3.41 \(\int \sin ^3(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=275 \[ -\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cos (c+d x)}{d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{3 b^4 \sec (c+d x)}{d} \]

[Out]

(4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (12*a^2*b^2*Co
s[c + d*x])/d - (3*b^4*Cos[c + d*x])/d + (a^4*Cos[c + d*x]^3)/(3*d) - (2*a^2*b^2*Cos[c + d*x]^3)/d + (b^4*Cos[
c + d*x]^3)/(3*d) + (6*a^2*b^2*Sec[c + d*x])/d - (3*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*
b*Sin[c + d*x])/d + (10*a*b^3*Sin[c + d*x])/d - (4*a^3*b*Sin[c + d*x]^3)/(3*d) + (10*a*b^3*Sin[c + d*x]^3)/(3*
d) + (2*a*b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/d

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Rubi [A]  time = 0.247616, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3517, 2633, 2592, 302, 206, 2590, 270, 288} \[ -\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cos (c+d x)}{d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{3 b^4 \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

(4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (12*a^2*b^2*Co
s[c + d*x])/d - (3*b^4*Cos[c + d*x])/d + (a^4*Cos[c + d*x]^3)/(3*d) - (2*a^2*b^2*Cos[c + d*x]^3)/d + (b^4*Cos[
c + d*x]^3)/(3*d) + (6*a^2*b^2*Sec[c + d*x])/d - (3*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*
b*Sin[c + d*x])/d + (10*a*b^3*Sin[c + d*x])/d - (4*a^3*b*Sin[c + d*x]^3)/(3*d) + (10*a*b^3*Sin[c + d*x]^3)/(3*
d) + (2*a*b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/d

Rule 3517

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Expand[Sin[e
+ f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2592

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

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

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 270

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

Rule 288

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]

Rubi steps

\begin{align*} \int \sin ^3(c+d x) (a+b \tan (c+d x))^4 \, dx &=\int \left (a^4 \sin ^3(c+d x)+4 a^3 b \sin ^3(c+d x) \tan (c+d x)+6 a^2 b^2 \sin ^3(c+d x) \tan ^2(c+d x)+4 a b^3 \sin ^3(c+d x) \tan ^3(c+d x)+b^4 \sin ^3(c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^4 \int \sin ^3(c+d x) \, dx+\left (4 a^3 b\right ) \int \sin ^3(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sin ^3(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sin ^3(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sin ^3(c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac{a^4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{b^4 \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^4 \cos (c+d x)}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}+\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \left (-2+\frac{1}{x^2}+x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{b^4 \operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}-\frac{3}{x^2}-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^4 \cos (c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{3 b^4 \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}+\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \cos (c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{3 b^4 \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{10 a b^3 \sin (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \cos (c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{3 b^4 \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{10 a b^3 \sin (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}\\ \end{align*}

Mathematica [B]  time = 6.27546, size = 1017, normalized size = 3.7 \[ -\frac{\left (3 a^4-42 b^2 a^2+11 b^4\right ) (a+b \tan (c+d x))^4 \cos ^5(c+d x)}{4 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a b \left (a^2-b^2\right ) \sin (3 (c+d x)) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (a^4-6 b^2 a^2+b^4\right ) \cos (3 (c+d x)) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \left (2 a^3 b-5 a b^3\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \left (2 a^3 b-5 a b^3\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (36 a^2 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-17 b^4 \sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (17 b^4 \sin \left (\frac{1}{2} (c+d x)\right )-36 a^2 b^2 \sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a b \left (5 a^2-9 b^2\right ) \sin (c+d x) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^2 \left (17 b^2-36 a^2\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (b^4+12 a b^3\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (b^4-12 a b^3\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^4 \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

-(b^2*(-36*a^2 + 17*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - ((
3*a^4 - 42*a^2*b^2 + 11*b^4)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^4)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)
+ ((a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x]^4*Cos[3*(c + d*x)]*(a + b*Tan[c + d*x])^4)/(12*d*(a*Cos[c + d*x] + b*S
in[c + d*x])^4) - (2*(2*a^3*b - 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c
+ d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*(2*a^3*b - 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2
] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((12*a*b^3 + b^4)*Cos[
c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d
*x])^4) + (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/
2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-12*a*b^3 + b^4)*Cos[c + d*x
]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)
+ (Cos[c + d*x]^4*(36*a^2*b^2*Sin[(c + d*x)/2] - 17*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(
c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-36*a^2*b^2*Sin[(c + d
*x)/2] + 17*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c
+ d*x] + b*Sin[c + d*x])^4) - (a*b*(5*a^2 - 9*b^2)*Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Tan[c + d*x])^4)/(d*(a*C
os[c + d*x] + b*Sin[c + d*x])^4) + (a*b*(a^2 - b^2)*Cos[c + d*x]^4*Sin[3*(c + d*x)]*(a + b*Tan[c + d*x])^4)/(3
*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)

