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

Optimal. Leaf size=163 \[ -\frac {256 c^4 \tan (e+f x) (a \sec (e+f x)+a)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{105 f}-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{21 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}{9 f} \]

[Out]

-8/21*c^2*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f-2/9*c*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2)*t
an(f*x+e)/f-256/315*c^4*(a+a*sec(f*x+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-64/105*c^3*(a+a*sec(f*x+e))*(c-c*
sec(f*x+e))^(1/2)*tan(f*x+e)/f

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Rubi [A]  time = 0.28, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3955, 3953} \[ -\frac {256 c^4 \tan (e+f x) (a \sec (e+f x)+a)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{105 f}-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{21 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}{9 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*c^4*(a + a*Sec[e + f*x])*Tan[e + f*x])/(315*f*Sqrt[c - c*Sec[e + f*x]]) - (64*c^3*(a + a*Sec[e + f*x])*S
qrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(105*f) - (8*c^2*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)*Tan[e +
f*x])/(21*f) - (2*c*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(9*f)

Rule 3953

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)]

Rule 3955

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

Rubi steps

\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{7/2} \, dx &=-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{3} (4 c) \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{21} \left (32 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac {64 c^3 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{105 f}-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{105} \left (128 c^3\right ) \int \sec (e+f x) (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \, dx\\ &=-\frac {256 c^4 (a+a \sec (e+f x)) \tan (e+f x)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{105 f}-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}\\ \end {align*}

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Mathematica [A]  time = 0.91, size = 86, normalized size = 0.53 \[ \frac {a c^3 \cos ^2\left (\frac {1}{2} (e+f x)\right ) (1617 \cos (e+f x)-642 \cos (2 (e+f x))+319 \cos (3 (e+f x))-782) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^4(e+f x) \sqrt {c-c \sec (e+f x)}}{315 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c^3*Cos[(e + f*x)/2]^2*(-782 + 1617*Cos[e + f*x] - 642*Cos[2*(e + f*x)] + 319*Cos[3*(e + f*x)])*Cot[(e + f*
x)/2]*Sec[e + f*x]^4*Sqrt[c - c*Sec[e + f*x]])/(315*f)

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fricas [A]  time = 0.46, size = 119, normalized size = 0.73 \[ \frac {2 \, {\left (319 \, a c^{3} \cos \left (f x + e\right )^{5} + 317 \, a c^{3} \cos \left (f x + e\right )^{4} - 158 \, a c^{3} \cos \left (f x + e\right )^{3} - 26 \, a c^{3} \cos \left (f x + e\right )^{2} + 95 \, a c^{3} \cos \left (f x + e\right ) - 35 \, a c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{315 \, f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/315*(319*a*c^3*cos(f*x + e)^5 + 317*a*c^3*cos(f*x + e)^4 - 158*a*c^3*cos(f*x + e)^3 - 26*a*c^3*cos(f*x + e)^
2 + 95*a*c^3*cos(f*x + e) - 35*a*c^3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^4*sin(f*x + e))

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giac [A]  time = 3.36, size = 111, normalized size = 0.68 \[ \frac {32 \, \sqrt {2} {\left (105 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} c^{2} + 189 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{3} + 135 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{4} + 35 \, c^{5}\right )} a c^{3}}{315 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {9}{2}} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

32/315*sqrt(2)*(105*(c*tan(1/2*f*x + 1/2*e)^2 - c)^3*c^2 + 189*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^3 + 135*(c*t
an(1/2*f*x + 1/2*e)^2 - c)*c^4 + 35*c^5)*a*c^3/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(9/2)*f)

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maple [A]  time = 1.22, size = 83, normalized size = 0.51 \[ \frac {2 a \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) \left (319 \left (\cos ^{3}\left (f x +e \right )\right )-321 \left (\cos ^{2}\left (f x +e \right )\right )+165 \cos \left (f x +e \right )-35\right )}{315 f \left (-1+\cos \left (f x +e \right )\right )^{5} \cos \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(7/2),x)

[Out]

2/315*a/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)*sin(f*x+e)^3*(319*cos(f*x+e)^3-321*cos(f*x+e)^2+165*cos(f*x+e)-
35)/(-1+cos(f*x+e))^5/cos(f*x+e)

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maxima [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))*(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 9.13, size = 483, normalized size = 2.96 \[ \frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,2{}\mathrm {i}}{f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,638{}\mathrm {i}}{315\,f}\right )}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,32{}\mathrm {i}}{9\,f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{9\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,96{}\mathrm {i}}{7\,f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{63\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,64{}\mathrm {i}}{5\,f}-\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,736{}\mathrm {i}}{105\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,8{}\mathrm {i}}{3\,f}-\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1256{}\mathrm {i}}{315\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(e + f*x))*(c - c/cos(e + f*x))^(7/2))/cos(e + f*x),x)

[Out]

((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*2i)/f + (a*c^3*exp(e*1i + f*x*1i)*638i)
/(315*f)))/(exp(e*1i + f*x*1i) - 1) - ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*3
2i)/(9*f) + (a*c^3*exp(e*1i + f*x*1i)*32i)/(9*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^4) + ((c
- c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*96i)/(7*f) + (a*c^3*exp(e*1i + f*x*1i)*32i
)/(63*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^3) - ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i
+ f*x*1i)/2))^(1/2)*((a*c^3*64i)/(5*f) - (a*c^3*exp(e*1i + f*x*1i)*736i)/(105*f)))/((exp(e*1i + f*x*1i) - 1)*(
exp(e*2i + f*x*2i) + 1)^2) + ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*8i)/(3*f)
- (a*c^3*exp(e*1i + f*x*1i)*1256i)/(315*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1))

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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))*(c-c*sec(f*x+e))**(7/2),x)

[Out]

Timed out

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