3.2.86 \(\int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx\) [186]

Optimal. Leaf size=171 \[ \frac {a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f} \]

[Out]

1/8*a*(8*c^3+12*c^2*d+12*c*d^2+3*d^3)*arctanh(sin(f*x+e))/f+1/6*a*(3*c^3+16*c^2*d+12*c*d^2+4*d^3)*tan(f*x+e)/f
+1/24*a*d*(6*c^2+20*c*d+9*d^2)*sec(f*x+e)*tan(f*x+e)/f+1/12*a*(3*c+4*d)*(c+d*sec(f*x+e))^2*tan(f*x+e)/f+1/4*a*
(c+d*sec(f*x+e))^3*tan(f*x+e)/f

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Rubi [A]
time = 0.20, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4087, 4082, 3872, 3855, 3852, 8} \begin {gather*} \frac {a d \left (6 c^2+20 c d+9 d^2\right ) \tan (e+f x) \sec (e+f x)}{24 f}+\frac {a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac {a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}+\frac {a (3 c+4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3,x]

[Out]

(a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*ArcTanh[Sin[e + f*x]])/(8*f) + (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^
3)*Tan[e + f*x])/(6*f) + (a*d*(6*c^2 + 20*c*d + 9*d^2)*Sec[e + f*x]*Tan[e + f*x])/(24*f) + (a*(3*c + 4*d)*(c +
d*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + (a*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(4*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3872

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

Rule 4082

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Dist[1/(n + 1), Int[(d
*Csc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e,
f, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rule 4087

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

Rubi steps

\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx &=\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{4} \int \sec (e+f x) (c+d \sec (e+f x))^2 (a (4 c+3 d)+a (3 c+4 d) \sec (e+f x)) \, dx\\ &=\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{12} \int \sec (e+f x) (c+d \sec (e+f x)) \left (a \left (12 c^2+15 c d+8 d^2\right )+a \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x)\right ) \, dx\\ &=\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{24} \int \sec (e+f x) \left (3 a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )+4 a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \sec (e+f x)\right ) \, dx\\ &=\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{8} \left (a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )\right ) \int \sec (e+f x) \, dx+\frac {1}{6} \left (a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )\right ) \int \sec ^2(e+f x) \, dx\\ &=\frac {a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}-\frac {\left (a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{6 f}\\ &=\frac {a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{6 f}+\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {a (3 c+4 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {a (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 103, normalized size = 0.60 \begin {gather*} \frac {a \left (3 \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left (24 (c+d)^3+9 d (2 c+d)^2 \sec (e+f x)+6 d^3 \sec ^3(e+f x)+8 d^2 (3 c+d) \tan ^2(e+f x)\right )\right )}{24 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3,x]

[Out]

(a*(3*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(24*(c + d)^3 + 9*d*(2*c + d)
^2*Sec[e + f*x] + 6*d^3*Sec[e + f*x]^3 + 8*d^2*(3*c + d)*Tan[e + f*x]^2)))/(24*f)

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Maple [A]
time = 0.30, size = 223, normalized size = 1.30

