3.2.8 \(\int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx\) [108]

Optimal. Leaf size=43 \[ \frac {a (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {4038} \begin {gather*} \frac {a \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4038

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

Rubi steps

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

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Mathematica [A]
time = 0.31, size = 73, normalized size = 1.70 \begin {gather*} \frac {c (-1+2 \cos (e+f x)) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*
Sec[e + f*x]])/(4*f)

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Maple [A]
time = 2.67, size = 72, normalized size = 1.67

method	result	size
default	\(-\frac {\sin \left (f x +e \right ) \left (3 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{2 f \left (-1+\cos \left (f x +e \right )\right )^{2}}\)	\(72\)
risch	\(\frac {2 i c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{2 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) f}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (40) = 80\).
time = 0.56, size = 324, normalized size = 7.53 \begin {gather*} \frac {2 \, {\left (2 \, c \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - {\left (c \sin \left (3 \, f x + 3 \, e\right ) - c \sin \left (2 \, f x + 2 \, e\right ) + c \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (c \cos \left (3 \, f x + 3 \, e\right ) - c \cos \left (2 \, f x + 2 \, e\right ) + c \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - {\left (2 \, c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} \sin \left (3 \, f x + 3 \, e\right ) + {\left (2 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (2 \, f x + 2 \, e\right ) - c \sin \left (f x + 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2*(2*c*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 2*c*cos(2*f*x + 2*e)*sin(f*x + e) - (c*sin(3*f*x + 3*e) - c*sin(2*f
*x + 2*e) + c*sin(f*x + e))*cos(4*f*x + 4*e) + (c*cos(3*f*x + 3*e) - c*cos(2*f*x + 2*e) + c*cos(f*x + e))*sin(
4*f*x + 4*e) - (2*c*cos(2*f*x + 2*e) + c)*sin(3*f*x + 3*e) + (2*c*cos(f*x + e) + c)*sin(2*f*x + 2*e) - c*sin(f
*x + 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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (40) = 80\).
time = 2.76, size = 85, normalized size = 1.98 \begin {gather*} \frac {{\left (2 \, c \cos \left (f x + e\right ) - 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 )}}}{2 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}} \sec {\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (37) = 74\).
time = 1.57, size = 83, normalized size = 1.93 \begin {gather*} \frac {2 \, {\left (2 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3} + c^{4}\right )} \sqrt {-a c} {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

2*(2*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^3 + c^4)*sqrt(-a*c)*abs(c)*sgn(tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/
2*e))/((c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^2*f)

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Mupad [B]
time = 2.64, size = 78, normalized size = 1.81 \begin {gather*} \frac {c\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\left (\sin \left (e+f\,x\right )-\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )}{f\,{\sin \left (2\,e+2\,f\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(e + f*x))^(1/2)*(c - c/cos(e + f*x))^(3/2))/cos(e + f*x),x)

[Out]

(c*(c - c/cos(e + f*x))^(1/2)*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*(sin(e + f*x) - sin(2*e + 2*f*x) + s
in(3*e + 3*f*x)))/(f*sin(2*e + 2*f*x)^2)

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