3.2.7 \(\int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx\) [107]

Optimal. Leaf size=43 \[ \frac {a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]

[Out]

1/3*a*(c-c*sec(f*x+e))^(5/2)*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))^{5/2}}{3 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])^(5/2),x]

[Out]

(a*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*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))^{5/2} \, dx &=\frac {a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(43)=86\).
time = 0.47, size = 87, normalized size = 2.02 \begin {gather*} \frac {c^2 (5-6 \cos (e+f x)+3 \cos (2 (e+f x))) \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 (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}}{12 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])^(5/2),x]

[Out]

(c^2*(5 - 6*Cos[e + f*x] + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Se
c[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(12*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(37)=74\).
time = 2.52, size = 82, normalized size = 1.91

method	result	size
default	\(-\frac {\sin \left (f x +e \right ) \left (7 \left (\cos ^{2}\left (f x +e \right )\right )-4 \cos \left (f x +e \right )+1\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}}{3 f \left (-1+\cos \left (f x +e \right )\right )^{3}}\)	\(82\)
risch	\(\frac {2 i c^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-6 \,{\mathrm e}^{4 i \left (f x +e \right )}+10 \,{\mathrm e}^{3 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} f}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c-c*sec(f*x+e))^(5/2)*(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/f*sin(f*x+e)*(7*cos(f*x+e)^2-4*cos(f*x+e)+1)*(c*(-1+cos(f*x+e))/cos(f*x+e))^(5/2)*(a*(cos(f*x+e)+1)/cos(f
*x+e))^(1/2)/(-1+cos(f*x+e))^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 688 vs. \(2 (40) = 80\).
time = 0.56, size = 688, normalized size = 16.00 \begin {gather*} \frac {2 \, {\left (30 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 9 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 3 \, c^{2} \sin \left (f x + e\right ) - {\left (3 \, c^{2} \sin \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 9 \, {\left (c^{2} \sin \left (4 \, f x + 4 \, e\right ) + c^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (5 \, f x + 5 \, e\right ) - 3 \, {\left (10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (3 \, c^{2} \cos \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (5 \, f x + 5 \, e\right ) + 3 \, {\left (10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (4 \, f x + 4 \, e\right ) - 10 \, {\left (3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a} \sqrt {c}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, f x + 4 \, e\right ) + 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (6 \, f x + 6 \, e\right ) + \cos \left (6 \, f x + 6 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + 9 \, \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, {\left (\sin \left (4 \, f x + 4 \, e\right ) + \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) + \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, \sin \left (4 \, f x + 4 \, e\right )^{2} + 18 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \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))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2/3*(30*c^2*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 9*c^2*cos(2*f*x + 2*e)*sin(f*x + e) - 3*c^2*sin(f*x + e) - (3*
c^2*sin(5*f*x + 5*e) - 6*c^2*sin(4*f*x + 4*e) + 10*c^2*sin(3*f*x + 3*e) - 6*c^2*sin(2*f*x + 2*e) + 3*c^2*sin(f
*x + e))*cos(6*f*x + 6*e) + 9*(c^2*sin(4*f*x + 4*e) + c^2*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - 3*(10*c^2*sin(3
*f*x + 3*e) + 3*c^2*sin(f*x + e))*cos(4*f*x + 4*e) + (3*c^2*cos(5*f*x + 5*e) - 6*c^2*cos(4*f*x + 4*e) + 10*c^2
*cos(3*f*x + 3*e) - 6*c^2*cos(2*f*x + 2*e) + 3*c^2*cos(f*x + e))*sin(6*f*x + 6*e) - 3*(3*c^2*cos(4*f*x + 4*e)
+ 3*c^2*cos(2*f*x + 2*e) + c^2)*sin(5*f*x + 5*e) + 3*(10*c^2*cos(3*f*x + 3*e) + 3*c^2*cos(f*x + e) + 2*c^2)*si
n(4*f*x + 4*e) - 10*(3*c^2*cos(2*f*x + 2*e) + c^2)*sin(3*f*x + 3*e) + 3*(3*c^2*cos(f*x + e) + 2*c^2)*sin(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*cos(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (40) = 80\).
time = 2.68, size = 101, normalized size = 2.35 \begin {gather*} \frac {{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\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 )}}}{3 \, f \cos \left (f x + e\right )^{2} \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))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))**(5/2)*(a+a*sec(f*x+e))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (37) = 74\).
time = 1.73, size = 104, normalized size = 2.42 \begin {gather*} \frac {8 \, {\left (3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} + 3 \, {\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 )}{3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} 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))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

8/3*(3*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^2 + 3*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^3 + c^4)*sqrt(-a*c)*abs(c)*sg
n(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)^3*f)

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Mupad [B]
time = 3.80, size = 136, normalized size = 3.16 \begin {gather*} \frac {2\,c^2\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (10\,\sin \left (e+f\,x\right )-12\,\sin \left (2\,e+2\,f\,x\right )+13\,\sin \left (3\,e+3\,f\,x\right )-6\,\sin \left (4\,e+4\,f\,x\right )+3\,\sin \left (5\,e+5\,f\,x\right )\right )}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-2\,\cos \left (4\,e+4\,f\,x\right )-\cos \left (6\,e+6\,f\,x\right )+2\right )} \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))^(5/2))/cos(e + f*x),x)

[Out]

(2*c^2*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*((c*(cos(e + f*x) - 1))/cos(e + f*x))^(1/2)*(10*sin(e + f*x
) - 12*sin(2*e + 2*f*x) + 13*sin(3*e + 3*f*x) - 6*sin(4*e + 4*f*x) + 3*sin(5*e + 5*f*x)))/(3*f*(cos(2*e + 2*f*
x) - 2*cos(4*e + 4*f*x) - cos(6*e + 6*f*x) + 2))

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