3.34 \(\int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx\)

Optimal. Leaf size=129 \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d} \]

[Out]

(-25*a^4*Log[1 - Sin[c + d*x]])/d - (16*a^4*Sin[c + d*x])/d - (9*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*
x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d) + a^6/(d*(a - a*Sin[c + d*x])^2) - (11*a^5)/(d*(a - a*Sin[c + d*x]))

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Rubi [A]  time = 0.0947062, antiderivative size = 129, 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, 77} \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^5,x]

[Out]

(-25*a^4*Log[1 - Sin[c + d*x]])/d - (16*a^4*Sin[c + d*x])/d - (9*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*
x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d) + a^6/(d*(a - a*Sin[c + d*x])^2) - (11*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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5 (a+x)}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-16 a^3+\frac{2 a^6}{(a-x)^3}-\frac{11 a^5}{(a-x)^2}+\frac{25 a^4}{a-x}-9 a^2 x-4 a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{25 a^4 \log (1-\sin (c+d x))}{d}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.469245, size = 83, normalized size = 0.64 \[ -\frac{a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+54 \sin ^2(c+d x)+192 \sin (c+d x)+\frac{120-132 \sin (c+d x)}{(\sin (c+d x)-1)^2}+300 \log (1-\sin (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^5,x]

[Out]

-(a^4*(300*Log[1 - Sin[c + d*x]] + (120 - 132*Sin[c + d*x])/(-1 + Sin[c + d*x])^2 + 192*Sin[c + d*x] + 54*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 = 0.1, size = 387, normalized size = 3. \begin{align*} -{\frac{5\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d}}-4\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}-5\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4\,d}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-3\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{25\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-25\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}+25\,{\frac{{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-6\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-12\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-25\,{\frac{{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{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)^5,x)

[Out]

-5/2/d*a^4*sin(d*x+c)^7-4/d*a^4*sin(d*x+c)^6-5/d*a^4*sin(d*x+c)^5-3/4/d*a^4*sin(d*x+c)^8+1/4/d*a^4*sin(d*x+c)^
10/cos(d*x+c)^4-3/4/d*a^4*sin(d*x+c)^10/cos(d*x+c)^2+1/d*a^4*sin(d*x+c)^9/cos(d*x+c)^4-5/2/d*a^4*sin(d*x+c)^9/
cos(d*x+c)^2+3/2/d*a^4*sin(d*x+c)^8/cos(d*x+c)^4-3/d*a^4*sin(d*x+c)^8/cos(d*x+c)^2+1/d*a^4*sin(d*x+c)^7/cos(d*
x+c)^4-3/2/d*a^4*sin(d*x+c)^7/cos(d*x+c)^2+1/4/d*a^4*tan(d*x+c)^4-1/2/d*a^4*tan(d*x+c)^2-25/3*a^4*sin(d*x+c)^3
/d-25*a^4*sin(d*x+c)/d+25/d*a^4*ln(sec(d*x+c)+tan(d*x+c))-6*a^4*sin(d*x+c)^4/d-12*a^4*sin(d*x+c)^2/d-25/d*a^4*
ln(cos(d*x+c))

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Maxima [A]  time = 1.10668, size = 147, normalized size = 1.14 \begin{align*} -\frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 54 \, a^{4} \sin \left (d x + c\right )^{2} + 300 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 192 \, a^{4} \sin \left (d x + c\right ) - \frac{12 \,{\left (11 \, a^{4} \sin \left (d x + c\right ) - 10 \, a^{4}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \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)^5,x, algorithm="maxima")

[Out]

-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 54*a^4*sin(d*x + c)^2 + 300*a^4*log(sin(d*x + c) - 1) +
192*a^4*sin(d*x + c) - 12*(11*a^4*sin(d*x + c) - 10*a^4)/(sin(d*x + c)^2 - 2*sin(d*x + c) + 1))/d

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Fricas [A]  time = 1.6719, size = 394, normalized size = 3.05 \begin{align*} -\frac{24 \, a^{4} \cos \left (d x + c\right )^{6} - 272 \, a^{4} \cos \left (d x + c\right )^{4} - 2393 \, a^{4} \cos \left (d x + c\right )^{2} + 1906 \, a^{4} + 2400 \,{\left (a^{4} \cos \left (d x + c\right )^{2} + 2 \, a^{4} \sin \left (d x + c\right ) - 2 \, a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \,{\left (8 \, a^{4} \cos \left (d x + c\right )^{4} - 96 \, a^{4} \cos \left (d x + c\right )^{2} + 181 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \,{\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((a+a*sin(d*x+c))^4*tan(d*x+c)^5,x, algorithm="fricas")

[Out]

-1/96*(24*a^4*cos(d*x + c)^6 - 272*a^4*cos(d*x + c)^4 - 2393*a^4*cos(d*x + c)^2 + 1906*a^4 + 2400*(a^4*cos(d*x
+ c)^2 + 2*a^4*sin(d*x + c) - 2*a^4)*log(-sin(d*x + c) + 1) - 10*(8*a^4*cos(d*x + c)^4 - 96*a^4*cos(d*x + c)^
2 + 181*a^4)*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((a+a*sin(d*x+c))**4*tan(d*x+c)**5,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)^5,x, algorithm="giac")

[Out]

Timed out