\(\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx\) [22]
Optimal result
Integrand size = 21, antiderivative size = 113 \[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {3}{8} \left (a^2-5 b^2\right ) x-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}+\frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}
\]
[Out]
3/8*(a^2-5*b^2)*x-2*a*b*ln(cos(d*x+c))/d+b^2*tan(d*x+c)/d+1/8*cos(d*x+c)^2*(7*b-5*a*tan(d*x+c))*(a+b*tan(d*x+c
))/d+1/4*cos(d*x+c)^3*sin(d*x+c)*(a+b*tan(d*x+c))^2/d
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1659, 1824, 649, 209,
266} \[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {3}{8} x \left (a^2-5 b^2\right )+\frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {b^2 \tan (c+d x)}{d}
\]
[In]
Int[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]
[Out]
(3*(a^2 - 5*b^2)*x)/8 - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (Cos[c + d*x]^2*(7*b - 5*a*Tan[c
+ d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(4*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 266
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 649
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[(-a)*c]
Rule 1659
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] + Dist[1/(2*a*c*(p
+ 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p
+ 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Rule 1824
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 3597
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]
Rubi steps \begin{align*}
\text {integral}& = \frac {b \text {Subst}\left (\int \frac {x^4 (a+x)^2}{\left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}-\frac {\text {Subst}\left (\int \frac {(a+x) \left (a b^4+3 b^4 x-4 a b^2 x^2-4 b^2 x^3\right )}{\left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d} \\ & = \frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {\text {Subst}\left (\int \frac {b^4 \left (3 a^2-7 b^2\right )+16 a b^4 x+8 b^4 x^2}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {\text {Subst}\left (\int \left (8 b^4+\frac {3 b^4 \left (a^2-5 b^2\right )+16 a b^4 x}{b^2+x^2}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {b^2 \tan (c+d x)}{d}+\frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {\text {Subst}\left (\int \frac {3 b^4 \left (a^2-5 b^2\right )+16 a b^4 x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {b^2 \tan (c+d x)}{d}+\frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {(2 a b) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}+\frac {\left (3 b \left (a^2-5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 d} \\ & = \frac {3}{8} \left (a^2-5 b^2\right ) x-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}+\frac {\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac {\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(250\) vs. \(2(113)=226\).
Time = 3.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.21
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {b \left (\frac {3 \left (a^2-b^2\right ) \arctan (\tan (c+d x))}{b}+\frac {4 \left (-2 a^2+3 b^2\right ) \arctan (\tan (c+d x))}{b}+16 a \cos ^2(c+d x)-4 a \cos ^4(c+d x)+4 \left (2 a+\frac {a^2-3 b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+4 \left (2 a+\frac {-a^2+3 b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {2 \left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{b}+\frac {3 (a-b) (a+b) \sin (2 (c+d x))}{2 b}+\frac {2 \left (-2 a^2+3 b^2\right ) \sin (2 (c+d x))}{b}+8 b \tan (c+d x)\right )}{8 d}
\]
[In]
Integrate[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]
[Out]
(b*((3*(a^2 - b^2)*ArcTan[Tan[c + d*x]])/b + (4*(-2*a^2 + 3*b^2)*ArcTan[Tan[c + d*x]])/b + 16*a*Cos[c + d*x]^2
- 4*a*Cos[c + d*x]^4 + 4*(2*a + (a^2 - 3*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 4*(2*a + (-a^2 +
3*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (2*(a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/b + (3*(a -
b)*(a + b)*Sin[2*(c + d*x)])/(2*b) + (2*(-2*a^2 + 3*b^2)*Sin[2*(c + d*x)])/b + 8*b*Tan[c + d*x]))/(8*d)
Maple [A] (verified)
Time = 1.87 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.24
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+b^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) |
\(140\) |
| default |
\(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+b^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) |
\(140\) |
| risch |
\(2 i a b x +\frac {3 a^{2} x}{8}-\frac {15 b^{2} x}{8}+\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )} a b}{8 d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{4 d}+\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )} a b}{8 d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{4 d}+\frac {4 i a b c}{d}+\frac {2 i b^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {a b \cos \left (4 d x +4 c \right )}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2}}{32 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{32 d}\) |
