∫ sec(e+fx)(a+a sec(e+fx))3∕2 ___________________________________________

3.120 \(\int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{24 c f (c-c \sec (e+f x))^{5/2}}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{6 f (c-c \sec (e+f x))^{7/2}} \]

[Out]

-1/6*(a+a*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/2)-1/24*(a+a*sec(f*x+e))^(3/2)*tan(f*x+e)/c/f/(c-

c*sec(f*x+e))^(5/2)

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Rubi [A] time = 0.30, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3951, 3950} \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{24 c f (c-c \sec (e+f x))^{5/2}}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{6 f (c-c \sec (e+f x))^{7/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(7/2),x]

[Out]

-((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(6*f*(c - c*Sec[e + f*x])^(7/2)) - ((a + a*Sec[e + f*x])^(3/2)*Tan[

e + f*x])/(24*c*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]

Rule 3951

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] +

Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x]

/; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2

*m + 1, 0] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{6 f (c-c \sec (e+f x))^{7/2}}+\frac {\int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx}{6 c}\\ &=-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{6 f (c-c \sec (e+f x))^{7/2}}-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{24 c f (c-c \sec (e+f x))^{5/2}}\\ \end {align*}

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Mathematica [A] time = 0.56, size = 80, normalized size = 0.91 \[ \frac {a (3 \cos (e+f x)-3 \cos (2 (e+f x))-4) \tan \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sec (e+f x)+1)}}{6 c^3 f (\cos (e+f x)-1)^3 \sqrt {c-c \sec (e+f x)}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(7/2),x]

[Out]

(a*(-4 + 3*Cos[e + f*x] - 3*Cos[2*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(6*c^3*f*(-1 + Cos[

e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])

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fricas [A] time = 0.42, size = 133, normalized size = 1.51 \[ \frac {{\left (6 \, a \cos \left (f x + e\right )^{3} - 3 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{6 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )} \]

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))^(7/2),x, algorithm="fricas")

[Out]

1/6*(6*a*cos(f*x + e)^3 - 3*a*cos(f*x + e)^2 + a*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c

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

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))^(7/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/

x/2)1/2*a^2*(1/12*(3*a^3*(-a*tan(1/2*(f*x+exp(1)))^2+a)-a^4)/(-a*tan(1/2*(f*x+exp(1)))^2)^3-1/12*a)/c^3/sqrt(-

a*c)/f/abs(a)/sign(tan(1/2*(f*x+exp(1)))^2-1)

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maple [A] time = 2.18, size = 83, normalized size = 0.94 \[ \frac {\left (5 \cos \left (f x +e \right )-1\right ) \left (\sin ^{3}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, a}{24 f \left (-1+\cos \left (f x +e \right )\right ) \cos \left (f x +e \right )^{3} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}} \]

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))^(7/2),x)

[Out]

1/24/f*(5*cos(f*x+e)-1)*sin(f*x+e)^3*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)/(-1+cos(f*x+e))/cos(f*x+e)^3/(c*(-1+c

os(f*x+e))/cos(f*x+e))^(7/2)*a

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maxima [B] time = 2.07, size = 1559, normalized size = 17.72 \[ \text {result too large to display} \]

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))^(7/2),x, algorithm="maxima")

[Out]

2/3*(3*(a*sin(4*f*x + 4*e) + a*sin(2*f*x + 2*e))*cos(6*f*x + 6*e) + 3*(a*sin(6*f*x + 6*e) + 9*a*sin(4*f*x + 4*

e) + 9*a*sin(2*f*x + 2*e) - 4*a*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/2*arctan2(sin(2*f*

x + 2*e), cos(2*f*x + 2*e))) + 4*(2*a*sin(6*f*x + 6*e) + 15*a*sin(4*f*x + 4*e) + 15*a*sin(2*f*x + 2*e) + 3*a*s

in(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 3*

(a*sin(6*f*x + 6*e) + 9*a*sin(4*f*x + 4*e) + 9*a*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x

+ 2*e))) - 3*(a*cos(4*f*x + 4*e) + a*cos(2*f*x + 2*e))*sin(6*f*x + 6*e) + 3*a*sin(4*f*x + 4*e) + 3*a*sin(2*f*

x + 2*e) - 3*(a*cos(6*f*x + 6*e) + 9*a*cos(4*f*x + 4*e) + 9*a*cos(2*f*x + 2*e) - 4*a*cos(3/2*arctan2(sin(2*f*x

