\(\int \csc (e+f x) (a+b \sec ^2(e+f x))^p \, dx\) [136]

Optimal result

Integrand size = 21, antiderivative size = 77 \[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right ) \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{f} \]

[Out]

-AppellF1(1/2,1,-p,3/2,sec(f*x+e)^2,-b*sec(f*x+e)^2/a)*sec(f*x+e)*(a+b*sec(f*x+e)^2)^p/f/((1+b*sec(f*x+e)^2/a)
^p)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4219, 441, 440} \[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=-\frac {\sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (\frac {b \sec ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right )}{f} \]

[In]

Int[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]

[Out]

-((AppellF1[1/2, 1, -p, 3/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^p)/(
f*(1 + (b*Sec[e + f*x]^2)/a)^p))

Rule 440

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

Rule 441

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

Rule 4219

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Cos[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^
n)^p/x^(m + 1)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (Gt
Q[m, 0] || EqQ[n, 2] || EqQ[n, 4])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\left (\left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right ) \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1532\) vs. \(2(77)=154\).

Time = 16.05 (sec) , antiderivative size = 1532, normalized size of antiderivative = 19.90 \[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\frac {(a+2 b+a \cos (2 (e+f x)))^p \csc (e+f x) \sec ^2(e+f x)^p \left (a+b \sec ^2(e+f x)\right )^p \left (\frac {2 \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}-\operatorname {AppellF1}\left (1,\frac {1}{2},-p,2,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \tan ^2(e+f x) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}\right )}{2 f \left (-a p (a+2 b+a \cos (2 (e+f x)))^{-1+p} \sec ^2(e+f x)^p \sin (2 (e+f x)) \left (\frac {2 \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}-\operatorname {AppellF1}\left (1,\frac {1}{2},-p,2,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \tan ^2(e+f x) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}\right )+p (a+2 b+a \cos (2 (e+f x)))^p \sec ^2(e+f x)^p \tan (e+f x) \left (\frac {2 \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}-\operatorname {AppellF1}\left (1,\frac {1}{2},-p,2,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \tan ^2(e+f x) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}\right )+\frac {1}{2} (a+2 b+a \cos (2 (e+f x)))^p \sec ^2(e+f x)^p \left (\frac {2 \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {4 (a+b) p \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-1-p} \sqrt {\csc ^2(e+f x)} \sqrt {\sec ^2(e+f x)}}{b (1+2 p)}+\frac {2 \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \left (-\frac {2 (a+b) \left (-\frac {1}{2}-p\right ) p \operatorname {AppellF1}\left (\frac {1}{2}-p,-\frac {1}{2},1-p,\frac {3}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \csc ^2(e+f x)}{b \left (\frac {1}{2}-p\right )}-\frac {\left (-\frac {1}{2}-p\right ) \operatorname {AppellF1}\left (\frac {1}{2}-p,\frac {1}{2},-p,\frac {3}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \csc ^2(e+f x)}{\frac {1}{2}-p}\right ) \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {2 \operatorname {AppellF1}\left (-\frac {1}{2}-p,-\frac {1}{2},-p,\frac {1}{2}-p,-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right ) \left (1+\frac {(a+b) \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)} \tan (e+f x)}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {2 b p \operatorname {AppellF1}\left (1,\frac {1}{2},-p,2,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \sec ^2(e+f x) \tan ^3(e+f x) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-1-p}}{a+b}-2 \operatorname {AppellF1}\left (1,\frac {1}{2},-p,2,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \sec ^2(e+f x) \tan (e+f x) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}-\tan ^2(e+f x) \left (\frac {b p \operatorname {AppellF1}\left (2,\frac {1}{2},1-p,3,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \sec ^2(e+f x) \tan (e+f x)}{a+b}-\frac {1}{2} \operatorname {AppellF1}\left (2,\frac {3}{2},-p,3,-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \sec ^2(e+f x) \tan (e+f x)\right ) \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}\right )\right )} \]

[In]

Integrate[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]

[Out]

((a + 2*b + a*Cos[2*(e + f*x)])^p*Csc[e + f*x]*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*((2*AppellF1[-1/2 -
p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + (
(a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*
x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b + b*Tan[e + f*x]^2)/(a + b))^p))/(2*f*(-(a*p*(a + 2*b + a*Cos[2*(e + f
*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2,
-(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e
+ f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b
+ b*Tan[e + f*x]^2)/(a + b))^p)) + p*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]*((2*Appe
llF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 +
2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b
*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b + b*Tan[e + f*x]^2)/(a + b))^p) + ((a + 2*b + a*Cos[2*(e +
f*x)])^p*(Sec[e + f*x]^2)^p*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]
^2)/b)]*Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2])
+ (4*(a + b)*p*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e +
f*x]*(1 + ((a + b)*Cot[e + f*x]^2)/b)^(-1 - p)*Sqrt[Csc[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])/(b*(1 + 2*p)) + (2*(
(-2*(a + b)*(-1/2 - p)*p*AppellF1[1/2 - p, -1/2, 1 - p, 3/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b
)]*Cot[e + f*x]*Csc[e + f*x]^2)/(b*(1/2 - p)) - ((-1/2 - p)*AppellF1[1/2 - p, 1/2, -p, 3/2 - p, -Cot[e + f*x]^
2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]^2)/(1/2 - p))*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1
+ ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]
^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)
/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*b*p*AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*S
ec[e + f*x]^2*Tan[e + f*x]^3*((a + b + b*Tan[e + f*x]^2)/(a + b))^(-1 - p))/(a + b) - (2*AppellF1[1, 1/2, -p,
2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/((a + b + b*Tan[e + f*x]^2)/(a
+ b))^p - (Tan[e + f*x]^2*((b*p*AppellF1[2, 1/2, 1 - p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Se
c[e + f*x]^2*Tan[e + f*x])/(a + b) - (AppellF1[2, 3/2, -p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*
Sec[e + f*x]^2*Tan[e + f*x])/2))/((a + b + b*Tan[e + f*x]^2)/(a + b))^p))/2))

Maple [F]

\[\int \csc \left (f x +e \right ) \left (a +b \sec \left (f x +e \right )^{2}\right )^{p}d x\]

[In]

int(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x)

[Out]

int(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x)

Fricas [F]

\[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right ) \,d x } \]

[In]

integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e), x)

Sympy [F(-1)]

Timed out. \[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\text {Timed out} \]

[In]

integrate(csc(f*x+e)*(a+b*sec(f*x+e)**2)**p,x)

[Out]

Timed out

Maxima [F]

\[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right ) \,d x } \]

[In]

integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e)^2 + a)^p*csc(f*x + e), x)

Giac [F]

\[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right ) \,d x } \]

[In]

integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e)^2 + a)^p*csc(f*x + e), x)

Mupad [F(-1)]

Timed out. \[ \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p}{\sin \left (e+f\,x\right )} \,d x \]

[In]

int((a + b/cos(e + f*x)^2)^p/sin(e + f*x),x)

[Out]

int((a + b/cos(e + f*x)^2)^p/sin(e + f*x), x)