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

Optimal. Leaf size=42 \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]

[Out]

-1/4*(a+a*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(5/2)

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Rubi [A]  time = 0.15, antiderivative size = 42, 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 = {3950} \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(4*f*(c - c*Sec[e + f*x])^(5/2))

Rule 3950

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

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 63, normalized size = 1.50 \[ \frac {a \tan \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \sec (e+f x)}}{c^3 f (\sec (e+f x)-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]]*Tan[(e + f*x)/2])/(c^3*f*(-1 + Sec[e + f*x
])^3)

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fricas [B]  time = 0.44, size = 95, normalized size = 2.26 \[ \frac {a \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)a^2*(1/4*a^3/(-a*tan(1/2*(f*x+exp(1)))^2)^2-1/4*a)/c^2/sqrt(-a*c)/f/abs(a)/sign(tan(1/2*(f*x+exp(1)))^2-1)

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maple [A]  time = 1.94, size = 73, normalized size = 1.74 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (\sin ^{3}\left (f x +e \right )\right ) a}{4 f \left (-1+\cos \left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \cos \left (f x +e \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.98, size = 533, normalized size = 12.69 \[ \frac {2 \, {\left (6 \, a \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 6 \, a \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, a \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - {\left (a \sin \left (3 \, f x + 3 \, e\right ) + a \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (a \cos \left (3 \, f x + 3 \, e\right ) + a \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - {\left (6 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} \sin \left (3 \, f x + 3 \, e\right ) - a \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c^{3} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \sin \left (2 \, f x + 2 \, e\right )^{2} - 48 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, c^{3} \sin \left (f x + e\right )^{2} - 8 \, c^{3} \cos \left (f x + e\right ) + c^{3} - 2 \, {\left (4 \, c^{3} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 8 \, {\left (6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 12 \, {\left (4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (2 \, c^{3} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 16 \, {\left (3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(6*a*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 6*a*cos(f*x + e)*sin(2*f*x + 2*e) - 6*a*cos(2*f*x + 2*e)*sin(f*x +
e) - (a*sin(3*f*x + 3*e) + a*sin(f*x + e))*cos(4*f*x + 4*e) + (a*cos(3*f*x + 3*e) + a*cos(f*x + e))*sin(4*f*x
+ 4*e) - (6*a*cos(2*f*x + 2*e) + a)*sin(3*f*x + 3*e) - a*sin(f*x + e))*sqrt(a)*sqrt(c)/((c^3*cos(4*f*x + 4*e)^
2 + 16*c^3*cos(3*f*x + 3*e)^2 + 36*c^3*cos(2*f*x + 2*e)^2 + 16*c^3*cos(f*x + e)^2 + c^3*sin(4*f*x + 4*e)^2 + 1
6*c^3*sin(3*f*x + 3*e)^2 + 36*c^3*sin(2*f*x + 2*e)^2 - 48*c^3*sin(2*f*x + 2*e)*sin(f*x + e) + 16*c^3*sin(f*x +
e)^2 - 8*c^3*cos(f*x + e) + c^3 - 2*(4*c^3*cos(3*f*x + 3*e) - 6*c^3*cos(2*f*x + 2*e) + 4*c^3*cos(f*x + e) - c
^3)*cos(4*f*x + 4*e) - 8*(6*c^3*cos(2*f*x + 2*e) - 4*c^3*cos(f*x + e) + c^3)*cos(3*f*x + 3*e) - 12*(4*c^3*cos(
f*x + e) - c^3)*cos(2*f*x + 2*e) - 4*(2*c^3*sin(3*f*x + 3*e) - 3*c^3*sin(2*f*x + 2*e) + 2*c^3*sin(f*x + e))*si
n(4*f*x + 4*e) - 16*(3*c^3*sin(2*f*x + 2*e) - 2*c^3*sin(f*x + e))*sin(3*f*x + 3*e))*f)

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mupad [B]  time = 4.83, size = 165, normalized size = 3.93 \[ -\frac {2\,a\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (6\,\sin \left (e+f\,x\right )-8\,\sin \left (2\,e+2\,f\,x\right )+7\,\sin \left (3\,e+3\,f\,x\right )-4\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (5\,e+5\,f\,x\right )\right )}{c^3\,f\,\left (48\,\cos \left (e+f\,x\right )+15\,\cos \left (2\,e+2\,f\,x\right )-40\,\cos \left (3\,e+3\,f\,x\right )+26\,\cos \left (4\,e+4\,f\,x\right )-8\,\cos \left (5\,e+5\,f\,x\right )+\cos \left (6\,e+6\,f\,x\right )-42\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*((c*(cos(e + f*x) - 1))/cos(e + f*x))^(1/2)*(6*sin(e + f*x)
- 8*sin(2*e + 2*f*x) + 7*sin(3*e + 3*f*x) - 4*sin(4*e + 4*f*x) + sin(5*e + 5*f*x)))/(c^3*f*(48*cos(e + f*x) +
15*cos(2*e + 2*f*x) - 40*cos(3*e + 3*f*x) + 26*cos(4*e + 4*f*x) - 8*cos(5*e + 5*f*x) + cos(6*e + 6*f*x) - 42))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sec(e + f*x) + 1))**(3/2)*sec(e + f*x)/(-c*(sec(e + f*x) - 1))**(5/2), x)

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