\(\int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx\) [179]

Optimal result

Integrand size = 27, antiderivative size = 468 \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} c^2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (c^2-d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4025, 186, 65, 212, 44} \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=-\frac {\left (c^2-d^2\right ) \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 c^2 \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\sqrt {2} c^2 \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {3 (c-d)^2 \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\left (c^2-d^2\right ) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {3 (c-d)^2 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {(c-d)^2 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt {a \sec (e+f x)+a}} \]

[In]

Int[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]

[Out]

-1/4*((c - d)^2*Tan[e + f*x])/(a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^2*Tan[e + f*x
])/(16*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - ((c^2 - d^2)*Tan[e + f*x])/(2*a^2*f*(1 + Sec[e + f
*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*f*Sqr
t[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[
a])]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^2*ArcTanh[Sqrt[a
- a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*
Sec[e + f*x]]) - ((c^2 - d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(
3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 186

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rule 212

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

Rule 4025

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

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

Mathematica [A] (verified)

Time = 6.25 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.56 \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {\cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} (c+d \sec (e+f x))^2 \left (\frac {\left (\left (-43 c^2+6 c d+5 d^2\right ) \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+32 \sqrt {2} c^2 \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {1+\sec (e+f x)}}{\sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )}}+\frac {1}{4} (c-d) (11 c+5 d+(15 c+d) \cos (e+f x)) \sec ^3\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {3}{2} (e+f x)\right )\right )\right )}{4 f (d+c \cos (e+f x))^2 (a (1+\sec (e+f x)))^{5/2}} \]

[In]

Integrate[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]

[Out]

(Cos[(e + f*x)/2]^4*Sqrt[Sec[e + f*x]]*(c + d*Sec[e + f*x])^2*((((-43*c^2 + 6*c*d + 5*d^2)*ArcSin[Tan[(e + f*x
)/2]] + 32*Sqrt[2]*c^2*ArcTan[Tan[(e + f*x)/2]/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]])*Sqrt[Cos[e + f*x]/(1 +
Cos[e + f*x])]*Sqrt[1 + Sec[e + f*x]])/Sqrt[Sec[(e + f*x)/2]^2] + ((c - d)*(11*c + 5*d + (15*c + d)*Cos[e + f*
x])*Sec[(e + f*x)/2]^3*Sqrt[Sec[e + f*x]]*(Sin[(e + f*x)/2] - Sin[(3*(e + f*x))/2]))/4))/(4*f*(d + c*Cos[e + f
*x])^2*(a*(1 + Sec[e + f*x]))^(5/2))

Maple [A] (warning: unable to verify)

Time = 4.48 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} c^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-4 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} c d \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} d^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-32 c^{2} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )-11 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+6 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c d \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+5 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+43 c^{2} \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )-6 c d \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )-5 d^{2} \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 a^{3} f}\)	\(523\)
parts	\(\frac {c^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-2 \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \csc \left (f x +e \right )^{3}+32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )+13 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-43 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}+\frac {d^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-5 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}+\frac {c d \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{16 f \,a^{3}}\)	\(638\)

[In]

int((c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/32/a^3/f*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(2*((1-cos(
f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*c^2*(-cot(f*x+e)+csc(f*x+e))-4*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*c*d*(-c
ot(f*x+e)+csc(f*x+e))+2*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*d^2*(-cot(f*x+e)+csc(f*x+e))-32*c^2*2^(1/2)*ar
ctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))-11*((1-cos(f*x+e))^2*csc(f*x+e
)^2-1)^(1/2)*c^2*(-cot(f*x+e)+csc(f*x+e))+6*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*c*d*(-cot(f*x+e)+csc(f*x+e
))+5*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*d^2*(-cot(f*x+e)+csc(f*x+e))+43*c^2*ln(csc(f*x+e)-cot(f*x+e)+((1-
cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))-6*c*d*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))-5
*d^2*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 13.49 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left ({\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 43 \, c^{2} - 6 \, c d - 5 \, d^{2} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {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 ) + 3 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 64 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 ) - 4 \, {\left ({\left (15 \, c^{2} - 14 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}, \frac {\sqrt {2} {\left ({\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 43 \, c^{2} - 6 \, c d - 5 \, d^{2} + 3 \, {\left (43 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \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 ) - 64 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 ) - 2 \, {\left ({\left (15 \, c^{2} - 14 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{2} - 6 \, c d - 5 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{32 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}\right ] \]

[In]

integrate((c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[1/64*(sqrt(2)*((43*c^2 - 6*c*d - 5*d^2)*cos(f*x + e)^3 + 3*(43*c^2 - 6*c*d - 5*d^2)*cos(f*x + e)^2 + 43*c^2 -
6*c*d - 5*d^2 + 3*(43*c^2 - 6*c*d - 5*d^2)*cos(f*x + e))*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(f*x + e
) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + 3*a*cos(f*x + e)^2 + 2*a*cos(f*x + e) - a)/(cos(f*x + e)^2 +
2*cos(f*x + e) + 1)) - 64*(c^2*cos(f*x + e)^3 + 3*c^2*cos(f*x + e)^2 + 3*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)) - 4*((15*c^2 - 14*c*d - d^2)*cos(f*x + e)^2 + (11*c^2 - 6*c*d - 5*d^2)*cos(f*x
+ e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + 3
*a^3*f*cos(f*x + e) + a^3*f), 1/32*(sqrt(2)*((43*c^2 - 6*c*d - 5*d^2)*cos(f*x + e)^3 + 3*(43*c^2 - 6*c*d - 5*d
^2)*cos(f*x + e)^2 + 43*c^2 - 6*c*d - 5*d^2 + 3*(43*c^2 - 6*c*d - 5*d^2)*cos(f*x + e))*sqrt(a)*arctan(sqrt(2)*
sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 64*(c^2*cos(f*x + e)^3 + 3*c^2*
cos(f*x + e)^2 + 3*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*((15*c^2 - 14*c*d - d^2)*cos(f*x + e)^2 + (11*c^2 - 6*c*d - 5*d^2)*cos(f*x + e))*
sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + 3*a^3*f
*cos(f*x + e) + a^3*f)]

Sympy [F]

\[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{2}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((c+d*sec(f*x+e))**2/(a+a*sec(f*x+e))**(5/2),x)

[Out]

Integral((c + d*sec(e + f*x))**2/(a*(sec(e + f*x) + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{2}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*sec(f*x + e) + c)^2/(a*sec(f*x + e) + a)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(5/2),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. \[ \int \frac {(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int((c + d/cos(e + f*x))^2/(a + a/cos(e + f*x))^(5/2),x)

[Out]

int((c + d/cos(e + f*x))^2/(a + a/cos(e + f*x))^(5/2), x)