method	result	size
derivativedivides	\(\frac {a \,c^{3} \tan \left (f x +e \right )+3 a \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a c \,d^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+a \,d^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f}\)	\(223\)
default	\(\frac {a \,c^{3} \tan \left (f x +e \right )+3 a \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a c \,d^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+a \,d^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f}\)	\(223\)
norman	\(\frac {-\frac {a \left (8 c^{3}+12 c^{2} d +12 c \,d^{2}+3 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a \left (8 c^{3}+36 c^{2} d +36 c \,d^{2}+13 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a \left (72 c^{3}+180 c^{2} d +84 c \,d^{2}+49 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}-\frac {a \left (72 c^{3}+252 c^{2} d +156 c \,d^{2}+31 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {a \left (8 c^{3}+12 c^{2} d +12 c \,d^{2}+3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {a \left (8 c^{3}+12 c^{2} d +12 c \,d^{2}+3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\)	\(259\)
risch	\(\frac {i a \left (16 d^{3}+24 c^{3}-36 c \,d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+216 c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+36 c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+216 c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+144 c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+36 c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}+36 c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+36 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-36 c^{2} d \,{\mathrm e}^{5 i \left (f x +e \right )}-36 c^{2} d \,{\mathrm e}^{7 i \left (f x +e \right )}+192 c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-36 c \,d^{2} {\mathrm e}^{7 i \left (f x +e \right )}+72 c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+72 c^{2} d +48 c \,d^{2}+33 d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+72 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+64 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-9 d^{3} {\mathrm e}^{7 i \left (f x +e \right )}+24 c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-33 d^{3} {\mathrm e}^{5 i \left (f x +e \right )}+72 c^{3} {\mathrm e}^{4 i \left (f x +e \right )}+48 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+9 d^{3} {\mathrm e}^{i \left (f x +e \right )}\right )}{12 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}+\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{2 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c \,d^{2}}{2 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{3}}{8 f}-\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2} d}{2 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c \,d^{2}}{2 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{3}}{8 f}\)	\(545\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))*(c+d*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(a*c^3*tan(f*x+e)+3*a*c^2*d*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+tan(f*x+e)))-3*a*c*d^2*(-2/3-1/3*
sec(f*x+e)^2)*tan(f*x+e)+a*d^3*(-(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+3/8*ln(sec(f*x+e)+tan(f*x+e)))+
a*c^3*ln(sec(f*x+e)+tan(f*x+e))+3*a*c^2*d*tan(f*x+e)+3*a*c*d^2*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+ta
n(f*x+e)))-a*d^3*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e))

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Maxima [A]
time = 0.30, size = 288, normalized size = 1.68 \begin {gather*} \frac {48 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a c d^{2} + 16 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a d^{3} - 3 \, a d^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, a c^{2} d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, a c d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 48 \, a c^{3} \tan \left (f x + e\right ) + 144 \, a c^{2} d \tan \left (f x + e\right )}{48 \, 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))*(c+d*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

1/48*(48*(tan(f*x + e)^3 + 3*tan(f*x + e))*a*c*d^2 + 16*(tan(f*x + e)^3 + 3*tan(f*x + e))*a*d^3 - 3*a*d^3*(2*(
3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(s
in(f*x + e) - 1)) - 36*a*c^2*d*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e)
- 1)) - 36*a*c*d^2*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 48
*a*c^3*log(sec(f*x + e) + tan(f*x + e)) + 48*a*c^3*tan(f*x + e) + 144*a*c^2*d*tan(f*x + e))/f

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Fricas [A]
time = 2.00, size = 220, normalized size = 1.29 \begin {gather*} \frac {3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (6 \, a d^{3} + 8 \, {\left (3 \, a c^{3} + 9 \, a c^{2} d + 6 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + 9 \, {\left (4 \, a c^{2} d + 4 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c+d*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

1/48*(3*(8*a*c^3 + 12*a*c^2*d + 12*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^4*log(sin(f*x + e) + 1) - 3*(8*a*c^3 + 12*a
*c^2*d + 12*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^4*log(-sin(f*x + e) + 1) + 2*(6*a*d^3 + 8*(3*a*c^3 + 9*a*c^2*d + 6
*a*c*d^2 + 2*a*d^3)*cos(f*x + e)^3 + 9*(4*a*c^2*d + 4*a*c*d^2 + a*d^3)*cos(f*x + e)^2 + 8*(3*a*c*d^2 + a*d^3)*
cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \left (\int c^{3} \sec {\left (e + f x \right )}\, dx + \int c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{3}{\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))*(c+d*sec(f*x+e))**3,x)

