\(\int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx\) [299]

Optimal result

Integrand size = 22, antiderivative size = 193 \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}+\frac {d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b} \]

[Out]

I*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b+d^2*arctanh(sin(b*x+a))/b^3-I*d*(d*x+c)*polylog(2,-I*exp(I*(b*x+a)))/b^2+
I*d*(d*x+c)*polylog(2,I*exp(I*(b*x+a)))/b^2+d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3-d^2*polylog(3,I*exp(I*(b*x+a)
))/b^3-d*(d*x+c)*sec(b*x+a)/b^2+1/2*(d*x+c)^2*sec(b*x+a)*tan(b*x+a)/b

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4498, 4266, 2611, 2320, 6724, 4271, 3855} \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}+\frac {d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \tan (a+b x) \sec (a+b x)}{2 b} \]

[In]

Int[(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x]^2,x]

[Out]

(I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b + (d^2*ArcTanh[Sin[a + b*x]])/b^3 - (I*d*(c + d*x)*PolyLog[2, (-I)*E
^(I*(a + b*x))])/b^2 + (I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 + (d^2*PolyLog[3, (-I)*E^(I*(a + b*x)
)])/b^3 - (d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (d*(c + d*x)*Sec[a + b*x])/b^2 + ((c + d*x)^2*Sec[a + b*x]
*Tan[a + b*x])/(2*b)

Rule 2320

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 2611

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4498

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]*Tan[(a_.) + (b_.)*(x_)]^(p_), x_Symbol] :> -Int[(c + d*
x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x] /; FreeQ[
{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rule 6724

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*} \text {integral}& = -\int (c+d x)^2 \sec (a+b x) \, dx+\int (c+d x)^2 \sec ^3(a+b x) \, dx \\ & = \frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^2 \sec (a+b x) \, dx+\frac {(2 d) \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}-\frac {(2 d) \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}+\frac {d^2 \int \sec (a+b x) \, dx}{b^2} \\ & = \frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {d \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {d \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}+\frac {\left (2 i d^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (2 i d^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = \frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (i d^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (i d^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = \frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {d^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3} \\ & = \frac {i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}+\frac {d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(526\) vs. \(2(193)=386\).

Time = 7.70 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.73 \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {i b c^2 \arctan \left (e^{i (a+b x)}\right )-\frac {2 i d^2 \arctan \left (e^{i (a+b x)}\right )}{b}-b c d x \log \left (1-i e^{i (a+b x)}\right )-\frac {1}{2} b d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+b c d x \log \left (1+i e^{i (a+b x)}\right )+\frac {1}{2} b d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+\frac {d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b}-\frac {d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b}}{b^2}-\frac {d (c+d x) \sec (a)}{b^2}+\frac {c^2+2 c d x+d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {-c d \sin \left (\frac {b x}{2}\right )-d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {-c^2-2 c d x-d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {c d \sin \left (\frac {b x}{2}\right )+d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )} \]

[In]

Integrate[(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x]^2,x]

[Out]

(I*b*c^2*ArcTan[E^(I*(a + b*x))] - ((2*I)*d^2*ArcTan[E^(I*(a + b*x))])/b - b*c*d*x*Log[1 - I*E^(I*(a + b*x))]
- (b*d^2*x^2*Log[1 - I*E^(I*(a + b*x))])/2 + b*c*d*x*Log[1 + I*E^(I*(a + b*x))] + (b*d^2*x^2*Log[1 + I*E^(I*(a
+ b*x))])/2 - I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] + I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] +
(d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b - (d^2*PolyLog[3, I*E^(I*(a + b*x))])/b)/b^2 - (d*(c + d*x)*Sec[a])/b
^2 + (c^2 + 2*c*d*x + d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])^2) + (-(c*d*Sin[(b*x)/2]) - d^2*
x*Sin[(b*x)/2])/(b^2*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-c^2 - 2*c*d*x - d^2*
x^2)/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])^2) + (c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2
] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (174 ) = 348\).