+ 2*e), cos(2*f*x + 2*e))) + a)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(2*a*cos(6*f*x + 6*e

) + 15*a*cos(4*f*x + 4*e) + 15*a*cos(2*f*x + 2*e) + 3*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +

2*a)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 3*(a*cos(6*f*x + 6*e) + 9*a*cos(4*f*x + 4*e) + 9*

a*cos(2*f*x + 2*e) + a)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((c^4*cos(6*f*x

+ 6*e)^2 + 225*c^4*cos(4*f*x + 4*e)^2 + 225*c^4*cos(2*f*x + 2*e)^2 + 36*c^4*cos(5/2*arctan2(sin(2*f*x + 2*e),

cos(2*f*x + 2*e)))^2 + 400*c^4*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 36*c^4*cos(1/2*arctan2

(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + c^4*sin(6*f*x + 6*e)^2 + 225*c^4*sin(4*f*x + 4*e)^2 + 450*c^4*sin(4*

f*x + 4*e)*sin(2*f*x + 2*e) + 225*c^4*sin(2*f*x + 2*e)^2 + 36*c^4*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x

+ 2*e)))^2 + 400*c^4*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 36*c^4*sin(1/2*arctan2(sin(2*f*x

+ 2*e), cos(2*f*x + 2*e)))^2 + 30*c^4*cos(2*f*x + 2*e) + c^4 + 2*(15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x

+ 2*e) + c^4)*cos(6*f*x + 6*e) + 30*(15*c^4*cos(2*f*x + 2*e) + c^4)*cos(4*f*x + 4*e) - 12*(c^4*cos(6*f*x + 6*e

) + 15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x + 2*e) - 20*c^4*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2

*e))) - 6*c^4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^4)*cos(5/2*arctan2(sin(2*f*x + 2*e), co

s(2*f*x + 2*e))) - 40*(c^4*cos(6*f*x + 6*e) + 15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x + 2*e) - 6*c^4*cos(1/

2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^4)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1

2*(c^4*cos(6*f*x + 6*e) + 15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x + 2*e) + c^4)*cos(1/2*arctan2(sin(2*f*x +

2*e), cos(2*f*x + 2*e))) + 30*(c^4*sin(4*f*x + 4*e) + c^4*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 12*(c^4*sin(6*

f*x + 6*e) + 15*c^4*sin(4*f*x + 4*e) + 15*c^4*sin(2*f*x + 2*e) - 20*c^4*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(

2*f*x + 2*e))) - 6*c^4*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x + 2*e),

cos(2*f*x + 2*e))) - 40*(c^4*sin(6*f*x + 6*e) + 15*c^4*sin(4*f*x + 4*e) + 15*c^4*sin(2*f*x + 2*e) - 6*c^4*sin

(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(

c^4*sin(6*f*x + 6*e) + 15*c^4*sin(4*f*x + 4*e) + 15*c^4*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), co

s(2*f*x + 2*e))))*f)

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mupad [B] time = 7.03, size = 273, normalized size = 3.10 \[ \frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^4\,f}-\frac {a\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,44{}\mathrm {i}}{3\,c^4\,f}+\frac {a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^4\,f}-\frac {a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^4\,f}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,28{}\mathrm {i}-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,28{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,12{}\mathrm {i}-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,2{}\mathrm {i}} \]

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))^(7/2)),x)

[Out]

((c - c/cos(e + f*x))^(1/2)*((a*exp(e*4i + f*x*4i)*(a + a/cos(e + f*x))^(1/2)*4i)/(c^4*f) - (a*cos(e + f*x)*ex

p(e*4i + f*x*4i)*(a + a/cos(e + f*x))^(1/2)*44i)/(3*c^4*f) + (a*exp(e*4i + f*x*4i)*cos(2*e + 2*f*x)*(a + a/cos

(e + f*x))^(1/2)*4i)/(c^4*f) - (a*exp(e*4i + f*x*4i)*cos(3*e + 3*f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/(c^4*f)))

/(exp(e*4i + f*x*4i)*sin(e + f*x)*28i - exp(e*4i + f*x*4i)*sin(2*e + 2*f*x)*28i + exp(e*4i + f*x*4i)*sin(3*e +

3*f*x)*12i - exp(e*4i + f*x*4i)*sin(4*e + 4*f*x)*2i)

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(7/2),x)

[Out]

Timed out

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