3.291 \(\int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx\)
Optimal. Leaf size=139 \[ \frac{6 i d^3 (c+d x) \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac{3 d^4 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}+\frac{2 i d (c+d x)^3}{b^2} \]
[Out]
((2*I)*d*(c + d*x)^3)/b^2 - (6*d^2*(c + d*x)^2*Log[1 + E^((2*I)*(a + b*x))])/b^3 + ((6*I)*d^3*(c + d*x)*PolyLo
g[2, -E^((2*I)*(a + b*x))])/b^4 - (3*d^4*PolyLog[3, -E^((2*I)*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x]^2)/
(2*b) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2
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Rubi [A] time = 0.257889, antiderivative size = 139, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used =
{4409, 4184, 3719, 2190, 2531, 2282, 6589} \[ \frac{6 i d^3 (c+d x) \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac{3 d^4 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}+\frac{2 i d (c+d x)^3}{b^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x],x]
[Out]
((2*I)*d*(c + d*x)^3)/b^2 - (6*d^2*(c + d*x)^2*Log[1 + E^((2*I)*(a + b*x))])/b^3 + ((6*I)*d^3*(c + d*x)*PolyLo
g[2, -E^((2*I)*(a + b*x))])/b^4 - (3*d^4*PolyLog[3, -E^((2*I)*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x]^2)/
(2*b) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2
Rule 4409
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
((c + d*x)^m*Sec[a + b*x]^n)/(b*n), x] - Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3719
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]
Rule 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps
\begin{align*} \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx &=\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{(2 d) \int (c+d x)^3 \sec ^2(a+b x) \, dx}{b}\\ &=\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{\left (6 d^2\right ) \int (c+d x)^2 \tan (a+b x) \, dx}{b^2}\\ &=\frac{2 i d (c+d x)^3}{b^2}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac{\left (12 i d^2\right ) \int \frac{e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx}{b^2}\\ &=\frac{2 i d (c+d x)^3}{b^2}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{\left (12 d^3\right ) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=\frac{2 i d (c+d x)^3}{b^2}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac{6 i d^3 (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac{\left (6 i d^4\right ) \int \text{Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^4}\\ &=\frac{2 i d (c+d x)^3}{b^2}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac{6 i d^3 (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac{\left (3 d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5}\\ &=\frac{2 i d (c+d x)^3}{b^2}-\frac{6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac{6 i d^3 (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}-\frac{3 d^4 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{b^5}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}\\ \end{align*}
Mathematica [B] time = 6.58545, size = 418, normalized size = 3.01 \[ -\frac{6 c d^3 \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt{\cot ^2(a)+1}}\right )}{b^4 \sqrt{\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac{i e^{-i a} d^4 \sec (a) \left (6 \left (1+e^{2 i a}\right ) b x \text{PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \text{PolyLog}\left (3,-e^{-2 i (a+b x)}\right )+2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )\right )}{2 b^5}-\frac{2 \sec (a) \sec (a+b x) \left (3 c^2 d^2 x \sin (b x)+c^3 d \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2}-\frac{6 c^2 d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left (\sin ^2(a)+\cos ^2(a)\right )}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x],x]
[Out]
((-I/2)*d^4*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))
*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^5
*E^(I*a)) + ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (6*c^2*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*
x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (6*c*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*
b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - A
rcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*
(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (2*Sec[a]*
Sec[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2
