3.862 \(\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx\)

Optimal. Leaf size=101 \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \sin (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]

[Out]

(-17*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (a^2*Sin[c + d*x])/d + a^4/(4*d*(a
- a*Sin[c + d*x])^2) - (7*a^3)/(4*d*(a - a*Sin[c + d*x]))

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Rubi [A]  time = 0.130506, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \sin (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]

[Out]

(-17*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (a^2*Sin[c + d*x])/d + a^4/(4*d*(a
- a*Sin[c + d*x])^2) - (7*a^3)/(4*d*(a - a*Sin[c + d*x]))

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^4}{a^4 (a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^4}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (-1+\frac{a^3}{2 (a-x)^3}-\frac{7 a^2}{4 (a-x)^2}+\frac{17 a}{8 (a-x)}+\frac{a}{8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (1+\sin (c+d x))}{8 d}-\frac{a^2 \sin (c+d x)}{d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{7 a^3}{4 d (a-a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.119892, size = 67, normalized size = 0.66 \[ -\frac{a^2 \left (8 \sin (c+d x)-\frac{14}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+17 \log (1-\sin (c+d x))-\log (\sin (c+d x)+1)\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]

[Out]

-(a^2*(17*Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 - 14/(-1 + Sin[c + d*x]) + 8
*Sin[c + d*x]))/(8*d)

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Maple [B]  time = 0.088, size = 213, normalized size = 2.1 \begin{align*}{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{8\,d}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{9\,{a}^{2}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{9\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{{a}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x)

[Out]

1/4/d*a^2*sin(d*x+c)^7/cos(d*x+c)^4-3/8/d*a^2*sin(d*x+c)^7/cos(d*x+c)^2-3/8*a^2*sin(d*x+c)^5/d-3/4*a^2*sin(d*x
+c)^3/d-9/4*a^2*sin(d*x+c)/d+9/4/d*a^2*ln(sec(d*x+c)+tan(d*x+c))+1/2/d*a^2*tan(d*x+c)^4-1/d*a^2*tan(d*x+c)^2-2
/d*a^2*ln(cos(d*x+c))+1/4/d*a^2*sin(d*x+c)^5/cos(d*x+c)^4-1/8/d*a^2*sin(d*x+c)^5/cos(d*x+c)^2

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Maxima [A]  time = 0.98556, size = 112, normalized size = 1.11 \begin{align*} \frac{a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - 8 \, a^{2} \sin \left (d x + c\right ) + \frac{2 \,{\left (7 \, a^{2} \sin \left (d x + c\right ) - 6 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/8*(a^2*log(sin(d*x + c) + 1) - 17*a^2*log(sin(d*x + c) - 1) - 8*a^2*sin(d*x + c) + 2*(7*a^2*sin(d*x + c) - 6
*a^2)/(sin(d*x + c)^2 - 2*sin(d*x + c) + 1))/d

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Fricas [A]  time = 1.46687, size = 377, normalized size = 3.73 \begin{align*} \frac{16 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} +{\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \,{\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{8 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/8*(16*a^2*cos(d*x + c)^2 - 4*a^2 + (a^2*cos(d*x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(sin(d*x + c) + 1) -
17*(a^2*cos(d*x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(-sin(d*x + c) + 1) - 2*(4*a^2*cos(d*x + c)^2 - a^2)*
sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*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(sec(d*x+c)**5*sin(d*x+c)**4*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.29202, size = 119, normalized size = 1.18 \begin{align*} \frac{2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 34 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 16 \, a^{2} \sin \left (d x + c\right ) + \frac{51 \, a^{2} \sin \left (d x + c\right )^{2} - 74 \, a^{2} \sin \left (d x + c\right ) + 27 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/16*(2*a^2*log(abs(sin(d*x + c) + 1)) - 34*a^2*log(abs(sin(d*x + c) - 1)) - 16*a^2*sin(d*x + c) + (51*a^2*sin
(d*x + c)^2 - 74*a^2*sin(d*x + c) + 27*a^2)/(sin(d*x + c) - 1)^2)/d