3.1.42 \(\int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx\) [42]

Optimal. Leaf size=73 \[ -\frac {15 x}{8 a}+\frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d} \]

[Out]

-15/8*x/a+15/8*tan(d*x+c)/a/d-5/8*sin(d*x+c)^2*tan(d*x+c)/a/d-1/4*sin(d*x+c)^4*tan(d*x+c)/a/d

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Rubi [A]
time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3254, 2671, 294, 327, 209} \begin {gather*} \frac {15 \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {15 x}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]

[Out]

(-15*x)/(8*a) + (15*Tan[c + d*x])/(8*a*d) - (5*Sin[c + d*x]^2*Tan[c + d*x])/(8*a*d) - (Sin[c + d*x]^4*Tan[c +
d*x])/(4*a*d)

Rule 209

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

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 3254

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \sin ^4(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 a d}\\ &=-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}+\frac {15 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=\frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac {15 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=-\frac {15 x}{8 a}+\frac {15 \tan (c+d x)}{8 a d}-\frac {5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac {\sin ^4(c+d x) \tan (c+d x)}{4 a d}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 44, normalized size = 0.60 \begin {gather*} -\frac {60 c+60 d x-16 \sin (2 (c+d x))+\sin (4 (c+d x))-32 \tan (c+d x)}{32 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]

[Out]

-1/32*(60*c + 60*d*x - 16*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] - 32*Tan[c + d*x])/(a*d)

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Maple [A]
time = 0.21, size = 57, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\tan \left (d x +c \right )-\frac {-\frac {9 \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {7 \tan \left (d x +c \right )}{8}}{\left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}-\frac {15 \arctan \left (\tan \left (d x +c \right )\right )}{8}}{d a}\)	\(57\)
default	\(\frac {\tan \left (d x +c \right )-\frac {-\frac {9 \left (\tan ^{3}\left (d x +c \right )\right )}{8}-\frac {7 \tan \left (d x +c \right )}{8}}{\left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}-\frac {15 \arctan \left (\tan \left (d x +c \right )\right )}{8}}{d a}\)	\(57\)
risch	\(-\frac {15 x}{8 a}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 a d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a d}+\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\sin \left (4 d x +4 c \right )}{32 a d}\)	\(83\)
norman	\(\frac {\frac {15 x}{8 a}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}-\frac {113 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {29 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {113 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {35 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}-\frac {15 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {75 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {135 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {75 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {135 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {75 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(tan(d*x+c)-(-9/8*tan(d*x+c)^3-7/8*tan(d*x+c))/(tan(d*x+c)^2+1)^2-15/8*arctan(tan(d*x+c)))

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Maxima [A]
time = 0.52, size = 72, normalized size = 0.99 \begin {gather*} \frac {\frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{a \tan \left (d x + c\right )^{4} + 2 \, a \tan \left (d x + c\right )^{2} + a} - \frac {15 \, {\left (d x + c\right )}}{a} + \frac {8 \, \tan \left (d x + c\right )}{a}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

1/8*((9*tan(d*x + c)^3 + 7*tan(d*x + c))/(a*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a) - 15*(d*x + c)/a + 8*tan(
d*x + c)/a)/d

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Fricas [A]
time = 0.38, size = 56, normalized size = 0.77 \begin {gather*} -\frac {15 \, d x \cos \left (d x + c\right ) + {\left (2 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{8 \, a d \cos \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

-1/8*(15*d*x*cos(d*x + c) + (2*cos(d*x + c)^4 - 9*cos(d*x + c)^2 - 8)*sin(d*x + c))/(a*d*cos(d*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1161 vs. \(2 (61) = 122\).
time = 7.85, size = 1161, normalized size = 15.90 \begin {gather*} \begin {cases} - \frac {15 d x \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {45 d x \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {30 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} + \frac {30 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} + \frac {45 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} + \frac {15 d x}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {30 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {80 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {36 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {80 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} - \frac {30 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 24 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 24 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 8 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{6}{\left (c \right )}}{- a \sin ^{2}{\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**6/(a-a*sin(d*x+c)**2),x)

[Out]

Piecewise((-15*d*x*tan(c/2 + d*x/2)**10/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(
c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 45*d*x*tan(c/2 + d*x/2)**
8/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x
/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 30*d*x*tan(c/2 + d*x/2)**6/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d
*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 -
8*a*d) + 30*d*x*tan(c/2 + d*x/2)**4/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2
+ d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) + 45*d*x*tan(c/2 + d*x/2)**2/(8
*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)*
*4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) + 15*d*x/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 1
6*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 30*tan(c/2 + d*
x/2)**9/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2
+ d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 80*tan(c/2 + d*x/2)**7/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a
*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2
- 8*a*d) - 36*tan(c/2 + d*x/2)**5/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 +
d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 80*tan(c/2 + d*x/2)**3/(8*a*d*t
an(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 2
4*a*d*tan(c/2 + d*x/2)**2 - 8*a*d) - 30*tan(c/2 + d*x/2)/(8*a*d*tan(c/2 + d*x/2)**10 + 24*a*d*tan(c/2 + d*x/2)
**8 + 16*a*d*tan(c/2 + d*x/2)**6 - 16*a*d*tan(c/2 + d*x/2)**4 - 24*a*d*tan(c/2 + d*x/2)**2 - 8*a*d), Ne(d, 0))
, (x*sin(c)**6/(-a*sin(c)**2 + a), True))

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Giac [A]
time = 0.43, size = 63, normalized size = 0.86 \begin {gather*} -\frac {\frac {15 \, {\left (d x + c\right )}}{a} - \frac {8 \, \tan \left (d x + c\right )}{a} - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} a}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm="giac")

[Out]

-1/8*(15*(d*x + c)/a - 8*tan(d*x + c)/a - (9*tan(d*x + c)^3 + 7*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*a))/d

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Mupad [B]
time = 13.72, size = 68, normalized size = 0.93 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d}-\frac {15\,x}{8\,a}+\frac {\frac {9\,{\mathrm {tan}\left (c+d\,x\right )}^3}{8}+\frac {7\,\mathrm {tan}\left (c+d\,x\right )}{8}}{d\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^6/(a - a*sin(c + d*x)^2),x)

[Out]

tan(c + d*x)/(a*d) - (15*x)/(8*a) + ((7*tan(c + d*x))/8 + (9*tan(c + d*x)^3)/8)/(d*(a + 2*a*tan(c + d*x)^2 + a
*tan(c + d*x)^4))

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