3.2.98 \(\int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx\) [198]

Optimal. Leaf size=117 \[ \frac {a^2 \tanh ^{-1}(\sin (e+f x))}{d^2 f}-\frac {2 a^2 \sqrt {c-d} (c+2 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{d^2 (c+d)^{3/2} f}-\frac {a^2 (c-d) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 231, normalized size of antiderivative = 1.97, number of steps used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4072, 100, 163, 65, 223, 209, 95, 211} \begin {gather*} \frac {2 a^3 \sqrt {c-d} (c+2 d) \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{d^2 f (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a^3 \tan (e+f x) \text {ArcTan}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{d^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {a^2 (c-d) \tan (e+f x)}{d f (c+d) (c+d \sec (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 211

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4072

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.55, size = 312, normalized size = 2.67 \begin {gather*} \frac {a^2 (d+c \cos (e+f x)) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (1+\sec (e+f x))^2 \left (-\left ((d+c \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+(d+c \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {2 \left (c^2+c d-2 d^2\right ) \text {ArcTan}\left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (d+c \cos (e+f x)) (i \cos (e)+\sin (e))}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {(c-d) d (d \sin (e)-c \sin (f x))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )}\right )}{4 d^2 f (c+d \sec (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^2*(-((d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2]
- Sin[(e + f*x)/2]]) + (d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (2*(c^2 + c*d - 2*d^2)
*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[
e])^2])]*(d + c*Cos[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c
- d)*d*(d*Sin[e] - c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2]))))/(4*d^2*f*(c + d*Sec
[e + f*x])^2)

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Maple [A]
time = 0.38, size = 157, normalized size = 1.34

method	result	size
derivativedivides	\(\frac {8 a^{2} \left (\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 d^{2}}+\frac {\left (c -d \right ) \left (\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 d^{2}}-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 d^{2}}\right )}{f}\)	\(157\)
default	\(\frac {8 a^{2} \left (\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 d^{2}}+\frac {\left (c -d \right ) \left (\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 d^{2}}-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 d^{2}}\right )}{f}\)	\(157\)
risch	\(-\frac {2 i a^{2} \left (c -d \right ) \left (d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}{d f \left (c +d \right ) c \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}+\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}+\frac {2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right )}{\left (c +d \right )^{2} f d}-\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}-\frac {2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right )}{\left (c +d \right )^{2} f d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{d^{2} f}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{d^{2} f}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/f*a^2*(1/8/d^2*ln(tan(1/2*f*x+1/2*e)+1)+1/4*(c-d)/d^2*(d/(c+d)*tan(1/2*f*x+1/2*e)/(c*tan(1/2*f*x+1/2*e)^2-d*
tan(1/2*f*x+1/2*e)^2-c-d)-(c+2*d)/(c+d)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/
2)))-1/8/d^2*ln(tan(1/2*f*x+1/2*e)-1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (112) = 224\).
time = 1.69, size = 589, normalized size = 5.03 \begin {gather*} \left [\frac {{\left (a^{2} c d + 2 \, a^{2} d^{2} + {\left (a^{2} c^{2} + 2 \, a^{2} c d\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{2} + c d + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + {\left (a^{2} c d + a^{2} d^{2} + {\left (a^{2} c^{2} + a^{2} c d\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (a^{2} c d + a^{2} d^{2} + {\left (a^{2} c^{2} + a^{2} c d\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (a^{2} c d - a^{2} d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) + {\left (c d^{3} + d^{4}\right )} f\right )}}, -\frac {2 \, {\left (a^{2} c d + 2 \, a^{2} d^{2} + {\left (a^{2} c^{2} + 2 \, a^{2} c d\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (d \cos \left (f x + e\right ) + c\right )} \sqrt {-\frac {c - d}{c + d}}}{{\left (c - d\right )} \sin \left (f x + e\right )}\right ) - {\left (a^{2} c d + a^{2} d^{2} + {\left (a^{2} c^{2} + a^{2} c d\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + {\left (a^{2} c d + a^{2} d^{2} + {\left (a^{2} c^{2} + a^{2} c d\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (a^{2} c d - a^{2} d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) + {\left (c d^{3} + d^{4}\right )} f\right )}}\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))^2/(c+d*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx\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))**2/(c+d*sec(f*x+e))**2,x)

[Out]

a**2*(Integral(sec(e + f*x)/(c**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**2), x) + Integral(2*sec(e + f*x)**
2/(c**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**2), x) + Integral(sec(e + f*x)**3/(c**2 + 2*c*d*sec(e + f*x)
+ d**2*sec(e + f*x)**2), x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (108) = 216\).
time = 0.53, size = 230, normalized size = 1.97 \begin {gather*} \frac {\frac {a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{d^{2}} - \frac {a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{d^{2}} + \frac {2 \, {\left (a^{2} c^{2} + a^{2} c d - 2 \, a^{2} d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c d^{2} + d^{3}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {2 \, {\left (a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )} {\left (c d + d^{2}\right )}}}{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))^2/(c+d*sec(f*x+e))^2,x, algorithm="giac")

[Out]

