\(\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx\) [95]

Optimal result

Integrand size = 30, antiderivative size = 103 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]

[Out]

a^2*c^2*ln(cos(f*x+e))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)+1/2*a^2*c^2*tan(f*x+e)^3/f/(
a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3990, 3554, 3556} \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]

[In]

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

[Out]

(a^2*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^2*c^2*Tan[
e + f*x]^3)/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3990

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 c^2 \tan (e+f x)\right ) \int \tan ^3(e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\left (a^2 c^2 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a^2 c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.64 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \left (2 \log (\cos (e+f x))+\sec ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]

[In]

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

[Out]

(a^2*c^2*(2*Log[Cos[e + f*x]] + Sec[e + f*x]^2)*Tan[e + f*x])/(2*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e
+ f*x]])

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {a \left (2 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+2 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-2 \cos \left (f x +e \right )^{2} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+\sin \left (f x +e \right )^{2}\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right ) c \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \csc \left (f x +e \right )}{2 f \left (\cos \left (f x +e \right )-1\right )}\)	\(143\)
risch	\(-\frac {a c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+{\mathrm e}^{4 i \left (f x +e \right )} f x +2 \,{\mathrm e}^{4 i \left (f x +e \right )} e +2 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+2 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +2 i {\mathrm e}^{2 i \left (f x +e \right )}+4 \,{\mathrm e}^{2 i \left (f x +e \right )} e +i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+f x +2 e \right )}{\left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\)	\(238\)

[In]

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

[Out]

1/2/f*a*(2*cos(f*x+e)^2*ln(-cot(f*x+e)+csc(f*x+e)-1)+2*cos(f*x+e)^2*ln(-cot(f*x+e)+csc(f*x+e)+1)-2*cos(f*x+e)^
2*ln(2/(cos(f*x+e)+1))+sin(f*x+e)^2)*(-c*(sec(f*x+e)-1))^(1/2)*(sec(f*x+e)-1)*c*(a*(sec(f*x+e)+1))^(1/2)/(cos(
f*x+e)-1)*csc(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 346, normalized size of antiderivative = 3.36 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\left [\frac {\sqrt {-a c} a c \cos \left (f x + e\right ) \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right ) - a c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{2 \, f \cos \left (f x + e\right )}, \frac {2 \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right ) \cos \left (f x + e\right ) - a c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{2 \, f \cos \left (f x + e\right )}\right ] \]

[In]

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

[Out]

[1/2*(sqrt(-a*c)*a*c*cos(f*x + e)*log(1/2*(a*c*cos(f*x + e)^4 - (cos(f*x + e)^3 + cos(f*x + e))*sqrt(-a*c)*sqr
t((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin(f*x + e) + a*c)/cos(f*x + e)^
2) - a*c*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin(f*x + e))/(f*cos(
f*x + e)), 1/2*(2*sqrt(a*c)*a*c*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e)
- c)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e)/(a*c*cos(f*x + e)^2 + a*c))*cos(f*x + e) - a*c*sqrt((a*cos(f*x +
e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e))]

Sympy [F]

\[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a*(sec(e + f*x) + 1))**(3/2)*(-c*(sec(e + f*x) - 1))**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (93) = 186\).

Time = 0.40 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.63 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=-\frac {{\left ({\left (f x + e\right )} a c \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (f x + e\right )} a c \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a c - 2 \, a c \sin \left (2 \, f x + 2 \, e\right ) - {\left (a c \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a c \cos \left (2 \, f x + 2 \, e\right )^{2} + a c \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a c \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a c \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c + 2 \, {\left (2 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a c - a c \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a c \sin \left (2 \, f x + 2 \, e\right ) + a c \cos \left (2 \, f x + 2 \, e\right )\right )} \sin \left (4 \, f x + 4 \, e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]

[In]

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

[Out]

-((f*x + e)*a*c*cos(4*f*x + 4*e)^2 + 4*(f*x + e)*a*c*cos(2*f*x + 2*e)^2 + (f*x + e)*a*c*sin(4*f*x + 4*e)^2 + 4
*(f*x + e)*a*c*sin(2*f*x + 2*e)^2 + 4*(f*x + e)*a*c*cos(2*f*x + 2*e) + (f*x + e)*a*c - 2*a*c*sin(2*f*x + 2*e)
- (a*c*cos(4*f*x + 4*e)^2 + 4*a*c*cos(2*f*x + 2*e)^2 + a*c*sin(4*f*x + 4*e)^2 + 4*a*c*sin(4*f*x + 4*e)*sin(2*f
*x + 2*e) + 4*a*c*sin(2*f*x + 2*e)^2 + 4*a*c*cos(2*f*x + 2*e) + a*c + 2*(2*a*c*cos(2*f*x + 2*e) + a*c)*cos(4*f
*x + 4*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + 2*(2*(f*x + e)*a*c*cos(2*f*x + 2*e) + (f*x + e)*a
*c - a*c*sin(2*f*x + 2*e))*cos(4*f*x + 4*e) + 2*(2*(f*x + e)*a*c*sin(2*f*x + 2*e) + a*c*cos(2*f*x + 2*e))*sin(
4*f*x + 4*e))*sqrt(a)*sqrt(c)/((2*(2*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + cos(4*f*x + 4*e)^2 + 4*cos(2*f*x
+ 2*e)^2 + sin(4*f*x + 4*e)^2 + 4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*sin(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*
e) + 1)*f)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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