Time = 1.25 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.03

method	result	size
risch	\(-\frac {i d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 i c d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {i c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{3}}-\frac {d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{3}}+\frac {c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}-\frac {d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}+\frac {d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}+\frac {c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {i c^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {i d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {i c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 i d^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {i \left (x^{2} d^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+2 c d x b \,{\mathrm e}^{3 i \left (x b +a \right )}+c^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}-x^{2} d^{2} b \,{\mathrm e}^{i \left (x b +a \right )}-2 c d x b \,{\mathrm e}^{i \left (x b +a \right )}-2 i d^{2} x \,{\mathrm e}^{3 i \left (x b +a \right )}-c^{2} b \,{\mathrm e}^{i \left (x b +a \right )}-2 i d c \,{\mathrm e}^{3 i \left (x b +a \right )}-2 i d^{2} x \,{\mathrm e}^{i \left (x b +a \right )}-2 i d c \,{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}\)	\(584\)

[In]

int((d*x+c)^2*sec(b*x+a)*tan(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-I/b^2*d^2*polylog(2,-I*exp(I*(b*x+a)))*x+I/b^3*d^2*a^2*arctan(exp(I*(b*x+a)))-2*I/b^2*c*d*a*arctan(exp(I*(b*x
+a)))-I/b^2*c*d*polylog(2,-I*exp(I*(b*x+a)))+1/2/b^3*a^2*d^2*ln(1-I*exp(I*(b*x+a)))-d^2*polylog(3,I*exp(I*(b*x
+a)))/b^3-1/b*c*d*ln(1-I*exp(I*(b*x+a)))*x-1/2/b^3*a^2*d^2*ln(1+I*exp(I*(b*x+a)))+1/b^2*c*d*ln(1+I*exp(I*(b*x+
a)))*a-1/2/b*d^2*ln(1-I*exp(I*(b*x+a)))*x^2+1/2/b*d^2*ln(1+I*exp(I*(b*x+a)))*x^2+1/b*c*d*ln(1+I*exp(I*(b*x+a))
)*x+d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3-1/b^2*c*d*ln(1-I*exp(I*(b*x+a)))*a+I/b*c^2*arctan(exp(I*(b*x+a)))+I/b
^2*d^2*polylog(2,I*exp(I*(b*x+a)))*x+I/b^2*c*d*polylog(2,I*exp(I*(b*x+a)))-2*I/b^3*d^2*arctan(exp(I*(b*x+a)))-
I/b^2/(exp(2*I*(b*x+a))+1)^2*(x^2*d^2*b*exp(3*I*(b*x+a))+2*c*d*x*b*exp(3*I*(b*x+a))+c^2*b*exp(3*I*(b*x+a))-x^2
*d^2*b*exp(I*(b*x+a))-2*c*d*x*b*exp(I*(b*x+a))-2*I*d^2*x*exp(3*I*(b*x+a))-c^2*b*exp(I*(b*x+a))-2*I*d*c*exp(3*I
*(b*x+a))-2*I*d^2*x*exp(I*(b*x+a))-2*I*c*d*exp(I*(b*x+a)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (165) = 330\).

Time = 0.30 (sec) , antiderivative size = 795, normalized size of antiderivative = 4.12 \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} - 2\right )} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} - 2\right )} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} - 2\right )} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} - 2\right )} d^{2}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 4 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{3} \cos \left (b x + a\right )^{2}} \]

[In]

integrate((d*x+c)^2*sec(b*x+a)*tan(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\int \left (c + d x\right )^{2} \tan ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**2*sec(b*x+a)*tan(b*x+a)**2,x)

[Out]

Integral((c + d*x)**2*tan(a + b*x)**2*sec(a + b*x), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1891 vs. \(2 (165) = 330\).