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Maple [A]  time = 0.059, size = 412, normalized size = 1.5 \begin{align*}{\frac{{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5\,{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{3\,d\cos \left ( dx+c \right ) }}-{\frac{16\,{b}^{4}\cos \left ( dx+c \right ) }{3\,d}}-{\frac{5\,{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3\,d}}-2\,{\frac{{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{8\,{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+2\,{\frac{{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d}}+{\frac{10\,{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+10\,{\frac{{b}^{3}a\sin \left ( dx+c \right ) }{d}}-10\,{\frac{{b}^{3}a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d\cos \left ( dx+c \right ) }}+16\,{\frac{{a}^{2}{b}^{2}\cos \left ( dx+c \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+8\,{\frac{{a}^{2}{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{4\,b{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-4\,{\frac{b{a}^{3}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{b{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}-{\frac{2\,{a}^{4}\cos \left ( dx+c \right ) }{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3*(a+b*tan(d*x+c))^4,x)

[Out]

1/3/d*b^4*sin(d*x+c)^8/cos(d*x+c)^3-5/3/d*b^4*sin(d*x+c)^8/cos(d*x+c)-16/3*b^4*cos(d*x+c)/d-5/3/d*b^4*cos(d*x+
c)*sin(d*x+c)^6-2/d*b^4*cos(d*x+c)*sin(d*x+c)^4-8/3/d*b^4*cos(d*x+c)*sin(d*x+c)^2+2/d*b^3*a*sin(d*x+c)^7/cos(d
*x+c)^2+2/d*b^3*a*sin(d*x+c)^5+10/3*a*b^3*sin(d*x+c)^3/d+10*a*b^3*sin(d*x+c)/d-10/d*b^3*a*ln(sec(d*x+c)+tan(d*
x+c))+6/d*a^2*b^2*sin(d*x+c)^6/cos(d*x+c)+16*a^2*b^2*cos(d*x+c)/d+6/d*a^2*b^2*cos(d*x+c)*sin(d*x+c)^4+8/d*a^2*
b^2*cos(d*x+c)*sin(d*x+c)^2-4/3*a^3*b*sin(d*x+c)^3/d-4*a^3*b*sin(d*x+c)/d+4/d*b*a^3*ln(sec(d*x+c)+tan(d*x+c))-
1/3/d*cos(d*x+c)*sin(d*x+c)^2*a^4-2/3*a^4*cos(d*x+c)/d

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Maxima [A]  time = 1.14764, size = 294, normalized size = 1.07 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{4} - 2 \,{\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{3} b - 6 \,{\left (\cos \left (d x + c\right )^{3} - \frac{3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{2} b^{2} +{\left (4 \, \sin \left (d x + c\right )^{3} - \frac{6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} a b^{3} +{\left (\cos \left (d x + c\right )^{3} - \frac{9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} b^{4}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3*((cos(d*x + c)^3 - 3*cos(d*x + c))*a^4 - 2*(2*sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c
) - 1) + 6*sin(d*x + c))*a^3*b - 6*(cos(d*x + c)^3 - 3/cos(d*x + c) - 6*cos(d*x + c))*a^2*b^2 + (4*sin(d*x + c
)^3 - 6*sin(d*x + c)/(sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1) + 24*sin(d*x +
c))*a*b^3 + (cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d*x + c))*b^4)/d

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Fricas [A]  time = 2.13296, size = 528, normalized size = 1.92 \begin{align*} \frac{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (2 \, a^{3} b - 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (2 \, a^{3} b - 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + b^{4} + 9 \,{\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (2 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{5} + 3 \, a b^{3} \cos \left (d x + c\right ) - 2 \,{\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*((a^4 - 6*a^2*b^2 + b^4)*cos(d*x + c)^6 - 3*(a^4 - 12*a^2*b^2 + 3*b^4)*cos(d*x + c)^4 + 3*(2*a^3*b - 5*a*b
^3)*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(2*a^3*b - 5*a*b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + b^4 +
9*(2*a^2*b^2 - b^4)*cos(d*x + c)^2 + 2*(2*(a^3*b - a*b^3)*cos(d*x + c)^5 + 3*a*b^3*cos(d*x + c) - 2*(4*a^3*b
- 7*a*b^3)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^3)

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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(sin(d*x+c)**3*(a+b*tan(d*x+c))**4,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(sin(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

Timed out