3.430 \(\int \cos ^7(c+d x) (a+b \tan ^2(c+d x)) \, dx\)

Optimal. Leaf size=76 \[ -\frac{(a-b) \sin ^7(c+d x)}{7 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d} \]

[Out]

(a*Sin[c + d*x])/d - ((3*a - b)*Sin[c + d*x]^3)/(3*d) + ((3*a - 2*b)*Sin[c + d*x]^5)/(5*d) - ((a - b)*Sin[c +
d*x]^7)/(7*d)

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Rubi [A]  time = 0.0586209, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3676, 373} \[ -\frac{(a-b) \sin ^7(c+d x)}{7 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2),x]

[Out]

(a*Sin[c + d*x])/d - ((3*a - b)*Sin[c + d*x]^3)/(3*d) + ((3*a - 2*b)*Sin[c + d*x]^5)/(5*d) - ((a - b)*Sin[c +
d*x]^7)/(7*d)

Rule 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -
ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n/2] && IntegerQ[p]

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \cos ^7(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^2 \left (a-(a-b) x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a-(3 a-b) x^2+(3 a-2 b) x^4-(a-b) x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a \sin (c+d x)}{d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(a-b) \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.28332, size = 75, normalized size = 0.99 \[ \frac{\sin (c+d x) ((897 a-113 b) \cos (2 (c+d x))+6 (27 a-13 b) \cos (4 (c+d x))+15 a \cos (6 (c+d x))+2286 a-15 b \cos (6 (c+d x))+206 b)}{3360 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2),x]

[Out]

((2286*a + 206*b + (897*a - 113*b)*Cos[2*(c + d*x)] + 6*(27*a - 13*b)*Cos[4*(c + d*x)] + 15*a*Cos[6*(c + d*x)]
- 15*b*Cos[6*(c + d*x)])*Sin[c + d*x])/(3360*d)

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Maple [A]  time = 0.081, size = 92, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{\sin \left ( dx+c \right ) }{35} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +{\frac{\sin \left ( dx+c \right ) a}{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+b*tan(d*x+c)^2),x)

[Out]

1/d*(b*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+1/7*a*(16/5+cos(d*x+
c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.0054, size = 86, normalized size = 1.13 \begin{align*} -\frac{15 \,{\left (a - b\right )} \sin \left (d x + c\right )^{7} - 21 \,{\left (3 \, a - 2 \, b\right )} \sin \left (d x + c\right )^{5} + 35 \,{\left (3 \, a - b\right )} \sin \left (d x + c\right )^{3} - 105 \, a \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/105*(15*(a - b)*sin(d*x + c)^7 - 21*(3*a - 2*b)*sin(d*x + c)^5 + 35*(3*a - b)*sin(d*x + c)^3 - 105*a*sin(d*
x + c))/d

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Fricas [A]  time = 1.52069, size = 163, normalized size = 2.14 \begin{align*} \frac{{\left (15 \,{\left (a - b\right )} \cos \left (d x + c\right )^{6} + 3 \,{\left (6 \, a + b\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (6 \, a + b\right )} \cos \left (d x + c\right )^{2} + 48 \, a + 8 \, b\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/105*(15*(a - b)*cos(d*x + c)^6 + 3*(6*a + b)*cos(d*x + c)^4 + 4*(6*a + b)*cos(d*x + c)^2 + 48*a + 8*b)*sin(d
*x + c)/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+b*tan(d*x+c)**2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)^2),x, algorithm="giac")

[Out]

Timed out