3.102 \(\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=190 \[ -\frac{a^2 c^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}} \]

[Out]

(a^3*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*Sqrt
[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*
x])/(2*f*Sqrt[c - c*Sec[e + f*x]]) + (c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x
]])

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Rubi [A]  time = 0.362838, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3909, 3906, 3905, 3475} \[ -\frac{a^2 c^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]

[Out]

(a^3*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*Sqrt
[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*
x])/(2*f*Sqrt[c - c*Sec[e + f*x]]) + (c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x
]])

Rule 3909

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(3/2)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Si
mp[(-2*a^2*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[a, Int[Sqrt[
a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] &&
EqQ[a^2 - b^2, 0] &&  !LeQ[n, -2^(-1)]

Rule 3906

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

Rule 3905

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]

Rule 3475

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

Rubi steps

\begin{align*} \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx &=\frac{c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}+c \int (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}+(a c) \int (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{a^2 c^2 \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}+\left (a^2 c\right ) \int \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{a^2 c^2 \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{\left (a^3 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^3 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 \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 1.16559, size = 149, normalized size = 0.78 \[ \frac{a^2 c \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left (6 \cos (2 (e+f x))+3 i f x \cos (3 (e+f x))+\left (-9 \log \left (1+e^{2 i (e+f x)}\right )+9 i f x-6\right ) \cos (e+f x)-3 \log \left (1+e^{2 i (e+f x)}\right ) \cos (3 (e+f x))+2\right )}{24 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]

[Out]

(a^2*c*Csc[(e + f*x)/2]*(2 + 6*Cos[2*(e + f*x)] + (3*I)*f*x*Cos[3*(e + f*x)] + Cos[e + f*x]*(-6 + (9*I)*f*x -
9*Log[1 + E^((2*I)*(e + f*x))]) - 3*Cos[3*(e + f*x)]*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*
x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)

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Maple [A]  time = 0.281, size = 199, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}}{6\,f\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) }\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}} \left ( 6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) -7\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,\cos \left ( fx+e \right ) +2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x)

[Out]

-1/6/f*a^2*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*(c*(-1+cos(f*x+e))/cos(f*x+e))^(3/2)*(6*cos(f*x+e)^3*ln((1-co
s(f*x+e)+sin(f*x+e))/sin(f*x+e))+6*cos(f*x+e)^3*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-6*cos(f*x+e)^3*ln(2
/(1+cos(f*x+e)))-7*cos(f*x+e)^3-6*cos(f*x+e)^2+3*cos(f*x+e)+2)/sin(f*x+e)/cos(f*x+e)/(-1+cos(f*x+e))

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Maxima [B]  time = 2.19047, size = 1831, normalized size = 9.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

-1/3*(3*(f*x + e)*a^2*c*cos(6*f*x + 6*e)^2 + 27*(f*x + e)*a^2*c*cos(4*f*x + 4*e)^2 + 27*(f*x + e)*a^2*c*cos(2*
f*x + 2*e)^2 + 3*(f*x + e)*a^2*c*sin(6*f*x + 6*e)^2 + 27*(f*x + e)*a^2*c*sin(4*f*x + 4*e)^2 + 27*(f*x + e)*a^2
*c*sin(2*f*x + 2*e)^2 + 18*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + 3*(f*x + e)*a^2*c - 6*a^2*c*sin(2*f*x + 2*e) - 3
*(a^2*c*cos(6*f*x + 6*e)^2 + 9*a^2*c*cos(4*f*x + 4*e)^2 + 9*a^2*c*cos(2*f*x + 2*e)^2 + a^2*c*sin(6*f*x + 6*e)^
2 + 9*a^2*c*sin(4*f*x + 4*e)^2 + 18*a^2*c*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a^2*c*sin(2*f*x + 2*e)^2 + 6*a
^2*c*cos(2*f*x + 2*e) + a^2*c + 2*(3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*cos(6*f*x + 6*
e) + 6*(3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*cos(4*f*x + 4*e) + 6*(a^2*c*sin(4*f*x + 4*e) + a^2*c*sin(2*f*x + 2*e
))*sin(6*f*x + 6*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + 6*(3*(f*x + e)*a^2*c*cos(4*f*x + 4*e) +
3*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + (f*x + e)*a^2*c - a^2*c*sin(4*f*x + 4*e) - a^2*c*sin(2*f*x + 2*e))*cos(6
*f*x + 6*e) + 18*(3*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + (f*x + e)*a^2*c)*cos(4*f*x + 4*e) - 6*(a^2*c*sin(6*f*x
+ 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*e))*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 4*(a^2*c*sin(6*f*x + 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*e))*cos(3/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) - 6*(a^2*c*sin(6*f*x + 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*
e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*(3*(f*x + e)*a^2*c*sin(4*f*x + 4*e) + 3*(f*x + e)
*a^2*c*sin(2*f*x + 2*e) + a^2*c*cos(4*f*x + 4*e) + a^2*c*cos(2*f*x + 2*e))*sin(6*f*x + 6*e) + 6*(9*(f*x + e)*a
^2*c*sin(2*f*x + 2*e) - a^2*c)*sin(4*f*x + 4*e) + 6*(a^2*c*cos(6*f*x + 6*e) + 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2
*c*cos(2*f*x + 2*e) + a^2*c)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(a^2*c*cos(6*f*x + 6*e)
+ 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) + 6*(a^2*c*cos(6*f*x + 6*e) + 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*sin(1/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((2*(3*cos(4*f*x + 4*e) + 3*cos(2*f*x + 2*e) + 1)*cos
(6*f*x + 6*e) + cos(6*f*x + 6*e)^2 + 6*(3*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 9*cos(4*f*x + 4*e)^2 + 9*co
s(2*f*x + 2*e)^2 + 6*(sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + sin(6*f*x + 6*e)^2 + 9*sin(4*f*x
+ 4*e)^2 + 18*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e) + 1)*f)

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Fricas [A]  time = 1.72729, size = 1152, normalized size = 6.06 \begin{align*} \left [\frac{{\left (a^{2} c \cos \left (f x + e\right )^{2} - 5 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \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 ) + 3 \,{\left (a^{2} c \cos \left (f x + e\right )^{3} + a^{2} c \cos \left (f x + e\right )^{2}\right )} \sqrt{-a c} \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 )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, \frac{{\left (a^{2} c \cos \left (f x + e\right )^{2} - 5 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \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 ) + 6 \,{\left (a^{2} c \cos \left (f x + e\right )^{3} + a^{2} c \cos \left (f x + e\right )^{2}\right )} \sqrt{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 )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

[1/6*((a^2*c*cos(f*x + e)^2 - 5*a^2*c*cos(f*x + e) - 2*a^2*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) + 3*(a^2*c*cos(f*x + e)^3 + a^2*c*cos(f*x + e)^2)*sqrt(-a*c)*log(
1/2*(a*c*cos(f*x + e)^4 - (cos(f*x + e)^3 + cos(f*x + e))*sqrt(-a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*s
qrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin(f*x + e) + a*c)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 + f*cos(f*x + e)
^2), 1/6*((a^2*c*cos(f*x + e)^2 - 5*a^2*c*cos(f*x + e) - 2*a^2*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) + 6*(a^2*c*cos(f*x + e)^3 + a^2*c*cos(f*x + e)^2)*sqrt(a*c)*a
rctan(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)*s
in(f*x + e)/(a*c*cos(f*x + e)^2 + a*c)))/(f*cos(f*x + e)^3 + f*cos(f*x + e)^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(5/2)*(c-c*sec(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out