3.2.100 \(\int (a+b \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx\) [200]

3.2.100.1 Optimal result

Integrand size = 25, antiderivative size = 380 \[ \int (a+b \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx=-\frac {2 (a-b) \sqrt {a+b} (3 b c+4 a d) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 b f}+\frac {2 \sqrt {a+b} \left (a b (6 c-4 d)-b^2 (3 c-d)+3 a^2 d\right ) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 b f}-\frac {2 a \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {2 b d \sqrt {a+b \sec (e+f x)} \tan (e+f x)}{3 f} \]

output

-2/3*(a-b)*(4*a*d+3*b*c)*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b) 
^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b 
*(1+sec(f*x+e))/(a-b))^(1/2)/b/f+2/3*(a*b*(6*c-4*d)-b^2*(3*c-d)+3*a^2*d)*c 
ot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2) 
)*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/ 
2)/b/f-2*a*c*cot(f*x+e)*EllipticPi((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),(a+b 
)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1 
+sec(f*x+e))/(a-b))^(1/2)/f+2/3*b*d*(a+b*sec(f*x+e))^(1/2)*tan(f*x+e)/f
 

3.2.100.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(6063\) vs. \(2(380)=760\).

Time = 23.57 (sec) , antiderivative size = 6063, normalized size of antiderivative = 15.96 \[ \int (a+b \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.2.100.3 Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4406, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4406

\(\displaystyle \frac {2}{3} \int \frac {3 c a^2+b (3 b c+4 a d) \sec ^2(e+f x)+\left (3 d a^2+6 b c a+b^2 d\right ) \sec (e+f x)}{2 \sqrt {a+b \sec (e+f x)}}dx+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 c a^2+b (3 b c+4 a d) \sec ^2(e+f x)+\left (3 d a^2+6 b c a+b^2 d\right ) \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \frac {3 c a^2+b (3 b c+4 a d) \csc \left (e+f x+\frac {\pi }{2}\right )^2+\left (3 d a^2+6 b c a+b^2 d\right ) \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{3} \left (\int \frac {3 c a^2+\left (3 d a^2+6 b c a+b^2 d-b (3 b c+4 a d)\right ) \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx+b (4 a d+3 b c) \int \frac {\sec (e+f x) (\sec (e+f x)+1)}{\sqrt {a+b \sec (e+f x)}}dx\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (\int \frac {3 c a^2+\left (3 d a^2+6 b c a+b^2 d-b (3 b c+4 a d)\right ) \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b (4 a d+3 b c) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {1}{3} \left (\left (3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right ) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx+3 a^2 c \int \frac {1}{\sqrt {a+b \sec (e+f x)}}dx+b (4 a d+3 b c) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (\left (3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right ) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+3 a^2 c \int \frac {1}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b (4 a d+3 b c) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {1}{3} \left (\left (3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right ) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b (4 a d+3 b c) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {6 a c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {1}{3} \left (b (4 a d+3 b c) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b f}-\frac {6 a c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{3} \left (\frac {2 \sqrt {a+b} \left (3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b f}-\frac {2 (a-b) \sqrt {a+b} (4 a d+3 b c) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b f}-\frac {6 a c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}\right )+\frac {2 b d \tan (e+f x) \sqrt {a+b \sec (e+f x)}}{3 f}\)

input

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

output

((-2*(a - b)*Sqrt[a + b]*(3*b*c + 4*a*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqr 
t[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + 
f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) + (2*Sqrt[a 
 + b]*(a*b*(6*c - 4*d) - b^2*(3*c - d) + 3*a^2*d)*Cot[e + f*x]*EllipticF[A 
rcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - 
 Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - 
(6*a*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Se 
c[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a 
+ b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f)/3 + (2*b*d*Sqrt[a + b*Se 
c[e + f*x]]*Tan[e + f*x])/(3*f)
 

3.2.100.3.1 Defintions of rubi rules used

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 3042

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

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4406

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d 
_.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m 
 - 1)/(f*m)), x] + Simp[1/m   Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2*c*m 
 + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m 
 - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b* 
c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

3.2.100.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2983\) vs. \(2(345)=690\).

Time = 24.79 (sec) , antiderivative size = 2984, normalized size of antiderivative = 7.85

method	result	size
parts	\(\text {Expression too large to display}\)	\(2984\)
default	\(\text {Expression too large to display}\)	\(2988\)

input

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

output

2*c/f*(EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a* 
cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*a^2*co 
s(f*x+e)^2-2*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b) 
*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)* 
a*b*cos(f*x+e)^2-EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/( 
a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1 
/2)*b^2*cos(f*x+e)^2+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos( 
f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b)) 
^(1/2))*a*b*cos(f*x+e)^2+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*( 
cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a 
+b))^(1/2))*b^2*cos(f*x+e)^2-2*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^( 
1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticPi(cot(f*x+e)-csc(f*x+e),-1 
,((a-b)/(a+b))^(1/2))*a^2*cos(f*x+e)^2+2*EllipticF(cot(f*x+e)-csc(f*x+e),( 
(a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f 
*x+e)/(cos(f*x+e)+1))^(1/2)*a^2*cos(f*x+e)-4*EllipticF(cot(f*x+e)-csc(f*x+ 
e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c 
os(f*x+e)/(cos(f*x+e)+1))^(1/2)*a*b*cos(f*x+e)-2*EllipticF(cot(f*x+e)-csc( 
f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2 
)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*b^2*cos(f*x+e)+2*(1/(a+b)*(b+a*cos(f*x 
+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(...
 

3.2.100.5 Fricas [F]

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

input

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

output

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

3.2.100.6 Sympy [F]

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

input

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

output

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

3.2.100.7 Maxima [F]

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

input

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

output

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

3.2.100.8 Giac [F]

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

input

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

output

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

3.2.100.9 Mupad [F(-1)]

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

input

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

output

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