[Out]

a*(Integral(c**3*sec(e + f*x), x) + Integral(c**3*sec(e + f*x)**2, x) + Integral(d**3*sec(e + f*x)**4, x) + In
tegral(d**3*sec(e + f*x)**5, x) + Integral(3*c*d**2*sec(e + f*x)**3, x) + Integral(3*c*d**2*sec(e + f*x)**4, x
) + Integral(3*c**2*d*sec(e + f*x)**2, x) + Integral(3*c**2*d*sec(e + f*x)**3, x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (161) = 322\).
time = 0.54, size = 380, normalized size = 2.22 \begin {gather*} \frac {3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 36 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 9 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 180 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 84 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 49 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 72 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 252 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 156 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 31 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 108 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 108 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 39 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, 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))*(c+d*sec(f*x+e))^3,x, algorithm="giac")

[Out]

1/24*(3*(8*a*c^3 + 12*a*c^2*d + 12*a*c*d^2 + 3*a*d^3)*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3*(8*a*c^3 + 12*a*c
^2*d + 12*a*c*d^2 + 3*a*d^3)*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(24*a*c^3*tan(1/2*f*x + 1/2*e)^7 + 36*a*c^
2*d*tan(1/2*f*x + 1/2*e)^7 + 36*a*c*d^2*tan(1/2*f*x + 1/2*e)^7 + 9*a*d^3*tan(1/2*f*x + 1/2*e)^7 - 72*a*c^3*tan
(1/2*f*x + 1/2*e)^5 - 180*a*c^2*d*tan(1/2*f*x + 1/2*e)^5 - 84*a*c*d^2*tan(1/2*f*x + 1/2*e)^5 - 49*a*d^3*tan(1/
2*f*x + 1/2*e)^5 + 72*a*c^3*tan(1/2*f*x + 1/2*e)^3 + 252*a*c^2*d*tan(1/2*f*x + 1/2*e)^3 + 156*a*c*d^2*tan(1/2*
f*x + 1/2*e)^3 + 31*a*d^3*tan(1/2*f*x + 1/2*e)^3 - 24*a*c^3*tan(1/2*f*x + 1/2*e) - 108*a*c^2*d*tan(1/2*f*x + 1
/2*e) - 108*a*c*d^2*tan(1/2*f*x + 1/2*e) - 39*a*d^3*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^4)/f

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Mupad [B]
time = 5.34, size = 255, normalized size = 1.49 \begin {gather*} \frac {\left (-2\,a\,c^3-3\,a\,c^2\,d-3\,a\,c\,d^2-\frac {3\,a\,d^3}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (6\,a\,c^3+15\,a\,c^2\,d+7\,a\,c\,d^2+\frac {49\,a\,d^3}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-6\,a\,c^3-21\,a\,c^2\,d-13\,a\,c\,d^2-\frac {31\,a\,d^3}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (2\,a\,c^3+9\,a\,c^2\,d+9\,a\,c\,d^2+\frac {13\,a\,d^3}{4}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{2\,\left (4\,c^3+6\,c^2\,d+6\,c\,d^2+\frac {3\,d^3}{2}\right )}\right )\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{4\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(e + f*x))*(c + d/cos(e + f*x))^3)/cos(e + f*x),x)

[Out]

(tan(e/2 + (f*x)/2)*(2*a*c^3 + (13*a*d^3)/4 + 9*a*c*d^2 + 9*a*c^2*d) - tan(e/2 + (f*x)/2)^7*(2*a*c^3 + (3*a*d^
3)/4 + 3*a*c*d^2 + 3*a*c^2*d) - tan(e/2 + (f*x)/2)^3*(6*a*c^3 + (31*a*d^3)/12 + 13*a*c*d^2 + 21*a*c^2*d) + tan
(e/2 + (f*x)/2)^5*(6*a*c^3 + (49*a*d^3)/12 + 7*a*c*d^2 + 15*a*c^2*d))/(f*(6*tan(e/2 + (f*x)/2)^4 - 4*tan(e/2 +
(f*x)/2)^2 - 4*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1)) + (a*atanh((tan(e/2 + (f*x)/2)*(12*c*d^2 + 1
2*c^2*d + 8*c^3 + 3*d^3))/(2*(6*c*d^2 + 6*c^2*d + 4*c^3 + (3*d^3)/2)))*(12*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/
(4*f)

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