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

Optimal. Leaf size=164 \[ -\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}} \]

[Out]

-1/32*a^2*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))/c^(7/2)/f*2^(1/2)-1/3*(a^2+a^2*sec(f*x
+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/2)+1/4*a^2*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(5/2)-1/16*a^2*tan(f*x+e)/c^2
/f/(c-c*sec(f*x+e))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3957, 3796, 3795, 203} \[ -\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(16*Sqrt[2]*c^(7/2)*f) - ((a^2 + a^2*
Sec[e + f*x])*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^(7/2)) + (a^2*Tan[e + f*x])/(4*c*f*(c - c*Sec[e + f*x])^
(5/2)) - (a^2*Tan[e + f*x])/(16*c^2*f*(c - c*Sec[e + f*x])^(3/2))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3795

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]

Rule 3796

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

Rule 3957

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

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}-\frac {a \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{5/2}} \, dx}{2 c}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}} \, dx}{8 c^2}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{32 c^3}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{16 c^3 f}\\ &=-\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 5.90, size = 398, normalized size = 2.43 \[ \frac {a^2 \csc \left (\frac {e}{2}\right ) e^{-\frac {1}{2} i (e+f x)} \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {3}{2}}(e+f x) (\sec (e+f x)+1)^2 \left (e^{\frac {i e}{2}} \sqrt {\sec (e+f x)} \left (e^{\frac {i f x}{2}} \sin \left (\frac {f x}{2}\right ) \left (-43 \sin ^2\left (\frac {1}{2} (e+f x)\right )+14 \sin ^2\left (\frac {e}{2}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )+34\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )-\frac {1}{8} \cos \left (\frac {e}{2}\right ) e^{\frac {i f x}{2}} \sin \left (\frac {1}{2} (e+f x)\right ) (36 \cos (e+f x)-43 \cos (2 (e+f x))-57)-\frac {7}{2} \sin (e) e^{\frac {i f x}{2}} \sin (f x) \csc \left (\frac {f x}{2}\right ) \sin ^6\left (\frac {1}{2} (e+f x)\right )+4 i e^{i f x}-4 i\right )-3 \sin \left (\frac {e}{2}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \sin ^6\left (\frac {1}{2} (e+f x)\right ) \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )\right )}{24 c^3 f (\sec (e+f x)-1)^3 \sqrt {c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Csc[e/2]*Sec[(e + f*x)/2]^3*Sec[e + f*x]^(3/2)*(1 + Sec[e + f*x])^2*(-3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I
)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x
))])]*Sin[e/2]*Sin[(e + f*x)/2]^6 + E^((I/2)*e)*Sqrt[Sec[e + f*x]]*(-4*I + (4*I)*E^(I*f*x) - (E^((I/2)*f*x)*Co
s[e/2]*(-57 + 36*Cos[e + f*x] - 43*Cos[2*(e + f*x)])*Sin[(e + f*x)/2])/8 - (7*E^((I/2)*f*x)*Csc[(f*x)/2]*Sin[e
]*Sin[f*x]*Sin[(e + f*x)/2]^6)/2 + E^((I/2)*f*x)*Sin[(f*x)/2]*Sin[(e + f*x)/2]^2*(34 - 43*Sin[(e + f*x)/2]^2 +
14*Sin[e/2]^2*Sin[(e + f*x)/2]^4)))*Tan[(e + f*x)/2])/(24*c^3*E^((I/2)*(e + f*x))*f*(-1 + Sec[e + f*x])^3*Sqr
t[c - c*Sec[e + f*x]])

