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

Optimal. Leaf size=88 \[ -\frac {a^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {7 a^4 \sin ^2(c+d x)}{2 d}-\frac {8 a^4 \sin (c+d x)}{d}-\frac {8 a^4 \log (1-\sin (c+d x))}{d} \]

[Out]

-8*a^4*ln(1-sin(d*x+c))/d-8*a^4*sin(d*x+c)/d-7/2*a^4*sin(d*x+c)^2/d-4/3*a^4*sin(d*x+c)^3/d-1/4*a^4*sin(d*x+c)^
4/d

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Rubi [A]  time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2707, 77} \[ -\frac {a^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {7 a^4 \sin ^2(c+d x)}{2 d}-\frac {8 a^4 \sin (c+d x)}{d}-\frac {8 a^4 \log (1-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))))

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]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 62, normalized size = 0.70 \[ -\frac {a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+42 \sin ^2(c+d x)+96 \sin (c+d x)+96 \log (1-\sin (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^4*(96*Log[1 - Sin[c + d*x]] + 96*Sin[c + d*x] + 42*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x
]^4))/d

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fricas [A]  time = 0.44, size = 74, normalized size = 0.84 \[ -\frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 48 \, a^{4} \cos \left (d x + c\right )^{2} + 96 \, a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 16 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 7 \, a^{4}\right )} \sin \left (d x + c\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*cos(d*x + c)^4 - 48*a^4*cos(d*x + c)^2 + 96*a^4*log(-sin(d*x + c) + 1) - 16*(a^4*cos(d*x + c)^2 -
7*a^4)*sin(d*x + c))/d

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.16, size = 101, normalized size = 1.15 \[ -\frac {a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {7 a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {8 a^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {4 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 a^{4} \sin \left (d x +c \right )}{d}+\frac {8 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.29, size = 70, normalized size = 0.80 \[ -\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 42 \, a^{4} \sin \left (d x + c\right )^{2} + 96 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 96 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 42*a^4*sin(d*x + c)^2 + 96*a^4*log(sin(d*x + c) - 1) + 9
6*a^4*sin(d*x + c))/d

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mupad [B]  time = 6.63, size = 131, normalized size = 1.49 \[ \frac {8\,a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}-\frac {28\,a^4\,\sin \left (c+d\,x\right )}{3\,d}-\frac {16\,a^4\,\ln \left (\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2}{d}-\frac {a^4\,{\cos \left (c+d\,x\right )}^4}{4\,d}+\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*a^4*log(1/cos(c/2 + (d*x)/2)^2))/d - (28*a^4*sin(c + d*x))/(3*d) - (16*a^4*log((cos(c/2 + (d*x)/2) - sin(c/
2 + (d*x)/2))/cos(c/2 + (d*x)/2)))/d + (4*a^4*cos(c + d*x)^2)/d - (a^4*cos(c + d*x)^4)/(4*d) + (4*a^4*cos(c +
d*x)^2*sin(c + d*x))/(3*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int 4 \sin {\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int \tan {\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**4*tan(d*x+c),x)

[Out]

a**4*(Integral(4*sin(c + d*x)*tan(c + d*x), x) + Integral(6*sin(c + d*x)**2*tan(c + d*x), x) + Integral(4*sin(
c + d*x)**3*tan(c + d*x), x) + Integral(sin(c + d*x)**4*tan(c + d*x), x) + Integral(tan(c + d*x), x))

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