\(\int \frac {\tan ^m(c+d x) (A+B \tan (c+d x)+C \tan ^2(c+d x))}{\sqrt {b \tan (c+d x)}} \, dx\) [48]

Optimal result

Integrand size = 41, antiderivative size = 170 \[ \int \frac {\tan ^m(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right )}{\sqrt {b \tan (c+d x)}} \, dx=\frac {2 C \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {2 (A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {2 B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 m),\frac {1}{4} (7+2 m),-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (3+2 m) \sqrt {b \tan (c+d x)}} \]

[Out]

2*C*tan(d*x+c)^(1+m)/d/(1+2*m)/(b*tan(d*x+c))^(1/2)+2*(A-C)*hypergeom([1, 1/4+1/2*m],[5/4+1/2*m],-tan(d*x+c)^2
)*tan(d*x+c)^(1+m)/d/(1+2*m)/(b*tan(d*x+c))^(1/2)+2*B*hypergeom([1, 3/4+1/2*m],[7/4+1/2*m],-tan(d*x+c)^2)*tan(
d*x+c)^(2+m)/d/(3+2*m)/(b*tan(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {20, 3711, 3619, 3557, 371} \[ \int \frac {\tan ^m(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right )}{\sqrt {b \tan (c+d x)}} \, dx=\frac {2 (A-C) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2 m+1),\frac {1}{4} (2 m+5),-\tan ^2(c+d x)\right )}{d (2 m+1) \sqrt {b \tan (c+d x)}}+\frac {2 B \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2 m+3),\frac {1}{4} (2 m+7),-\tan ^2(c+d x)\right )}{d (2 m+3) \sqrt {b \tan (c+d x)}}+\frac {2 C \tan ^{m+1}(c+d x)}{d (2 m+1) \sqrt {b \tan (c+d x)}} \]

[In]

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

[Out]

(2*C*Tan[c + d*x]^(1 + m))/(d*(1 + 2*m)*Sqrt[b*Tan[c + d*x]]) + (2*(A - C)*Hypergeometric2F1[1, (1 + 2*m)/4, (
5 + 2*m)/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + 2*m)*Sqrt[b*Tan[c + d*x]]) + (2*B*Hypergeometric2F1
[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(3 + 2*m)*Sqrt[b*Tan[c + d*x]])

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}& = \frac {\sqrt {\tan (c+d x)} \int \tan ^{-\frac {1}{2}+m}(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx}{\sqrt {b \tan (c+d x)}} \\ & = \frac {2 C \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {\sqrt {\tan (c+d x)} \int \tan ^{-\frac {1}{2}+m}(c+d x) (A-C+B \tan (c+d x)) \, dx}{\sqrt {b \tan (c+d x)}} \\ & = \frac {2 C \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {\left (B \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {1}{2}+m}(c+d x) \, dx}{\sqrt {b \tan (c+d x)}}+\frac {\left ((A-C) \sqrt {\tan (c+d x)}\right ) \int \tan ^{-\frac {1}{2}+m}(c+d x) \, dx}{\sqrt {b \tan (c+d x)}} \\ & = \frac {2 C \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {\left (B \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^{\frac {1}{2}+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {b \tan (c+d x)}}+\frac {\left ((A-C) \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^{-\frac {1}{2}+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {b \tan (c+d x)}} \\ & = \frac {2 C \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {2 (A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+2 m) \sqrt {b \tan (c+d x)}}+\frac {2 B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 m),\frac {1}{4} (7+2 m),-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (3+2 m) \sqrt {b \tan (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \frac {\tan ^m(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right )}{\sqrt {b \tan (c+d x)}} \, dx=\frac {2 \tan ^{1+m}(c+d x) \left (C (3+2 m)+(A-C) (3+2 m) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),-\tan ^2(c+d x)\right )+B (1+2 m) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 m),\frac {1}{4} (7+2 m),-\tan ^2(c+d x)\right ) \tan (c+d x)\right )}{d (1+2 m) (3+2 m) \sqrt {b \tan (c+d x)}} \]

[In]

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

[Out]

(2*Tan[c + d*x]^(1 + m)*(C*(3 + 2*m) + (A - C)*(3 + 2*m)*Hypergeometric2F1[1, (1 + 2*m)/4, (5 + 2*m)/4, -Tan[c
+ d*x]^2] + B*(1 + 2*m)*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[c + d*x]^2]*Tan[c + d*x]))/(d*(1
+ 2*m)*(3 + 2*m)*Sqrt[b*Tan[c + d*x]])

Maple [F]

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

[In]

int(tan(d*x+c)^m*(A+B*tan(d*x+c)+C*tan(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x)

[Out]

int(tan(d*x+c)^m*(A+B*tan(d*x+c)+C*tan(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x)

Fricas [F]

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

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c)+C*tan(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(tan(d*x+c)**m*(A+B*tan(d*x+c)+C*tan(d*x+c)**2)/(b*tan(d*x+c))**(1/2),x)

[Out]

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

Maxima [F(-1)]

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

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c)+C*tan(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

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

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c)+C*tan(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int((tan(c + d*x)^m*(A + B*tan(c + d*x) + C*tan(c + d*x)^2))/(b*tan(c + d*x))^(1/2),x)

[Out]

int((tan(c + d*x)^m*(A + B*tan(c + d*x) + C*tan(c + d*x)^2))/(b*tan(c + d*x))^(1/2), x)