\(\int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx\) [110]

Optimal result

Integrand size = 30, antiderivative size = 204 \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {c^4 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {8 c^4 \log (1+\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {4 c^4 \sec (e+f x) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {c^4 \sec ^2(e+f x) \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]

[Out]

c^4*ln(cos(f*x+e))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)+8*c^4*ln(1+sec(f*x+e))*tan(f*x+e
)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)-4*c^4*sec(f*x+e)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*se
c(f*x+e))^(1/2)+1/2*c^4*sec(f*x+e)^2*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3997, 84} \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {c^4 \tan (e+f x) \sec ^2(e+f x)}{2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {4 c^4 \tan (e+f x) \sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {8 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]

[In]

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

[Out]

(c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (8*c^4*Log[1 + Se
c[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Sec[e + f*x]*Tan[e +
f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]^2*Tan[e + f*x])/(2*f*Sqrt[a +
a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 3997

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

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

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.41 \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {c^4 \left (2 (\log (\cos (e+f x))+8 \log (1+\sec (e+f x)))-8 \sec (e+f x)+\sec ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]

[In]

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

[Out]

(c^4*(2*(Log[Cos[e + f*x]] + 8*Log[1 + Sec[e + f*x]]) - 8*Sec[e + f*x] + Sec[e + f*x]^2)*Tan[e + f*x])/(2*f*Sq
rt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right )^{3} c^{3} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (14 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+14 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+2 \cos \left (f x +e \right )^{2} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+9 \cos \left (f x +e \right )^{2}+8 \cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \cot \left (f x +e \right )}{2 f a \left (\cos \left (f x +e \right )-1\right )^{3}}\)	\(166\)
risch	\(\frac {c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, x}{\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}-\frac {2 c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (f x +e \right )}{\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}+\frac {2 i c^{3} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (4 \,{\mathrm e}^{2 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}+4\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}-\frac {16 i c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}+\frac {7 i c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\)	\(536\)

[In]

int((c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/f/a*(-c*(sec(f*x+e)-1))^(1/2)*(sec(f*x+e)-1)^3*c^3*(a*(sec(f*x+e)+1))^(1/2)*(14*cos(f*x+e)^2*ln(-cot(f*x+
e)+csc(f*x+e)+1)+14*cos(f*x+e)^2*ln(-cot(f*x+e)+csc(f*x+e)-1)+2*cos(f*x+e)^2*ln(2/(cos(f*x+e)+1))+9*cos(f*x+e)
^2+8*cos(f*x+e)-1)/(cos(f*x+e)-1)^3*cos(f*x+e)*cot(f*x+e)

Fricas [F]

\[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int { \frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(c^3*sec(f*x + e)^3 - 3*c^3*sec(f*x + e)^2 + 3*c^3*sec(f*x + e) - c^3)*sqrt(-c*sec(f*x + e) + c)/sqr
t(a*sec(f*x + e) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((c-c*sec(f*x+e))**(7/2)/(a+a*sec(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F(-2)]

Exception generated. \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c-c*sec(f*x+e))^(7/2)/(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:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(c-c \sec (e+f x))^{7/2}}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int((c - c/cos(e + f*x))^(7/2)/(a + a/cos(e + f*x))^(1/2),x)

[Out]

int((c - c/cos(e + f*x))^(7/2)/(a + a/cos(e + f*x))^(1/2), x)