Integrand size = 25, antiderivative size = 127 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {b \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),-\tan ^2(e+f x)\right ) \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)} \]
a*hypergeom([1, 1/2*n*p+1/2],[1/2*n*p+3/2],-tan(f*x+e)^2)*tan(f*x+e)*(c*(d *tan(f*x+e))^p)^n/f/(n*p+1)+b*hypergeom([1, 1/2*n*p+1],[1/2*n*p+2],-tan(f* x+e)^2)*tan(f*x+e)^2*(c*(d*tan(f*x+e))^p)^n/f/(n*p+2)
Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\frac {\tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (a (2+n p) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\tan ^2(e+f x)\right )+b (1+n p) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n p}{2},2+\frac {n p}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)\right )}{f (1+n p) (2+n p)} \]
(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a*(2 + n*p)*Hypergeometric2F1[1, ( 1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] + b*(1 + n*p)*Hypergeometric2F1[ 1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]))/(f*(1 + n*p)* (2 + n*p))
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4853, 2042, 557, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a+b \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^ndx\) |
\(\Big \downarrow \) 4853 |
\(\displaystyle \frac {\int \frac {\left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 2042 |
\(\displaystyle \frac {\tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n \int \frac {\tan ^{n p}(e+f x) (a+b \tan (e+f x))}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 557 |
\(\displaystyle \frac {\tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (a \int \frac {\tan ^{n p}(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)+b \int \frac {\tan ^{n p+1}(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)\right )}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (\frac {a \tan ^{n p+1}(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),-\tan ^2(e+f x)\right )}{n p+1}+\frac {b \tan ^{n p+2}(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),-\tan ^2(e+f x)\right )}{n p+2}\right )}{f}\) |
((c*(d*Tan[e + f*x])^p)^n*((a*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/ 2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n*p))/(1 + n*p) + (b*Hypergeometric2 F1[1, (2 + n*p)/2, (4 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(2 + n*p))/( 2 + n*p)))/(f*Tan[e + f*x]^(n*p))
3.14.26.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[(u_.)*((c_.)*((d_)*((a_.) + (b_.)*(x_)))^(q_))^(p_), x_Symbol] :> Simp[ (c*(d*(a + b*x))^q)^p/(a + b*x)^(p*q) Int[u*(a + b*x)^(p*q), x], x] /; Fr eeQ[{a, b, c, d, q, p}, x] && !IntegerQ[q] && !IntegerQ[p]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Tan[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x ]]
\[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \tan \left (f x +e \right )\right )d x\]
\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\int \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \]
\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
Timed out. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x)) \, dx=\int {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]