3.1.79 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{7/2} \, dx\) [79]

Optimal. Leaf size=171 \[ -\frac {256 c^4 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{429 f}-\frac {24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f} \]

[Out]

-24/143*c^2*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f-2/13*c*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^
(5/2)*tan(f*x+e)/f-256/3003*c^4*(a+a*sec(f*x+e))^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-64/429*c^3*(a+a*sec(f*x
+e))^3*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f

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Rubi [A]
time = 0.30, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {4040, 4038} \begin {gather*} -\frac {256 c^4 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}}{429 f}-\frac {24 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{143 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}{13 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*c^4*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(3003*f*Sqrt[c - c*Sec[e + f*x]]) - (64*c^3*(a + a*Sec[e + f*x]
)^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(429*f) - (24*c^2*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)
*Tan[e + f*x])/(143*f) - (2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(13*f)

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)]

Rule 4040

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))^3 (c-c \sec (e+f x))^{7/2} \, dx &=-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac {1}{13} (12 c) \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac {24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac {1}{143} \left (96 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac {64 c^3 (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{429 f}-\frac {24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac {1}{429} \left (128 c^3\right ) \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx\\ &=-\frac {256 c^4 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{429 f}-\frac {24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}\\ \end {align*}

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Mathematica [A]
time = 2.55, size = 88, normalized size = 0.51 \begin {gather*} \frac {4 a^3 c^3 \cos ^6\left (\frac {1}{2} (e+f x)\right ) (-3766+6285 \cos (e+f x)-2842 \cos (2 (e+f x))+835 \cos (3 (e+f x))) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^6(e+f x) \sqrt {c-c \sec (e+f x)}}{3003 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*c^3*Cos[(e + f*x)/2]^6*(-3766 + 6285*Cos[e + f*x] - 2842*Cos[2*(e + f*x)] + 835*Cos[3*(e + f*x)])*Cot[(
e + f*x)/2]*Sec[e + f*x]^6*Sqrt[c - c*Sec[e + f*x]])/(3003*f)

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Maple [A]
time = 2.40, size = 85, normalized size = 0.50

method	result	size
default	\(\frac {2 a^{3} \left (835 \left (\cos ^{3}\left (f x +e \right )\right )-1421 \left (\cos ^{2}\left (f x +e \right )\right )+945 \cos \left (f x +e \right )-231\right ) \left (\sin ^{7}\left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}}{3003 f \left (-1+\cos \left (f x +e \right )\right )^{7} \cos \left (f x +e \right )^{3}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/3003*a^3/f*(835*cos(f*x+e)^3-1421*cos(f*x+e)^2+945*cos(f*x+e)-231)*sin(f*x+e)^7*(c*(-1+cos(f*x+e))/cos(f*x+e
))^(7/2)/(-1+cos(f*x+e))^7/cos(f*x+e)^3

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]
time = 2.68, size = 174, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (835 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} + 1919 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 271 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 1637 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 103 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 973 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 21 \, a^{3} c^{3} \cos \left (f x + e\right ) - 231 \, a^{3} c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3003 \, f \cos \left (f x + e\right )^{6} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/3003*(835*a^3*c^3*cos(f*x + e)^7 + 1919*a^3*c^3*cos(f*x + e)^6 + 271*a^3*c^3*cos(f*x + e)^5 - 1637*a^3*c^3*c
os(f*x + e)^4 - 103*a^3*c^3*cos(f*x + e)^3 + 973*a^3*c^3*cos(f*x + e)^2 + 21*a^3*c^3*cos(f*x + e) - 231*a^3*c^
3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^6*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)*(a+a*sec(f*x+e))**3*(c-c*sec(f*x+e))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7317 deep

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Giac [A]
time = 1.26, size = 109, normalized size = 0.64 \begin {gather*} \frac {128 \, \sqrt {2} {\left (429 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} c^{4} + 1001 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{5} + 819 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{6} + 231 \, c^{7}\right )} a^{3} c^{3}}{3003 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {13}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

128/3003*sqrt(2)*(429*(c*tan(1/2*f*x + 1/2*e)^2 - c)^3*c^4 + 1001*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^5 + 819*(
c*tan(1/2*f*x + 1/2*e)^2 - c)*c^6 + 231*c^7)*a^3*c^3/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(13/2)*f)

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Mupad [B]
time = 14.67, size = 710, normalized size = 4.15 \begin {gather*} \frac {\left (\frac {a^3\,c^3\,2{}\mathrm {i}}{f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1670{}\mathrm {i}}{3003\,f}\right )\,\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}}}}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}+\frac {\left (\frac {a^3\,c^3\,128{}\mathrm {i}}{13\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,128{}\mathrm {i}}{13\,f}\right )\,\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 ({\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 )}^6}-\frac {\left (\frac {a^3\,c^3\,384{}\mathrm {i}}{11\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,3456{}\mathrm {i}}{143\,f}\right )\,\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 ({\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 )}^5}-\frac {\left (\frac {a^3\,c^3\,8{}\mathrm {i}}{f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,2168{}\mathrm {i}}{3003\,f}\right )\,\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 ({\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 )}+\frac {\left (\frac {a^3\,c^3\,24{}\mathrm {i}}{f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,5464{}\mathrm {i}}{1001\,f}\right )\,\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 ({\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 {\left (\frac {a^3\,c^3\,160{}\mathrm {i}}{3\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,11360{}\mathrm {i}}{429\,f}\right )\,\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 ({\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 {\left (\frac {a^3\,c^3\,320{}\mathrm {i}}{7\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,46400{}\mathrm {i}}{3003\,f}\right )\,\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 ({\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^(7/2))/cos(e + f*x),x)

[Out]

(((a^3*c^3*2i)/f + (a^3*c^3*exp(e*1i + f*x*1i)*1670i)/(3003*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*
x*1i)/2))^(1/2))/(exp(e*1i + f*x*1i) - 1) + (((a^3*c^3*128i)/(13*f) + (a^3*c^3*exp(e*1i + f*x*1i)*128i)/(13*f)
)*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i)
+ 1)^6) - (((a^3*c^3*384i)/(11*f) + (a^3*c^3*exp(e*1i + f*x*1i)*3456i)/(143*f))*(c - c/(exp(- e*1i - f*x*1i)/
2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^5) - (((a^3*c^3*8i)/f + (
a^3*c^3*exp(e*1i + f*x*1i)*2168i)/(3003*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((e
xp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)) + (((a^3*c^3*24i)/f + (a^3*c^3*exp(e*1i + f*x*1i)*5464i)/(100
1*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x
*2i) + 1)^2) + (((a^3*c^3*160i)/(3*f) + (a^3*c^3*exp(e*1i + f*x*1i)*11360i)/(429*f))*(c - c/(exp(- e*1i - f*x*
1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^4) - (((a^3*c^3*320i)
/(7*f) + (a^3*c^3*exp(e*1i + f*x*1i)*46400i)/(3003*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))
^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^3)

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