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Maple [B] time = 0.193, size = 489, normalized size = 3.5 \begin{align*} 2\,{\frac{b{d}^{4}{x}^{4}{{\rm e}^{2\,i \left ( bx+a \right ) }}+4\,bc{d}^{3}{x}^{3}{{\rm e}^{2\,i \left ( bx+a \right ) }}+6\,b{c}^{2}{d}^{2}{x}^{2}{{\rm e}^{2\,i \left ( bx+a \right ) }}+4\,b{c}^{3}dx{{\rm e}^{2\,i \left ( bx+a \right ) }}-2\,i{d}^{4}{x}^{3}{{\rm e}^{2\,i \left ( bx+a \right ) }}+b{c}^{4}{{\rm e}^{2\,i \left ( bx+a \right ) }}-6\,ic{d}^{3}{x}^{2}{{\rm e}^{2\,i \left ( bx+a \right ) }}-6\,i{c}^{2}{d}^{2}x{{\rm e}^{2\,i \left ( bx+a \right ) }}-2\,i{d}^{4}{x}^{3}-2\,i{c}^{3}d{{\rm e}^{2\,i \left ( bx+a \right ) }}-6\,ic{d}^{3}{x}^{2}-6\,i{c}^{2}{d}^{2}x-2\,i{c}^{3}d}{{b}^{2} \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) ^{2}}}-6\,{\frac{{c}^{2}{d}^{2}\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) }{{b}^{3}}}+12\,{\frac{{c}^{2}{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+12\,{\frac{{d}^{4}{a}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{5}}}+{\frac{24\,i{d}^{3}cax}{{b}^{3}}}+{\frac{12\,i{d}^{3}c{a}^{2}}{{b}^{4}}}-{\frac{8\,i{d}^{4}{a}^{3}}{{b}^{5}}}-6\,{\frac{{d}^{4}\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ){x}^{2}}{{b}^{3}}}+{\frac{6\,i{d}^{4}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) x}{{b}^{4}}}-3\,{\frac{{d}^{4}{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{5}}}-24\,{\frac{c{d}^{3}a\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{4}}}-12\,{\frac{c{d}^{3}\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) x}{{b}^{3}}}+{\frac{12\,ic{d}^{3}{x}^{2}}{{b}^{2}}}+{\frac{6\,ic{d}^{3}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{4}}}+{\frac{4\,i{d}^{4}{x}^{3}}{{b}^{2}}}-{\frac{12\,i{d}^{4}{a}^{2}x}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x)
[Out]
2*(b*d^4*x^4*exp(2*I*(b*x+a))+4*b*c*d^3*x^3*exp(2*I*(b*x+a))+6*b*c^2*d^2*x^2*exp(2*I*(b*x+a))+4*b*c^3*d*x*exp(
2*I*(b*x+a))-2*I*d^4*x^3*exp(2*I*(b*x+a))+b*c^4*exp(2*I*(b*x+a))-6*I*c*d^3*x^2*exp(2*I*(b*x+a))-6*I*c^2*d^2*x*
exp(2*I*(b*x+a))-2*I*d^4*x^3-2*I*c^3*d*exp(2*I*(b*x+a))-6*I*c*d^3*x^2-6*I*c^2*d^2*x-2*I*c^3*d)/b^2/(exp(2*I*(b
*x+a))+1)^2-6*d^2/b^3*c^2*ln(exp(2*I*(b*x+a))+1)+12*d^2/b^3*c^2*ln(exp(I*(b*x+a)))+12*d^4/b^5*a^2*ln(exp(I*(b*
x+a)))+24*I*d^3/b^3*c*a*x+12*I*d^3/b^4*c*a^2-8*I*d^4/b^5*a^3-6*d^4/b^3*ln(exp(2*I*(b*x+a))+1)*x^2+6*I*d^4/b^4*
polylog(2,-exp(2*I*(b*x+a)))*x-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5-24*d^3/b^4*c*a*ln(exp(I*(b*x+a)))-12*d^3
/b^3*c*ln(exp(2*I*(b*x+a))+1)*x+12*I*d^3/b^2*c*x^2+6*I*d^3/b^4*c*polylog(2,-exp(2*I*(b*x+a)))+4*I*d^4/b^2*x^3-
12*I*d^4/b^4*a^2*x
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Maxima [B] time = 2.31788, size = 4641, normalized size = 33.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="maxima")
[Out]
1/2*(c^4*tan(b*x + a)^2 - 4*a*c^3*d*tan(b*x + a)^2/b + 6*a^2*c^2*d^2*tan(b*x + a)^2/b^2 - 4*a^3*c*d^3*tan(b*x
+ a)^2/b^3 + a^4*d^4*tan(b*x + a)^2/b^4 + 8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 +
(2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x +
a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*c^3*d/((2*(2*cos(2*b*x + 2*a
) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*
sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b) - 24*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*
(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a
)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a)
)*a*c^2*d^2/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*
b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^2) + 24*
(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x +
2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)
*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*a^2*c*d^3/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*
a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2
+ 4*cos(2*b*x + 2*a) + 1)*b^3) - 8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x
+ a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2
*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*a^3*d^4/((2*(2*cos(2*b*x + 2*a) + 1)*
cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b
*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^4) + 6*(8*(b*x + a)^2*cos(2*b*x + 2*a)^2 + 8*(b*x
+ a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)*sin(
2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b
*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x +
2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + 4*((b*x + a)^2*sin(2*b*x + 2
*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/((2*(2*co
s(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(
4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^2) - 12*(8*(b*x + a)^2*cos(2*
b*x + 2*a)^2 + 8*(b*x + a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x + a)^2*cos(2*b*x +
2*a) + (b*x + a)*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x
+ 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*
a)^2 + 4*cos(2*b*x + 2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + 4*((b*x
+ a)^2*sin(2*b*x + 2*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x + a)*sin(2*b*x + 2*