(a^2*log(abs(tan(1/2*f*x + 1/2*e) + 1))/d^2 - a^2*log(abs(tan(1/2*f*x + 1/2*e) - 1))/d^2 + 2*(a^2*c^2 + a^2*c*
d - 2*a^2*d^2)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(2*c - 2*d) + arctan((c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f
*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c*d^2 + d^3)*sqrt(-c^2 + d^2)) + 2*(a^2*c*tan(1/2*f*x + 1/2*e) - a^2*d*tan(1
/2*f*x + 1/2*e))/((c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)*(c*d + d^2)))/f

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Mupad [B]
time = 4.79, size = 2563, normalized size = 21.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*atan(((a^2*((a^2*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^2*c^4*d^4))/(2*c*d^4 +
d^5 + c^2*d^3) - (32*a^2*tan(e/2 + (f*x)/2)*(2*c*d^8 - 4*c^3*d^6 + 2*c^5*d^4))/(d^2*(2*c*d^3 + d^4 + c^2*d^2)
)))/d^2 + (32*tan(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3
+ d^4 + c^2*d^2))*1i)/d^2 - (a^2*((a^2*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^2*c^
4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (32*a^2*tan(e/2 + (f*x)/2)*(2*c*d^8 - 4*c^3*d^6 + 2*c^5*d^4))/(d^2*(2*c*d^
3 + d^4 + c^2*d^2))))/d^2 - (32*tan(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*
c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2))*1i)/d^2)/((64*(2*a^6*d^4 - a^6*c^4 - 5*a^6*c*d^3 + a^6*c^3*d + 3*a^6*c^2*
d^2))/(2*c*d^4 + d^5 + c^2*d^3) + (a^2*((a^2*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a
^2*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) - (32*a^2*tan(e/2 + (f*x)/2)*(2*c*d^8 - 4*c^3*d^6 + 2*c^5*d^4))/(d^2*(2
*c*d^3 + d^4 + c^2*d^2))))/d^2 + (32*tan(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7
*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2)))/d^2 + (a^2*((a^2*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*
a^2*c^3*d^5 + a^2*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (32*a^2*tan(e/2 + (f*x)/2)*(2*c*d^8 - 4*c^3*d^6 + 2*c^
5*d^4))/(d^2*(2*c*d^3 + d^4 + c^2*d^2))))/d^2 - (32*tan(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 +
a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2)))/d^2))*2i)/(d^2*f) + (a^2*atan(((a^2*((32*tan(e/2 + (
f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2) + (a^2*
((c + d)^3*(c - d))^(1/2)*(c + 2*d)*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^2*c^4*d^
4))/(2*c*d^4 + d^5 + c^2*d^3) - (32*a^2*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*(2*c*d^8 - 4*c^
3*d^6 + 2*c^5*d^4))/((2*c*d^3 + d^4 + c^2*d^2)*(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2))))/(3*c*d^4 + d^5 + 3*c^2
*d^3 + c^3*d^2))*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*1i)/(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2) + (a^2*((32*tan
(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2)
- (a^2*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^
2*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (32*a^2*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*(2*c*d^
8 - 4*c^3*d^6 + 2*c^5*d^4))/((2*c*d^3 + d^4 + c^2*d^2)*(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2))))/(3*c*d^4 + d^5
+ 3*c^2*d^3 + c^3*d^2))*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*1i)/(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2))/((64*(
2*a^6*d^4 - a^6*c^4 - 5*a^6*c*d^3 + a^6*c^3*d + 3*a^6*c^2*d^2))/(2*c*d^4 + d^5 + c^2*d^3) + (a^2*((32*tan(e/2
+ (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2) + (a
^2*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^2*c^4
*d^4))/(2*c*d^4 + d^5 + c^2*d^3) - (32*a^2*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*(2*c*d^8 - 4
*c^3*d^6 + 2*c^5*d^4))/((2*c*d^3 + d^4 + c^2*d^2)*(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2))))/(3*c*d^4 + d^5 + 3*
c^2*d^3 + c^3*d^2))*((c + d)^3*(c - d))^(1/2)*(c + 2*d))/(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2) - (a^2*((32*tan
(e/2 + (f*x)/2)*(2*a^4*c^5 - 5*a^4*d^5 + 9*a^4*c*d^4 + a^4*c^2*d^3 - 7*a^4*c^3*d^2))/(2*c*d^3 + d^4 + c^2*d^2)
- (a^2*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*((32*(3*a^2*d^8 - 2*a^2*c*d^7 - 4*a^2*c^2*d^6 + 2*a^2*c^3*d^5 + a^
2*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (32*a^2*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d))^(1/2)*(c + 2*d)*(2*c*d^
8 - 4*c^3*d^6 + 2*c^5*d^4))/((2*c*d^3 + d^4 + c^2*d^2)*(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2))))/(3*c*d^4 + d^5
+ 3*c^2*d^3 + c^3*d^2))*((c + d)^3*(c - d))^(1/2)*(c + 2*d))/(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2)))*((c + d)
^3*(c - d))^(1/2)*(c + 2*d)*2i)/(f*(3*c*d^4 + d^5 + 3*c^2*d^3 + c^3*d^2)) - (2*a^2*tan(e/2 + (f*x)/2)*(c - d))
/(d*f*(c + d)*(c + d - tan(e/2 + (f*x)/2)^2*(c - d)))

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