3.35 \(\int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx\)

Optimal. Leaf size=107 \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{4 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^5}{d (a-a \sin (c+d x))}+\frac{12 a^4 \sin (c+d x)}{d}+\frac{16 a^4 \log (1-\sin (c+d x))}{d} \]

[Out]

(16*a^4*Log[1 - Sin[c + d*x]])/d + (12*a^4*Sin[c + d*x])/d + (4*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)
/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d) + (4*a^5)/(d*(a - a*Sin[c + d*x]))

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Rubi [A]  time = 0.0786493, antiderivative size = 107, 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 = {2707, 88} \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{4 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^5}{d (a-a \sin (c+d x))}+\frac{12 a^4 \sin (c+d x)}{d}+\frac{16 a^4 \log (1-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]

[Out]

(16*a^4*Log[1 - Sin[c + d*x]])/d + (12*a^4*Sin[c + d*x])/d + (4*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)
/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d) + (4*a^5)/(d*(a - a*Sin[c + d*x]))

Rule 2707

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

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 (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 (a+x)^2}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (12 a^3+\frac{4 a^5}{(a-x)^2}-\frac{16 a^4}{a-x}+8 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{16 a^4 \log (1-\sin (c+d x))}{d}+\frac{12 a^4 \sin (c+d x)}{d}+\frac{4 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^5}{d (a-a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.155479, size = 76, normalized size = 0.71 \[ \frac{a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+48 \sin ^2(c+d x)+144 \sin (c+d x)+\frac{48}{1-\sin (c+d x)}+192 \log (1-\sin (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]

[Out]

(a^4*(192*Log[1 - Sin[c + d*x]] + 48/(1 - Sin[c + d*x]) + 144*Sin[c + d*x] + 48*Sin[c + d*x]^2 + 16*Sin[c + d*
x]^3 + 3*Sin[c + d*x]^4))/(12*d)

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Maple [B]  time = 0.084, size = 245, normalized size = 2.3 \begin{align*}{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{2\,d}}+{\frac{15\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{15\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+16\,{\frac{{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d}}+{\frac{16\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+16\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}-16\,{\frac{{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x)

[Out]

1/2/d*a^4*sin(d*x+c)^8/cos(d*x+c)^2+1/2/d*a^4*sin(d*x+c)^6+15/4*a^4*sin(d*x+c)^4/d+15/2*a^4*sin(d*x+c)^2/d+16/
d*a^4*ln(cos(d*x+c))+2/d*a^4*sin(d*x+c)^7/cos(d*x+c)^2+2/d*a^4*sin(d*x+c)^5+16/3*a^4*sin(d*x+c)^3/d+16*a^4*sin
(d*x+c)/d-16/d*a^4*ln(sec(d*x+c)+tan(d*x+c))+3/d*a^4*sin(d*x+c)^6/cos(d*x+c)^2+2/d*a^4*sin(d*x+c)^5/cos(d*x+c)
^2+1/2/d*a^4*tan(d*x+c)^2

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Maxima [A]  time = 1.06484, size = 115, normalized size = 1.07 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 48 \, a^{4} \sin \left (d x + c\right )^{2} + 192 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a^{4} \sin \left (d x + c\right ) - \frac{48 \, a^{4}}{\sin \left (d x + c\right ) - 1}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="maxima")

[Out]

1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 48*a^4*sin(d*x + c)^2 + 192*a^4*log(sin(d*x + c) - 1) + 1
44*a^4*sin(d*x + c) - 48*a^4/(sin(d*x + c) - 1))/d

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Fricas [A]  time = 1.5324, size = 293, normalized size = 2.74 \begin{align*} \frac{104 \, a^{4} \cos \left (d x + c\right )^{4} - 976 \, a^{4} \cos \left (d x + c\right )^{2} + 689 \, a^{4} + 1536 \,{\left (a^{4} \sin \left (d x + c\right ) - a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (24 \, a^{4} \cos \left (d x + c\right )^{4} - 304 \, a^{4} \cos \left (d x + c\right )^{2} - 1073 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \,{\left (d \sin \left (d x + c\right ) - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="fricas")

[Out]

1/96*(104*a^4*cos(d*x + c)^4 - 976*a^4*cos(d*x + c)^2 + 689*a^4 + 1536*(a^4*sin(d*x + c) - a^4)*log(-sin(d*x +
c) + 1) + (24*a^4*cos(d*x + c)^4 - 304*a^4*cos(d*x + c)^2 - 1073*a^4)*sin(d*x + c))/(d*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((a+a*sin(d*x+c))**4*tan(d*x+c)**3,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((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="giac")

[Out]

Timed out