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 517, normalized size = 3.15 \[ \left [-\frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 4 \, {\left (7 \, a^{2} \cos \left (f x + e\right )^{4} + 29 \, a^{2} \cos \left (f x + e\right )^{3} + 25 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{192 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}, \frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, {\left (7 \, a^{2} \cos \left (f x + e\right )^{4} + 29 \, a^{2} \cos \left (f x + e\right )^{3} + 25 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{96 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/192*(3*sqrt(2)*(a^2*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 3*a^2*cos(f*x + e) - a^2)*sqrt(-c)*log((2*sqrt
(2)*(cos(f*x + e)^2 + cos(f*x + e))*sqrt(-c)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) + (3*c*cos(f*x + e) + c)*
sin(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))*sin(f*x + e) - 4*(7*a^2*cos(f*x + e)^4 + 29*a^2*cos(f*x + e)^
3 + 25*a^2*cos(f*x + e)^2 + 3*a^2*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)), 1/96*(3*sqrt(2)*(a^2*cos(f*x + e)^3
- 3*a^2*cos(f*x + e)^2 + 3*a^2*cos(f*x + e) - a^2)*sqrt(c)*arctan(sqrt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x +
e))*cos(f*x + e)/(sqrt(c)*sin(f*x + e)))*sin(f*x + e) + 2*(7*a^2*cos(f*x + e)^4 + 29*a^2*cos(f*x + e)^3 + 25*a
^2*cos(f*x + e)^2 + 3*a^2*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))]

________________________________________________________________________________________

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))^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)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/2*(1/48*(-8*a^2*c^2*sqrt(c*tan((f*x+exp(1))/2)^2-c)*(c*tan
((f*x+exp(1))/2)^2-c)-3*a^2*c*sqrt(c*tan((f*x+exp(1))/2)^2-c)*(c*tan((f*x+exp(1))/2)^2-c)^2+3*a^2*c^3*sqrt(c*t
an((f*x+exp(1))/2)^2-c))/(c*tan((f*x+exp(1))/2)^2)^3-1/16*a^2*sqrt(c)*atan(sqrt(c*tan((f*x+exp(1))/2)^2-c)/sqr
t(c)))/sqrt(2)/c^4/sign(tan((f*x+exp(1))/2))/sign(tan((f*x+exp(1))/2)^2-1)

________________________________________________________________________________________

maple [B]  time = 2.46, size = 402, normalized size = 2.45 \[ -\frac {a^{2} \left (-1+\cos \left (f x +e \right )\right )^{4} \left (5 \left (\cos ^{3}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+15 \left (\cos ^{2}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+3 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+3 \left (\cos ^{3}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )+27 \cos \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}-9 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-9 \left (\cos ^{2}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )+17 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+9 \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+9 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )-3 \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-3 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )\right )}{6 f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sin \left (f x +e \right )^{7} \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a^2/f*(-1+cos(f*x+e))^4*(5*cos(f*x+e)^3*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)+15*cos(f*x+e)^2*(-2*cos(f*x+
e)/(1+cos(f*x+e)))^(3/2)+3*cos(f*x+e)^3*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)+3*cos(f*x+e)^3*arctan(1/(-2*cos(f
*x+e)/(1+cos(f*x+e)))^(1/2))+27*cos(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)-9*cos(f*x+e)^2*(-2*cos(f*x+e)/
(1+cos(f*x+e)))^(1/2)-9*cos(f*x+e)^2*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))+17*(-2*cos(f*x+e)/(1+cos(f
*x+e)))^(3/2)+9*cos(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)+9*cos(f*x+e)*arctan(1/(-2*cos(f*x+e)/(1+cos(f*
x+e)))^(1/2))-3*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)-3*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)))/(c*(-1+
cos(f*x+e))/cos(f*x+e))^(7/2)/sin(f*x+e)^7/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(sec(e + f*x)/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 + 3*c**3*sqrt(-c*sec(e + f*x) + c
)*sec(e + f*x)**2 - 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x) + Inte
gral(2*sec(e + f*x)**2/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 + 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec
(e + f*x)**2 - 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x) + Integral(
sec(e + f*x)**3/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 + 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*
x)**2 - 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x))

________________________________________________________________________________________