3.1.84 \(\int \frac {\tan (c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [84]

Optimal. Leaf size=105 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{16 a^4 d}+\frac {1}{8 d (a+a \sin (c+d x))^4}-\frac {1}{12 a d (a+a \sin (c+d x))^3}-\frac {1}{16 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {1}{16 d \left (a^4+a^4 \sin (c+d x)\right )} \]

[Out]

1/16*arctanh(sin(d*x+c))/a^4/d+1/8/d/(a+a*sin(d*x+c))^4-1/12/a/d/(a+a*sin(d*x+c))^3-1/16/d/(a^2+a^2*sin(d*x+c)
)^2-1/16/d/(a^4+a^4*sin(d*x+c))

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Rubi [A]
time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2786, 78, 212} \begin {gather*} -\frac {1}{16 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{16 a^4 d}-\frac {1}{16 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {1}{12 a d (a \sin (c+d x)+a)^3}+\frac {1}{8 d (a \sin (c+d x)+a)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/(16*a^4*d) + 1/(8*d*(a + a*Sin[c + d*x])^4) - 1/(12*a*d*(a + a*Sin[c + d*x])^3) - 1/(16*
d*(a^2 + a^2*Sin[c + d*x])^2) - 1/(16*d*(a^4 + a^4*Sin[c + d*x]))

Rule 78

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2786

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

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Mathematica [A]
time = 0.18, size = 62, normalized size = 0.59 \begin {gather*} \frac {3 \tanh ^{-1}(\sin (c+d x))-\frac {4+19 \sin (c+d x)+12 \sin ^2(c+d x)+3 \sin ^3(c+d x)}{(1+\sin (c+d x))^4}}{48 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*ArcTanh[Sin[c + d*x]] - (4 + 19*Sin[c + d*x] + 12*Sin[c + d*x]^2 + 3*Sin[c + d*x]^3)/(1 + Sin[c + d*x])^4)/
(48*a^4*d)

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Maple [A]
time = 0.32, size = 79, normalized size = 0.75

method	result	size
derivativedivides	\(\frac {\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{32}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{32}}{d \,a^{4}}\)	\(79\)
default	\(\frac {\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{32}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{32}}{d \,a^{4}}\)	\(79\)
risch	\(-\frac {i \left (-3 \,{\mathrm e}^{i \left (d x +c \right )}+85 \,{\mathrm e}^{3 i \left (d x +c \right )}+24 i {\mathrm e}^{2 i \left (d x +c \right )}-80 i {\mathrm e}^{4 i \left (d x +c \right )}-85 \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{24 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d \,a^{4}}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d/a^4*(1/8/(1+sin(d*x+c))^4-1/12/(1+sin(d*x+c))^3-1/16/(1+sin(d*x+c))^2-1/16/(1+sin(d*x+c))+1/32*ln(1+sin(d*
x+c))-1/32*ln(sin(d*x+c)-1))

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Maxima [A]
time = 0.28, size = 121, normalized size = 1.15 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right )^{2} + 19 \, \sin \left (d x + c\right ) + 4\right )}}{a^{4} \sin \left (d x + c\right )^{4} + 4 \, a^{4} \sin \left (d x + c\right )^{3} + 6 \, a^{4} \sin \left (d x + c\right )^{2} + 4 \, a^{4} \sin \left (d x + c\right ) + a^{4}} - \frac {3 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {3 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4}}}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(2*(3*sin(d*x + c)^3 + 12*sin(d*x + c)^2 + 19*sin(d*x + c) + 4)/(a^4*sin(d*x + c)^4 + 4*a^4*sin(d*x + c)
^3 + 6*a^4*sin(d*x + c)^2 + 4*a^4*sin(d*x + c) + a^4) - 3*log(sin(d*x + c) + 1)/a^4 + 3*log(sin(d*x + c) - 1)/
a^4)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (95) = 190\).
time = 0.35, size = 198, normalized size = 1.89 \begin {gather*} \frac {24 \, \cos \left (d x + c\right )^{2} + 3 \, {\left (\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 22\right )} \sin \left (d x + c\right ) - 32}{96 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 8 \, a^{4} d \cos \left (d x + c\right )^{2} + 8 \, a^{4} d - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{2} - 2 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(24*cos(d*x + c)^2 + 3*(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)*log(
sin(d*x + c) + 1) - 3*(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)*log(-sin(d
*x + c) + 1) + 2*(3*cos(d*x + c)^2 - 22)*sin(d*x + c) - 32)/(a^4*d*cos(d*x + c)^4 - 8*a^4*d*cos(d*x + c)^2 + 8
*a^4*d - 4*(a^4*d*cos(d*x + c)^2 - 2*a^4*d)*sin(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tan {\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(c + d*x)/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a**4

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Giac [A]
time = 17.15, size = 91, normalized size = 0.87 \begin {gather*} \frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4}} - \frac {25 \, \sin \left (d x + c\right )^{4} + 124 \, \sin \left (d x + c\right )^{3} + 246 \, \sin \left (d x + c\right )^{2} + 252 \, \sin \left (d x + c\right ) + 57}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(12*log(abs(sin(d*x + c) + 1))/a^4 - 12*log(abs(sin(d*x + c) - 1))/a^4 - (25*sin(d*x + c)^4 + 124*sin(d*
x + c)^3 + 246*sin(d*x + c)^2 + 252*sin(d*x + c) + 57)/(a^4*(sin(d*x + c) + 1)^4))/d

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Mupad [B]
time = 10.07, size = 240, normalized size = 2.29 \begin {gather*} \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^4\,d}+\frac {-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{8}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}+\frac {10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+28\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+70\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+56\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+28\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a^4\right )} \end {gather*}

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]

atanh(tan(c/2 + (d*x)/2))/(8*a^4*d) + (tan(c/2 + (d*x)/2)^2 - tan(c/2 + (d*x)/2)/8 + (43*tan(c/2 + (d*x)/2)^3)
/24 + (10*tan(c/2 + (d*x)/2)^4)/3 + (43*tan(c/2 + (d*x)/2)^5)/24 + tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^7
/8)/(d*(28*a^4*tan(c/2 + (d*x)/2)^2 + 56*a^4*tan(c/2 + (d*x)/2)^3 + 70*a^4*tan(c/2 + (d*x)/2)^4 + 56*a^4*tan(c
/2 + (d*x)/2)^5 + 28*a^4*tan(c/2 + (d*x)/2)^6 + 8*a^4*tan(c/2 + (d*x)/2)^7 + a^4*tan(c/2 + (d*x)/2)^8 + a^4 +
8*a^4*tan(c/2 + (d*x)/2)))

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