a))*a*c*d^3/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*
b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^3) + 6*(
8*(b*x + a)^2*cos(2*b*x + 2*a)^2 + 8*(b*x + a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x
+ a)^2*cos(2*b*x + 2*a) + (b*x + a)*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*
x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a
) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x +
2*a) + 1) + 4*((b*x + a)^2*sin(2*b*x + 2*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x
+ a)*sin(2*b*x + 2*a))*a^2*d^4/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b
*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x +
2*a) + 1)*b^4) - 2*((6*(b*x + a)^2*d^4 + 12*(b*c*d^3 - a*d^4)*(b*x + a) + 6*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*
d^4)*(b*x + a))*cos(4*b*x + 4*a) + 12*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (6*
I*(b*x + a)^2*d^4 + (12*I*b*c*d^3 - 12*I*a*d^4)*(b*x + a))*sin(4*b*x + 4*a) + (12*I*(b*x + a)^2*d^4 + (24*I*b*
c*d^3 - 24*I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 4*((b*x + a
)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2)*cos(4*b*x + 4*a) + (2*I*(b*x + a)^4*d^4 + (8*I*b*c*d^3 - 4*(2*I*a +
1)*d^4)*(b*x + a)^3 - 12*(b*c*d^3 - a*d^4)*(b*x + a)^2)*cos(2*b*x + 2*a) - (6*b*c*d^3 + 6*(b*x + a)*d^4 - 6*a
*d^4 + 6*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) + 12*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x +
2*a) - (-6*I*b*c*d^3 - 6*I*(b*x + a)*d^4 + 6*I*a*d^4)*sin(4*b*x + 4*a) - (-12*I*b*c*d^3 - 12*I*(b*x + a)*d^4
+ 12*I*a*d^4)*sin(2*b*x + 2*a))*dilog(-e^(2*I*b*x + 2*I*a)) + (-3*I*(b*x + a)^2*d^4 + (-6*I*b*c*d^3 + 6*I*a*d^
4)*(b*x + a) + (-3*I*(b*x + a)^2*d^4 + (-6*I*b*c*d^3 + 6*I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (-6*I*(b*x + a
)^2*d^4 + (-12*I*b*c*d^3 + 12*I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 3*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*
(b*x + a))*sin(4*b*x + 4*a) + 6*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(2*
b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (-3*I*d^4*cos(4*b*x + 4*a) - 6*I*d^4*cos(2*b*x +
2*a) + 3*d^4*sin(4*b*x + 4*a) + 6*d^4*sin(2*b*x + 2*a) - 3*I*d^4)*polylog(3, -e^(2*I*b*x + 2*I*a)) + (-4*I*(b
*x + a)^3*d^4 + (-12*I*b*c*d^3 + 12*I*a*d^4)*(b*x + a)^2)*sin(4*b*x + 4*a) - (2*(b*x + a)^4*d^4 + (8*b*c*d^3 -
(8*a - 4*I)*d^4)*(b*x + a)^3 - (-12*I*b*c*d^3 + 12*I*a*d^4)*(b*x + a)^2)*sin(2*b*x + 2*a))/(-I*b^4*cos(4*b*x
+ 4*a) - 2*I*b^4*cos(2*b*x + 2*a) + b^4*sin(4*b*x + 4*a) + 2*b^4*sin(2*b*x + 2*a) - I*b^4))/b
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Fricas [C] time = 0.776583, size = 2211, normalized size = 15.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="fricas")
[Out]
1/2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*d^4*cos(b*x + a)^2*polyl
og(3, I*cos(b*x + a) + sin(b*x + a)) - 12*d^4*cos(b*x + a)^2*polylog(3, I*cos(b*x + a) - sin(b*x + a)) - 12*d^
4*cos(b*x + a)^2*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 12*d^4*cos(b*x + a)^2*polylog(3, -I*cos(b*x + a)
- sin(b*x + a)) + (-12*I*b*d^4*x - 12*I*b*c*d^3)*cos(b*x + a)^2*dilog(I*cos(b*x + a) + sin(b*x + a)) + (12*I*
b*d^4*x + 12*I*b*c*d^3)*cos(b*x + a)^2*dilog(I*cos(b*x + a) - sin(b*x + a)) + (12*I*b*d^4*x + 12*I*b*c*d^3)*co
s(b*x + a)^2*dilog(-I*cos(b*x + a) + sin(b*x + a)) + (-12*I*b*d^4*x - 12*I*b*c*d^3)*cos(b*x + a)^2*dilog(-I*co
s(b*x + a) - sin(b*x + a)) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2*log(cos(b*x + a) + I*sin(b
*x + a) + I) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2*log(cos(b*x + a) - I*sin(b*x + a) + I) -
6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4)*cos(b*x + a)^2*log(I*cos(b*x + a) + sin(b*x + a) + 1)
- 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4)*cos(b*x + a)^2*log(I*cos(b*x + a) - sin(b*x + a) +
1) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4)*cos(b*x + a)^2*log(-I*cos(b*x + a) + sin(b*x + a)
+ 1) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4)*cos(b*x + a)^2*log(-I*cos(b*x + a) - sin(b*x +
a) + 1) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2*log(-cos(b*x + a) + I*sin(b*x + a) + I) - 6*
(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2*log(-cos(b*x + a) - I*sin(b*x + a) + I) - 4*(b^3*d^4*x^3
+ 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*cos(b*x + a)*sin(b*x + a))/(b^5*cos(b*x + a)^2)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{4} \tan{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**4*sec(b*x+a)**2*tan(b*x+a),x)
[Out]
Integral((c + d*x)**4*tan(a + b*x)*sec(a + b*x)**2, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{4} \sec \left (b x + a\right )^{2} \tan \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="giac")
[Out]
integrate((d*x + c)^4*sec(b*x + a)^2*tan(b*x + a), x)