3.21 \(\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=93 \[ \frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d} \]

[Out]

(15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^
3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d

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Rubi [A]  time = 0.113988, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3791, 3767, 8, 3768, 3770} \[ \frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]

[Out]

(15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^
3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d

Rule 3791

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 3767

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 8

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

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+3 a^3 \sec ^4(c+d x)+a^3 \sec ^5(c+d x)\right ) \, dx\\ &=a^3 \int \sec ^2(c+d x) \, dx+a^3 \int \sec ^5(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \left (3 a^3\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{2} \left (3 a^3\right ) \int \sec (c+d x) \, dx-\frac{a^3 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a^3 \tan ^3(c+d x)}{d}+\frac{1}{8} \left (3 a^3\right ) \int \sec (c+d x) \, dx\\ &=\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a^3 \tan ^3(c+d x)}{d}\\ \end{align*}

Mathematica [B]  time = 6.39876, size = 877, normalized size = 9.43 \[ -\frac{15 \cos ^3(c+d x) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) (\sec (c+d x) a+a)^3 \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{64 d}+\frac{15 \cos ^3(c+d x) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) (\sec (c+d x) a+a)^3 \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{64 d}+\frac{3 \cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{3 \cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \left (19 \cos \left (\frac{c}{2}\right )-11 \sin \left (\frac{c}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \left (-19 \cos \left (\frac{c}{2}\right )-11 \sin \left (\frac{c}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4}-\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]

[Out]

(-15*Cos[c + d*x]^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/
(64*d) + (15*Cos[c + d*x]^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d
*x])^3)/(64*d) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(128*d*(Cos[c/2 + (d*x)/2] - Sin
[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(16*d*(Cos[c/2
] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c
+ d*x])^3*(19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])
^2) + (3*Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(8*d*(Cos[c/2] - Sin[c/2])*(
Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(128*
d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*S
in[(d*x)/2])/(16*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^3*Sec[c/
2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(-19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d
*x)/2] + Sin[c/2 + (d*x)/2])^2) + (3*Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/
(8*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

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Maple [A]  time = 0.033, size = 101, normalized size = 1.1 \begin{align*} 3\,{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{15\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+a*sec(d*x+c))^3,x)

[Out]

3*a^3*tan(d*x+c)/d+15/8*a^3*sec(d*x+c)*tan(d*x+c)/d+15/8/d*a^3*ln(sec(d*x+c)+tan(d*x+c))+1/d*a^3*tan(d*x+c)*se
c(d*x+c)^2+1/4*a^3*sec(d*x+c)^3*tan(d*x+c)/d

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Maxima [A]  time = 1.05683, size = 211, normalized size = 2.27 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3}{\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 )} + 16 \, a^{3} \tan \left (d x + c\right )}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/16*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^3 - a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 -
2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 12*a^3*(2*sin(d*x + c)/(sin(d*x +
c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 16*a^3*tan(d*x + c))/d

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Fricas [A]  time = 1.75651, size = 286, normalized size = 3.08 \begin{align*} \frac{15 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/16*(15*a^3*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 15*a^3*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*a^3*c
os(d*x + c)^3 + 15*a^3*cos(d*x + c)^2 + 8*a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*(a+a*sec(d*x+c))**3,x)

[Out]

a**3*(Integral(sec(c + d*x)**2, x) + Integral(3*sec(c + d*x)**3, x) + Integral(3*sec(c + d*x)**4, x) + Integra
l(sec(c + d*x)**5, x))

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Giac [A]  time = 1.35389, size = 165, normalized size = 1.77 \begin{align*} \frac{15 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 55 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 73 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 49 \, a^{3} \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 )}^{4}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(15*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(15*a^3*tan(1/2
*d*x + 1/2*c)^7 - 55*a^3*tan(1/2*d*x + 1/2*c)^5 + 73*a^3*tan(1/2*d*x + 1/2*c)^3 - 49*a^3*tan(1/2*d*x + 1/2*c))
/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d