3.264 \(\int (a+b \sec (c+d x)) \tan ^6(c+d x) \, dx\)

Optimal. Leaf size=102 \[ \frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d} \]

[Out]

-(a*x) - (5*b*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*b*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*b*Se
c[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)

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Rubi [A]  time = 0.0953595, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3881, 3770} \[ \frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^6,x]

[Out]

-(a*x) - (5*b*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*b*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*b*Se
c[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

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 (a+b \sec (c+d x)) \tan ^6(c+d x) \, dx &=\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{6} \int (6 a+5 b \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}+\frac{1}{24} \int (24 a+15 b \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{48} \int (48 a+15 b \sec (c+d x)) \, dx\\ &=-a x+\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac{1}{16} (5 b) \int \sec (c+d x) \, dx\\ &=-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{(16 a+5 b \sec (c+d x)) \tan (c+d x)}{16 d}-\frac{(8 a+5 b \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac{(6 a+5 b \sec (c+d x)) \tan ^5(c+d x)}{30 d}\\ \end{align*}

Mathematica [A]  time = 1.02845, size = 103, normalized size = 1.01 \[ \frac{\frac{1}{8} \tan (c+d x) \sec ^5(c+d x) (1168 a \cos (c+d x)+568 a \cos (3 (c+d x))+184 a \cos (5 (c+d x))+140 b \cos (2 (c+d x))+165 b \cos (4 (c+d x))+295 b)-240 a \tan ^{-1}(\tan (c+d x))-75 b \tanh ^{-1}(\sin (c+d x))}{240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^6,x]

[Out]

(-240*a*ArcTan[Tan[c + d*x]] - 75*b*ArcTanh[Sin[c + d*x]] + ((295*b + 1168*a*Cos[c + d*x] + 140*b*Cos[2*(c + d
*x)] + 568*a*Cos[3*(c + d*x)] + 165*b*Cos[4*(c + d*x)] + 184*a*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Tan[c + d*x])/
8)/(240*d)

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Maple [A]  time = 0.042, size = 178, normalized size = 1.8 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}a}{5\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{a\tan \left ( dx+c \right ) }{d}}-ax-{\frac{ac}{d}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}b}{16\,d}}+{\frac{5\,b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{5\,\sin \left ( dx+c \right ) b}{16\,d}}-{\frac{5\,b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))*tan(d*x+c)^6,x)

[Out]

1/5/d*tan(d*x+c)^5*a-1/3/d*a*tan(d*x+c)^3+1/d*a*tan(d*x+c)-a*x-1/d*a*c+1/6/d*b*sin(d*x+c)^7/cos(d*x+c)^6-1/24/
d*b*sin(d*x+c)^7/cos(d*x+c)^4+1/16/d*b*sin(d*x+c)^7/cos(d*x+c)^2+1/16/d*sin(d*x+c)^5*b+5/48/d*b*sin(d*x+c)^3+5
/16/d*sin(d*x+c)*b-5/16/d*b*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.46158, size = 181, normalized size = 1.77 \begin{align*} \frac{32 \,{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a - 5 \, b{\left (\frac{2 \,{\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*tan(d*x+c)^6,x, algorithm="maxima")

[Out]

1/480*(32*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a - 5*b*(2*(33*sin(d*x + c)^
5 - 40*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) + 15*log(s
in(d*x + c) + 1) - 15*log(sin(d*x + c) - 1)))/d

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Fricas [A]  time = 0.943063, size = 377, normalized size = 3.7 \begin{align*} -\frac{480 \, a d x \cos \left (d x + c\right )^{6} + 75 \, b \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 75 \, b \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (368 \, a \cos \left (d x + c\right )^{5} + 165 \, b \cos \left (d x + c\right )^{4} - 176 \, a \cos \left (d x + c\right )^{3} - 130 \, b \cos \left (d x + c\right )^{2} + 48 \, a \cos \left (d x + c\right ) + 40 \, b\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*tan(d*x+c)^6,x, algorithm="fricas")

[Out]

-1/480*(480*a*d*x*cos(d*x + c)^6 + 75*b*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 75*b*cos(d*x + c)^6*log(-sin(d*
x + c) + 1) - 2*(368*a*cos(d*x + c)^5 + 165*b*cos(d*x + c)^4 - 176*a*cos(d*x + c)^3 - 130*b*cos(d*x + c)^2 + 4
8*a*cos(d*x + c) + 40*b)*sin(d*x + c))/(d*cos(d*x + c)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right ) \tan ^{6}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*tan(d*x+c)**6,x)

[Out]

Integral((a + b*sec(c + d*x))*tan(c + d*x)**6, x)

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Giac [B]  time = 5.0167, size = 308, normalized size = 3.02 \begin{align*} -\frac{240 \,{\left (d x + c\right )} a + 75 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 75 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 75 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 425 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4128 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 990 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 4128 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 990 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 425 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 75 \, b \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 )}^{6}}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*tan(d*x+c)^6,x, algorithm="giac")

[Out]

-1/240*(240*(d*x + c)*a + 75*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 75*b*log(abs(tan(1/2*d*x + 1/2*c) - 1)) +
2*(240*a*tan(1/2*d*x + 1/2*c)^11 - 75*b*tan(1/2*d*x + 1/2*c)^11 - 1520*a*tan(1/2*d*x + 1/2*c)^9 + 425*b*tan(1/
2*d*x + 1/2*c)^9 + 4128*a*tan(1/2*d*x + 1/2*c)^7 - 990*b*tan(1/2*d*x + 1/2*c)^7 - 4128*a*tan(1/2*d*x + 1/2*c)^
5 - 990*b*tan(1/2*d*x + 1/2*c)^5 + 1520*a*tan(1/2*d*x + 1/2*c)^3 + 425*b*tan(1/2*d*x + 1/2*c)^3 - 240*a*tan(1/
2*d*x + 1/2*c) - 75*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d