3.5.11 \(\int (a+a \sec (c+d x)) (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [411]

Optimal. Leaf size=63 \[ a A x+\frac {a (2 A+2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d} \]

[Out]

a*A*x+1/2*a*(2*A+2*B+C)*arctanh(sin(d*x+c))/d+a*(B+C)*tan(d*x+c)/d+1/2*a*C*sec(d*x+c)*tan(d*x+c)/d

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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4133, 3855, 3852, 8} \begin {gather*} \frac {a (2 A+2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \tan (c+d x) \sec (c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

a*A*x + (a*(2*A + 2*B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c
+ d*x])/(2*d)

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 4133

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

Rubi steps

\begin {align*} \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+a (2 A+2 B+C) \sec (c+d x)+2 a (B+C) \sec ^2(c+d x)\right ) \, dx\\ &=a A x+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+(a (B+C)) \int \sec ^2(c+d x) \, dx+\frac {1}{2} (a (2 A+2 B+C)) \int \sec (c+d x) \, dx\\ &=a A x+\frac {a (2 A+2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}-\frac {(a (B+C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a A x+\frac {a (2 A+2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(305\) vs. \(2(63)=126\).
time = 1.92, size = 305, normalized size = 4.84 \begin {gather*} \frac {a \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (4 A x-\frac {2 (2 A+2 B+C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {2 (2 A+2 B+C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {C}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 (B+C) \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {C}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 (B+C) \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{2 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a*Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(4*A*x - (2*(2*A + 2*B + C)*Log[Cos[(c + d*x)/2] - S
in[(c + d*x)/2]])/d + (2*(2*A + 2*B + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + C/(d*(Cos[(c + d*x)/2]
- Sin[(c + d*x)/2])^2) + (4*(B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2
])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(
c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))

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Maple [A]
time = 0.61, size = 100, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {A a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \tan \left (d x +c \right )+a C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A a \left (d x +c \right )+B a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a C \tan \left (d x +c \right )}{d}\)	\(100\)
default	\(\frac {A a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \tan \left (d x +c \right )+a C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A a \left (d x +c \right )+B a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a C \tan \left (d x +c \right )}{d}\)	\(100\)
norman	\(\frac {a A x +a A x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (2 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-2 a A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a \left (2 B +C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a \left (2 A +2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a \left (2 A +2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\)	\(149\)
risch	\(a A x -\frac {i a \left (C \,{\mathrm e}^{3 i \left (d x +c \right )}-2 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2 C \,{\mathrm e}^{2 i \left (d x +c \right )}-C \,{\mathrm e}^{i \left (d x +c \right )}-2 B -2 C \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(A*a*ln(sec(d*x+c)+tan(d*x+c))+B*a*tan(d*x+c)+a*C*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c))
)+A*a*(d*x+c)+B*a*ln(sec(d*x+c)+tan(d*x+c))+a*C*tan(d*x+c))

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Maxima [A]
time = 0.33, size = 116, normalized size = 1.84 \begin {gather*} \frac {4 \, {\left (d x + c\right )} A a - C a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a \tan \left (d x + c\right ) + 4 \, C a \tan \left (d x + c\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/4*(4*(d*x + c)*A*a - C*a*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1
)) + 4*A*a*log(sec(d*x + c) + tan(d*x + c)) + 4*B*a*log(sec(d*x + c) + tan(d*x + c)) + 4*B*a*tan(d*x + c) + 4*
C*a*tan(d*x + c))/d

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Fricas [A]
time = 3.27, size = 109, normalized size = 1.73 \begin {gather*} \frac {4 \, A a d x \cos \left (d x + c\right )^{2} + {\left (2 \, A + 2 \, B + C\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + 2 \, B + C\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + C a\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/4*(4*A*a*d*x*cos(d*x + c)^2 + (2*A + 2*B + C)*a*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*A + 2*B + C)*a*cos
(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(2*(B + C)*a*cos(d*x + c) + C*a)*sin(d*x + c))/(d*cos(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

a*(Integral(A, x) + Integral(A*sec(c + d*x), x) + Integral(B*sec(c + d*x), x) + Integral(B*sec(c + d*x)**2, x)
+ Integral(C*sec(c + d*x)**2, x) + Integral(C*sec(c + d*x)**3, x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (59) = 118\).
time = 0.50, size = 141, normalized size = 2.24 \begin {gather*} \frac {2 \, {\left (d x + c\right )} A a + {\left (2 \, A a + 2 \, B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, A a + 2 \, B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/2*(2*(d*x + c)*A*a + (2*A*a + 2*B*a + C*a)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (2*A*a + 2*B*a + C*a)*log(ab
s(tan(1/2*d*x + 1/2*c) - 1)) - 2*(2*B*a*tan(1/2*d*x + 1/2*c)^3 + C*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*
x + 1/2*c) - 3*C*a*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2)/d

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Mupad [B]
time = 3.51, size = 176, normalized size = 2.79 \begin {gather*} \frac {\frac {C\,a\,\sin \left (c+d\,x\right )}{2}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{2}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )}-\frac {2\,\left (-A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}+B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}+\frac {C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{2}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

((C*a*sin(c + d*x))/2 + (B*a*sin(2*c + 2*d*x))/2 + (C*a*sin(2*c + 2*d*x))/2)/(d*(cos(2*c + 2*d*x)/2 + 1/2)) -
(2*(A*a*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*1i - A*a*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))
+ B*a*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*1i + (C*a*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/
2))*1i)/2))/d

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