\(\int \cos (e+f x) (a+b \sec ^2(e+f x))^{3/2} \, dx\) [244]

Optimal result

Integrand size = 23, antiderivative size = 228 \[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {b \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f}+\frac {(a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}}+\frac {b (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f \left (a+b-a \sin ^2(e+f x)\right )} \] Output:

b*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f+(a-b)*(cos(f*x+e) 
^2)^(1/2)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(sec(f*x+e)^2*(a+b-a*sin(f 
*x+e)^2))^(1/2)/f/((a+b-a*sin(f*x+e)^2)/(a+b))^(1/2)+b*(a+b)*(cos(f*x+e)^2 
)^(1/2)*EllipticF(sin(f*x+e),(a/(a+b))^(1/2))*((a+b-a*sin(f*x+e)^2)/(a+b)) 
^(1/2)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f/(a+b-a*sin(f*x+e)^2)
 

Mathematica [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx \] Input:

Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2), x]
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4636, 2057, 2058, 315, 25, 27, 399, 323, 321, 330, 327}

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.

\(\displaystyle \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \sec (e+f x)^2\right )^{3/2}}{\sec (e+f x)}dx\)

\(\Big \downarrow \) 4636

\(\displaystyle \frac {\int \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{3/2}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2057

\(\displaystyle \frac {\int \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{3/2}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^{3/2}}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}-\int -\frac {a \left (-\left ((a-b) \sin ^2(e+f x)\right )+a+b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\int \frac {a \left (-\left ((a-b) \sin ^2(e+f x)\right )+a+b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \int \frac {-\left ((a-b) \sin ^2(e+f x)\right )+a+b}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \left (\frac {b (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}+\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \left (\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \left (\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \left (\frac {(a-b) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (a \left (\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}+\frac {(a-b) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

Input:

Int[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x 
]^2)]*((b*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/Sqrt[1 - Sin[e + f* 
x]^2] + a*(((a - b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b 
- a*Sin[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (b*(a + b) 
*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a 
 + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))))/(f*Sqrt[a + b - a*Sin[e + f* 
x]^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4636

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
 Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x 
, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] 
 && IntegerQ[n/2] &&  !IntegerQ[p]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 9.90 (sec) , antiderivative size = 1987, normalized size of antiderivative = 8.71

method	result	size
default	\(\text {Expression too large to display}\)	\(1987\)

Input:

int(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/f/(2*I*a^(1/2)*b^(1/2)-a+b)/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*cos( 
f*x+e)^2*((1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x 
+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a 
^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e)))^(1/2)*a^3*EllipticE(((2*I*a 
^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(f*x+e)),(-(4*I*a^(3/2)*b^ 
(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*(cos(f*x+e)^3+2*c 
os(f*x+e)^2+cos(f*x+e))+(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b 
^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)* 
cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e)))^(1/2)*a^2*b*E 
llipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(f*x+e)),( 
-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*( 
cos(f*x+e)^3+2*cos(f*x+e)^2+cos(f*x+e))+(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f* 
x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I* 
a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e) 
))^(1/2)*a*b^2*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x+ 
e)-csc(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/( 
a+b)^2)^(1/2))*(-cos(f*x+e)^3-2*cos(f*x+e)^2-cos(f*x+e))+(1/(a+b)*(I*a^(1/ 
2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1 
/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a 
-b)/(1+cos(f*x+e)))^(1/2)*b^3*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b...
 

Fricas [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right ) \,d x } \] Input:

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

integral((b*cos(f*x + e)*sec(f*x + e)^2 + a*cos(f*x + e))*sqrt(b*sec(f*x + 
 e)^2 + a), x)
 

Sympy [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cos {\left (e + f x \right )}\, dx \] Input:

integrate(cos(f*x+e)*(a+b*sec(f*x+e)**2)**(3/2),x)
 

Output:

Integral((a + b*sec(e + f*x)**2)**(3/2)*cos(e + f*x), x)
 

Maxima [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right ) \,d x } \] Input:

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^(3/2)*cos(f*x + e), x)
 

Giac [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right ) \,d x } \] Input:

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^(3/2)*cos(f*x + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \cos \left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2} \,d x \] Input:

int(cos(e + f*x)*(a + b/cos(e + f*x)^2)^(3/2),x)
 

Output:

int(cos(e + f*x)*(a + b/cos(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sec \left (f x +e \right )^{2} b +a}\, \cos \left (f x +e \right ) \sec \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\sec \left (f x +e \right )^{2} b +a}\, \cos \left (f x +e \right )d x \right ) a \] Input:

int(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(sec(e + f*x)**2*b + a)*cos(e + f*x)*sec(e + f*x)**2,x)*b + int(sq 
rt(sec(e + f*x)**2*b + a)*cos(e + f*x),x)*a