3.32 \(\int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\)

Optimal. Leaf size=96 \[ \frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d} \]

[Out]

(35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^
4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.109265, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3791, 3770, 3767, 8, 3768} \[ \frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^4,x]

[Out]

(35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^
4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(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 3770

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

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]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx\\ &=a^4 \int \sec (c+d x) \, dx+a^4 \int \sec ^5(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^2(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^4(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \left (3 a^4\right ) \int \sec ^3(c+d x) \, dx+\left (3 a^4\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a^4\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (4 a^4\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{27 a^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{1}{8} \left (3 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{27 a^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{4 a^4 \tan ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [B]  time = 6.39469, size = 877, normalized size = 9.14 \[ -\frac{35 \cos ^4(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)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{35 \cos ^4(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)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{5 \cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 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{5 \cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \left (97 \cos \left (\frac{c}{2}\right )-65 \sin \left (\frac{c}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \left (-97 \cos \left (\frac{c}{2}\right )-65 \sin \left (\frac{c}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 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 ^4(c+d x) (\sec (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4,x]

[Out]

(-35*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/
(128*d) + (35*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c +
d*x])^4)/(128*d) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(256*d*(Cos[c/2 + (d*x)/2] - S
in[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(24*d*(Cos[c
/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec
[c + d*x])^4*(97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2
])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2]
)*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(2
56*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^
4*Sin[(d*x)/2])/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec
[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(-97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 +
(d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2
])/(12*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 = 0.037, size = 102, normalized size = 1.1 \begin{align*}{\frac{35\,{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{20\,{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{27\,{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{4\,{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4} \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)*(a+a*sec(d*x+c))^4,x)

[Out]

35/8/d*a^4*ln(sec(d*x+c)+tan(d*x+c))+20/3*a^4*tan(d*x+c)/d+27/8*a^4*sec(d*x+c)*tan(d*x+c)/d+4/3/d*a^4*tan(d*x+
c)*sec(d*x+c)^2+1/4*a^4*sec(d*x+c)^3*tan(d*x+c)/d

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Maxima [A]  time = 1.13003, size = 236, normalized size = 2.46 \begin{align*} \frac{64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} - 3 \, a^{4}{\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 )} - 72 \, a^{4}{\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 )} + 48 \, a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 192 \, a^{4} \tan \left (d x + c\right )}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

1/48*(64*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^4 - 3*a^4*(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)) - 72*a^4*(2*sin(d*x + c)/(sin(d*x
+ c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 48*a^4*log(sec(d*x + c) + tan(d*x + c)) + 192*
a^4*tan(d*x + c))/d

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Fricas [A]  time = 1.73283, size = 292, normalized size = 3.04 \begin{align*} \frac{105 \, a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (160 \, a^{4} \cos \left (d x + c\right )^{3} + 81 \, a^{4} \cos \left (d x + c\right )^{2} + 32 \, a^{4} \cos \left (d x + c\right ) + 6 \, a^{4}\right )} \sin \left (d x + c\right )}{48 \, 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)*(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/48*(105*a^4*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 105*a^4*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(160*a^
4*cos(d*x + c)^3 + 81*a^4*cos(d*x + c)^2 + 32*a^4*cos(d*x + c) + 6*a^4)*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^{4} \left (\int \sec{\left (c + d x \right )}\, dx + \int 4 \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 \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)*(a+a*sec(d*x+c))**4,x)

[Out]

a**4*(Integral(sec(c + d*x), x) + Integral(4*sec(c + d*x)**2, x) + Integral(6*sec(c + d*x)**3, x) + Integral(4
*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**5, x))

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Giac [A]  time = 1.3898, size = 165, normalized size = 1.72 \begin{align*} \frac{105 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (105 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 385 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 511 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 279 \, a^{4} \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}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/24*(105*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(105*a^4*tan
(1/2*d*x + 1/2*c)^7 - 385*a^4*tan(1/2*d*x + 1/2*c)^5 + 511*a^4*tan(1/2*d*x + 1/2*c)^3 - 279*a^4*tan(1/2*d*x +
1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d