B \tan ^{m+2}(c+d x) (b \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n+2),\frac {1}{2} (m+n+4),-\tan ^2(c+d x)\right )}{d (m+n+2)}+\frac {C \tan ^{m+1}(c+d x) (b \tan (c+d x))^n}{d (m+n+1)} \]

[In]

Int[Tan[c + d*x]^m*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(C*Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n)/(d*(1 + m + n)) + ((A - C)*Hypergeometric2F1[1, (1 + m + n)/2, (3
+ m + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n)/(d*(1 + m + n)) + (B*Hypergeometric2F1[1
, (2 + m + n)/2, (4 + m + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m)*(b*Tan[c + d*x])^n)/(d*(2 + m + n))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3619

Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*T
an[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ
[c^2 + d^2, 0] &&  !IntegerQ[2*m]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \left (\tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx \\ & = \frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\left (\tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) (A-C+B \tan (c+d x)) \, dx \\ & = \frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\left (B \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{1+m+n}(c+d x) \, dx+\left ((A-C) \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) \, dx \\ & = \frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {\left (B \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \text {Subst}\left (\int \frac {x^{1+m+n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left ((A-C) \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \text {Subst}\left (\int \frac {x^{m+n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x) (b \tan (c+d x))^n}{d (2+m+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\tan ^{1+m}(c+d x) (b \tan (c+d x))^n \left (\frac {C}{1+m+n}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\tan ^2(c+d x)\right )}{1+m+n}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),-\tan ^2(c+d x)\right ) \tan (c+d x)}{2+m+n}\right )}{d} \]

[In]

Integrate[Tan[c + d*x]^m*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n*(C/(1 + m + n) + ((A - C)*Hypergeometric2F1[1, (1 + m + n)/2, (3 + m
+ n)/2, -Tan[c + d*x]^2])/(1 + m + n) + (B*Hypergeometric2F1[1, (2 + m + n)/2, (4 + m + n)/2, -Tan[c + d*x]^2]
*Tan[c + d*x])/(2 + m + n)))/d

Maple [F]

\[\int \tan \left (d x +c \right )^{m} \left (b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )+C \tan \left (d x +c \right )^{2}\right )d x\]

[In]

int(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

int(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)

Fricas [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d*x + c)^m, x)

Sympy [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int \left (b \tan {\left (c + d x \right )}\right )^{n} \left (A + B \tan {\left (c + d x \right )} + C \tan ^{2}{\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}\, dx \]

[In]

integrate(tan(d*x+c)**m*(b*tan(d*x+c))**n*(A+B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

Integral((b*tan(c + d*x))**n*(A + B*tan(c + d*x) + C*tan(c + d*x)**2)*tan(c + d*x)**m, x)

Maxima [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d*x + c)^m, x)

Giac [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d*x + c)^m, x)

Mupad [F(-1)]

Timed out. \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n\,\left (C\,{\mathrm {tan}\left (c+d\,x\right )}^2+B\,\mathrm {tan}\left (c+d\,x\right )+A\right ) \,d x \]

[In]

int(tan(c + d*x)^m*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x)^2),x)

[Out]

int(tan(c + d*x)^m*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x)^2), x)