\(\int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx\) [67]

Optimal result

Integrand size = 28, antiderivative size = 119 \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {4 \sqrt {2} c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 c^2 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3989, 3972, 490, 536, 209} \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {4 \sqrt {2} c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}+\frac {2 c^2 \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}} \]

[In]

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

[Out]

(2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (4*Sqrt[2]*c^2*ArcTan[(Sqrt[a]*T
an[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (2*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x
]])

Rule 209

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

Rule 490

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(2*n -
1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rule 3989

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

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.84 \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 c^2 \left (\text {arctanh}\left (\sqrt {1-\sec (e+f x)}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )+\sqrt {1-\sec (e+f x)}\right ) \tan (e+f x)}{f \sqrt {1-\sec (e+f x)} \sqrt {a (1+\sec (e+f x))}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(102)=204\).

Time = 5.18 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.73

method	result	size
default	\(-\frac {2 c^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (2 \sqrt {2}\, \cos \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+2 \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )-\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )-\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\sin \left (f x +e \right )\right )}{f a \left (\cos \left (f x +e \right )+1\right )}\)	\(325\)
parts	\(-\frac {c^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )-2 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )\right )}{f a}-\frac {c^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )\right )}{f a}-\frac {2 c^{2} \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}}{f a}\)	\(350\)

[In]

int((c-c*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*c^2/f/a*(a*(sec(f*x+e)+1))^(1/2)*(2*2^(1/2)*cos(f*x+e)*ln(csc(f*x+e)-cot(f*x+e)+(cot(f*x+e)^2-2*csc(f*x+e)*
cot(f*x+e)+csc(f*x+e)^2-1)^(1/2))*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+2*2^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1))^(1
/2)*ln(csc(f*x+e)-cot(f*x+e)+(cot(f*x+e)^2-2*csc(f*x+e)*cot(f*x+e)+csc(f*x+e)^2-1)^(1/2))-arctanh(sin(f*x+e)/(
cos(f*x+e)+1)/(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2))*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)-arctanh(sin(f*
x+e)/(cos(f*x+e)+1)/(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2))*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-sin(f*x+e))/(cos(f*
x+e)+1)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.68 \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\left [\frac {2 \, c^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 2 \, \sqrt {2} {\left (a c^{2} \cos \left (f x + e\right ) + a c^{2}\right )} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - {\left (c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right )}{a f \cos \left (f x + e\right ) + a f}, \frac {2 \, {\left (c^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - {\left (c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) + \frac {2 \, \sqrt {2} {\left (a c^{2} \cos \left (f x + e\right ) + a c^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right )}{\sqrt {a}}\right )}}{a f \cos \left (f x + e\right ) + a f}\right ] \]

[In]

integrate((c-c*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[(2*c^2*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) + 2*sqrt(2)*(a*c^2*cos(f*x + e) + a*c^2)*sqrt(-1/
a)*log((2*sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*cos(f*x + e)*sin(f*x + e) + 3*cos(f*x + e
)^2 + 2*cos(f*x + e) - 1)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) - (c^2*cos(f*x + e) + c^2)*sqrt(-a)*log((2*a*
cos(f*x + e)^2 + 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e)
- a)/(cos(f*x + e) + 1)))/(a*f*cos(f*x + e) + a*f), 2*(c^2*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x +
e) - (c^2*cos(f*x + e) + c^2)*sqrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin
(f*x + e))) + 2*sqrt(2)*(a*c^2*cos(f*x + e) + a*c^2)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*co
s(f*x + e)/(sqrt(a)*sin(f*x + e)))/sqrt(a))/(a*f*cos(f*x + e) + a*f)]

Sympy [F]

\[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=c^{2} \left (\int \left (- \frac {2 \sec {\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \frac {1}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx\right ) \]

[In]

integrate((c-c*sec(f*x+e))**2/(a+a*sec(f*x+e))**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\int { \frac {{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}{\sqrt {a \sec \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((c-c*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c-c*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(c-c \sec (e+f x))^2}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^2}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int((c - c/cos(e + f*x))^2/(a + a/cos(e + f*x))^(1/2),x)

[Out]

int((c - c/cos(e + f*x))^2/(a + a/cos(e + f*x))^(1/2), x)