Time = 0.57 (sec) , antiderivative size = 1891, normalized size of antiderivative = 9.80 \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^2*sec(b*x+a)*tan(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/4*(c^2*(2*sin(b*x + a)/(sin(b*x + a)^2 - 1) + log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1)) - 2*a*c*d*(2*s
in(b*x + a)/(sin(b*x + a)^2 - 1) + log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1))/b + a^2*d^2*(2*sin(b*x + a)/
(sin(b*x + a)^2 - 1) + log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1))/b^2 - 4*(2*((b*x + a)^2*d^2 + 2*(b*c*d -
a*d^2)*(b*x + a) - 2*d^2 + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*cos(4*b*x + 4*a) + 2*((b*x
+ a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*cos(2*b*x + 2*a) + (I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^
2)*(b*x + a) - 2*I*d^2)*sin(4*b*x + 4*a) + 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2)*s
in(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) -
2*d^2 + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*cos(4*b*x + 4*a) + 2*((b*x + a)^2*d^2 + 2*(b*c
*d - a*d^2)*(b*x + a) - 2*d^2)*cos(2*b*x + 2*a) + (I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d
^2)*sin(4*b*x + 4*a) + 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2)*sin(2*b*x + 2*a))*arc
tan2(cos(b*x + a), -sin(b*x + a) + 1) - 4*((b*x + a)^2*d^2 - 2*I*b*c*d + 2*I*a*d^2 + 2*(b*c*d - (a + I)*d^2)*(
b*x + a))*cos(3*b*x + 3*a) + 4*((b*x + a)^2*d^2 + 2*I*b*c*d - 2*I*a*d^2 + 2*(b*c*d - (a - I)*d^2)*(b*x + a))*c
os(b*x + a) + 4*(b*c*d + (b*x + a)*d^2 - a*d^2 + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) + 2*(b*c*d +
(b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) + (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(4*b*x + 4*a) + 2*(I*b*c*d
+ I*(b*x + a)*d^2 - I*a*d^2)*sin(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 4*(b*c*d + (b*x + a)*d^2 - a*d^2 +
(b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) + 2*(b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) - (-I*b*
c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(4*b*x + 4*a) - 2*(-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(2*b*x + 2*a))
*dilog(-I*e^(I*b*x + I*a)) - (-I*(b*x + a)^2*d^2 - 2*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2 + (-I*(b*x + a)^2
*d^2 - 2*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2)*cos(4*b*x + 4*a) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d
^2)*(b*x + a) - 2*I*d^2)*cos(2*b*x + 2*a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(4*b*x
+ 4*a) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(
b*x + a)^2 + 2*sin(b*x + a) + 1) - (I*(b*x + a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a) - 2*I*d^2 + (I*(b*x +
a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(4*b*x + 4*a) - 2*(-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d
+ I*a*d^2)*(b*x + a) + 2*I*d^2)*cos(2*b*x + 2*a) - ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*si
n(4*b*x + 4*a) - 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)^
2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + 4*(I*d^2*cos(4*b*x + 4*a) + 2*I*d^2*cos(2*b*x + 2*a) - d^2*sin(4*b*
x + 4*a) - 2*d^2*sin(2*b*x + 2*a) + I*d^2)*polylog(3, I*e^(I*b*x + I*a)) + 4*(-I*d^2*cos(4*b*x + 4*a) - 2*I*d^
2*cos(2*b*x + 2*a) + d^2*sin(4*b*x + 4*a) + 2*d^2*sin(2*b*x + 2*a) - I*d^2)*polylog(3, -I*e^(I*b*x + I*a)) + 4
*(-I*(b*x + a)^2*d^2 - 2*b*c*d + 2*a*d^2 + 2*(-I*b*c*d + (I*a - 1)*d^2)*(b*x + a))*sin(3*b*x + 3*a) + 4*(I*(b*
x + a)^2*d^2 - 2*b*c*d + 2*a*d^2 + 2*(I*b*c*d + (-I*a - 1)*d^2)*(b*x + a))*sin(b*x + a))/(-4*I*b^2*cos(4*b*x +
4*a) - 8*I*b^2*cos(2*b*x + 2*a) + 4*b^2*sin(4*b*x + 4*a) + 8*b^2*sin(2*b*x + 2*a) - 4*I*b^2))/b

Giac [F]

\[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]

[In]

integrate((d*x+c)^2*sec(b*x+a)*tan(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^2*sec(b*x + a)*tan(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx=\text {Hanged} \]

[In]

int((tan(a + b*x)^2*(c + d*x)^2)/cos(a + b*x),x)

[Out]

\text{Hanged}