\(224\) |
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[In]
int(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
[Out]
1/d*(a^2*(-1/4*(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)+3/8*d*x+3/8*c)+2*a*b*(-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)
^2-ln(cos(d*x+c)))+b^2*(sin(d*x+c)^7/cos(d*x+c)+(sin(d*x+c)^5+5/4*sin(d*x+c)^3+15/8*sin(d*x+c))*cos(d*x+c)-15/
8*d*x-15/8*c))
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {8 \, a b \cos \left (d x + c\right )^{5} - 32 \, a b \cos \left (d x + c\right )^{3} + 32 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - {\left (6 \, {\left (a^{2} - 5 \, b^{2}\right )} d x - 13 \, a b\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{2} - 9 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )}
\]
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
-1/16*(8*a*b*cos(d*x + c)^5 - 32*a*b*cos(d*x + c)^3 + 32*a*b*cos(d*x + c)*log(-cos(d*x + c)) - (6*(a^2 - 5*b^2
)*d*x - 13*a*b)*cos(d*x + c) - 2*(2*(a^2 - b^2)*cos(d*x + c)^4 - (5*a^2 - 9*b^2)*cos(d*x + c)^2 + 8*b^2)*sin(d
*x + c))/(d*cos(d*x + c))
Sympy [F]
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sin ^{4}{\left (c + d x \right )}\, dx
\]
[In]
integrate(sin(d*x+c)**4*(a+b*tan(d*x+c))**2,x)
[Out]
Integral((a + b*tan(c + d*x))**2*sin(c + d*x)**4, x)
Maxima [A] (verification not implemented)
none
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {8 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 8 \, b^{2} \tan \left (d x + c\right ) + 3 \, {\left (a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )} + \frac {16 \, a b \tan \left (d x + c\right )^{2} - {\left (5 \, a^{2} - 9 \, b^{2}\right )} \tan \left (d x + c\right )^{3} + 12 \, a b - {\left (3 \, a^{2} - 7 \, b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d}
\]
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/8*(8*a*b*log(tan(d*x + c)^2 + 1) + 8*b^2*tan(d*x + c) + 3*(a^2 - 5*b^2)*(d*x + c) + (16*a*b*tan(d*x + c)^2 -
(5*a^2 - 9*b^2)*tan(d*x + c)^3 + 12*a*b - (3*a^2 - 7*b^2)*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 +
1))/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5604 vs. \(2 (107) = 214\).
Time = 2.74 (sec) , antiderivative size = 5604, normalized size of antiderivative = 49.59
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Too large to display}
\]
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
1/64*(3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*
tan(c))*tan(d*x)^5*tan(c)^5 + 24*a^2*d*x*tan(d*x)^5*tan(c)^5 - 120*b^2*d*x*tan(d*x)^5*tan(c)^5 + 3*pi*b^2*sgn(
-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^5 + 6*pi*b^2*sgn(2*tan(d
*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^
3 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*ta
n(c))*tan(d*x)^4*tan(c)^4 + 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(
c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^5 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*ta
n(d*x)^5*tan(c)^5 - 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 64*a*b*log(
4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*
tan(c)^5 + 48*a^2*d*x*tan(d*x)^5*tan(c)^3 - 240*b^2*d*x*tan(d*x)^5*tan(c)^3 + 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c
) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^3 - 24*a^2*d*x*tan(d*x)^4*tan(c)^4 + 120*b^
2*d*x*tan(d*x)^4*tan(c)^4 - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*t
an(d*x)^4*tan(c)^4 + 48*a^2*d*x*tan(d*x)^3*tan(c)^5 - 240*b^2*d*x*tan(d*x)^3*tan(c)^5 + 6*pi*b^2*sgn(-2*tan(d*
x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^5 + 44*a*b*tan(d*x)^5*tan(c)^5 +
3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)
)*tan(d*x)^5*tan(c) - 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 +
2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 12*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c
) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 12*b^2*arctan((tan(d*x) + tan(c))/(tan(
d*x)*tan(c) - 1))*tan(d*x)^5*tan(c)^3 - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^5*t
an(c)^3 - 128*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(
c)^2 + 1))*tan(d*x)^5*tan(c)^3 - 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)
*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1
))*tan(d*x)^4*tan(c)^4 + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 64*a*b
*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*
x)^4*tan(c)^4 + 24*a^2*tan(d*x)^5*tan(c)^4 - 120*b^2*tan(d*x)^5*tan(c)^4 + 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2
