\(\int \sec (a+b x) (c \sin (a+b x))^m \, dx\) [343]

Optimal result

Integrand size = 17, antiderivative size = 48 \[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m)} \]

[Out]

hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],sin(b*x+a)^2)*(c*sin(b*x+a))^(1+m)/b/c/(1+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2644, 371} \[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\frac {(c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1)} \]

[In]

Int[Sec[a + b*x]*(c*Sin[a + b*x])^m,x]

[Out]

(Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))

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 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^m}{1-\frac {x^2}{c^2}} \, dx,x,c \sin (a+b x)\right )}{b c} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},1+\frac {1+m}{2},\sin ^2(a+b x)\right ) \sin (a+b x) (c \sin (a+b x))^m}{b (1+m)} \]

[In]

Integrate[Sec[a + b*x]*(c*Sin[a + b*x])^m,x]

[Out]

(Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x])^m)/(b*(1 + m))

Maple [F]

\[\int \sec \left (b x +a \right ) \left (c \sin \left (b x +a \right )\right )^{m}d x\]

[In]

int(sec(b*x+a)*(c*sin(b*x+a))^m,x)

[Out]

int(sec(b*x+a)*(c*sin(b*x+a))^m,x)

Fricas [F]

\[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right ) \,d x } \]

[In]

integrate(sec(b*x+a)*(c*sin(b*x+a))^m,x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^m*sec(b*x + a), x)

Sympy [F]

\[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{m} \sec {\left (a + b x \right )}\, dx \]

[In]

integrate(sec(b*x+a)*(c*sin(b*x+a))**m,x)

[Out]

Integral((c*sin(a + b*x))**m*sec(a + b*x), x)

Maxima [F]

\[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right ) \,d x } \]

[In]

integrate(sec(b*x+a)*(c*sin(b*x+a))^m,x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^m*sec(b*x + a), x)

Giac [F]

\[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right ) \,d x } \]

[In]

integrate(sec(b*x+a)*(c*sin(b*x+a))^m,x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^m*sec(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sec (a+b x) (c \sin (a+b x))^m \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{\cos \left (a+b\,x\right )} \,d x \]

[In]

int((c*sin(a + b*x))^m/cos(a + b*x),x)

[Out]

int((c*sin(a + b*x))^m/cos(a + b*x), x)