3.171 \(\int \frac{\tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=126 \[ -\frac{a^3 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left (a^2-b^2\right )}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]

[Out]

((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*L
og[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)

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Rubi [A]  time = 0.190922, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2721, 1647, 801} \[ -\frac{a^3 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left (a^2-b^2\right )}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*L
og[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)

Rule 2721

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)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

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

Mathematica [A]  time = 0.484179, size = 117, normalized size = 0.93 \[ \frac{-\frac{4 a^3 \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}-\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{(a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{(a-b)^2}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.059, size = 164, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) a}{2\,d \left ( a+b \right ) ^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) b}{4\,d \left ( a+b \right ) ^{2}}}+{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) a}{2\,d \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) b}{4\,d \left ( a-b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53645, size = 192, normalized size = 1.52 \begin{align*} -\frac{\frac{4 \, a^{3} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (2 \, a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{{\left (2 \, a + b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (b \sin \left (d x + c\right ) - a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a^3*log(b*sin(d*x + c) + a)/(a^4 - 2*a^2*b^2 + b^4) - (2*a - b)*log(sin(d*x + c) + 1)/(a^2 - 2*a*b + b
^2) - (2*a + b)*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b^2) - 2*(b*sin(d*x + c) - a)/((a^2 - b^2)*sin(d*x + c)^2
- a^2 + b^2))/d

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Fricas [A]  time = 1.96149, size = 367, normalized size = 2.91 \begin{align*} -\frac{4 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, a^{3} + 2 \, a b^{2} + 2 \,{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**3/(a+b*sin(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**3/(a + b*sin(c + d*x)), x)

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Giac [A]  time = 1.98003, size = 239, normalized size = 1.9 \begin{align*} -\frac{\frac{4 \, a^{3} b \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac{{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{{\left (2 \, a + b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac{2 \,{\left (a^{3} \sin \left (d x + c\right )^{2} - a^{2} b \sin \left (d x + c\right ) + b^{3} \sin \left (d x + c\right ) - a b^{2}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\sin \left (d x + c\right )^{2} - 1\right )}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/4*(4*a^3*b*log(abs(b*sin(d*x + c) + a))/(a^4*b - 2*a^2*b^3 + b^5) - (2*a - b)*log(abs(sin(d*x + c) + 1))/(a
^2 - 2*a*b + b^2) - (2*a + b)*log(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) + 2*(a^3*sin(d*x + c)^2 - a^2*b*s
in(d*x + c) + b^3*sin(d*x + c) - a*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(sin(d*x + c)^2 - 1)))/d