- 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^5 + 12*b^2*arctan
((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^3*tan(c)^5 - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)
*tan(c) + 1))*tan(d*x)^3*tan(c)^5 - 128*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^5 + 24*a^2*tan(d*x)^4*tan(c)^5 - 120*b^2*tan(d*x)^4*tan
(c)^5 + 24*a^2*d*x*tan(d*x)^5*tan(c) - 120*b^2*d*x*tan(d*x)^5*tan(c) + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*t
an(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c) - 48*a^2*d*x*tan(d*x)^4*tan(c)^2 + 240*b^2*d*x*tan
(d*x)^4*tan(c)^2 - 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4
*tan(c)^2 + 96*a^2*d*x*tan(d*x)^3*tan(c)^3 - 480*b^2*d*x*tan(d*x)^3*tan(c)^3 + 12*pi*b^2*sgn(-2*tan(d*x)^2*tan
(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 24*a*b*tan(d*x)^5*tan(c)^3 - 48*a^2*d
*x*tan(d*x)^2*tan(c)^4 + 240*b^2*d*x*tan(d*x)^2*tan(c)^4 - 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(
c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 - 172*a*b*tan(d*x)^4*tan(c)^4 + 24*a^2*d*x*tan(d*x)*tan(c)^5
- 120*b^2*d*x*tan(d*x)*tan(c)^5 + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*ta
n(c))*tan(d*x)*tan(c)^5 + 24*a*b*tan(d*x)^3*tan(c)^5 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)
^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 + 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*
sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c) + 6*b^2*arctan((tan(
d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^5*tan(c) - 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) +
1))*tan(d*x)^5*tan(c) - 64*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan
(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c) - 12*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c)
+ 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^2 - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d
*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^2 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*ta
n(c)^2 + 128*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c
)^2 + 1))*tan(d*x)^4*tan(c)^2 + 40*a^2*tan(d*x)^5*tan(c)^2 - 200*b^2*tan(d*x)^5*tan(c)^2 + 6*pi*b^2*sgn(2*tan(
d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^3
+ 24*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^3*tan(c)^3 - 24*b^2*arctan(-(tan(d*x) - t
an(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 256*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1
)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 24*a^2*tan(d*x)^4*tan(c)^3 - 120*b^
2*tan(d*x)^4*tan(c)^3 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2
+ 2*tan(d*x) - 2*tan(c))*tan(c)^4 - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c
)^4 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^4 + 128*a*b*log(4*(tan(d*x)^
2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^4 + 2
4*a^2*tan(d*x)^3*tan(c)^4 - 120*b^2*tan(d*x)^3*tan(c)^4 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) -
1))*tan(d*x)*tan(c)^5 - 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c)^5 - 64*a*b*lo
g(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*
tan(c)^5 + 40*a^2*tan(d*x)^2*tan(c)^5 - 200*b^2*tan(d*x)^2*tan(c)^5 - 24*a^2*d*x*tan(d*x)^4 + 120*b^2*d*x*tan(
d*x)^4 - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 + 48*a^2*
d*x*tan(d*x)^3*tan(c) - 240*b^2*d*x*tan(d*x)^3*tan(c) + 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^
2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c) - 52*a*b*tan(d*x)^5*tan(c) - 96*a^2*d*x*tan(d*x)^2*tan(c)^2 + 480
*b^2*d*x*tan(d*x)^2*tan(c)^2 - 12*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c
))*tan(d*x)^2*tan(c)^2 - 280*a*b*tan(d*x)^4*tan(c)^2 + 48*a^2*d*x*tan(d*x)*tan(c)^3 - 240*b^2*d*x*tan(d*x)*tan
(c)^3 + 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^3 - 1
6*a*b*tan(d*x)^3*tan(c)^3 - 24*a^2*d*x*tan(c)^4 + 120*b^2*d*x*tan(c)^4 - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2
*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^4 - 280*a*b*tan(d*x)^2*tan(c)^4 - 52*a*b*tan(d*x)*tan(c)^5
- 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(
c))*tan(d*x)^2 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4 + 6*b^2*arctan(-(tan(d*x)
- tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4 + 64*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(
d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4 - 64*b^2*tan(d*x)^5 + 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c
)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c) + 12*b^2*arct
an((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^3*tan(c) - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)
*tan(c) + 1))*tan(d*x)^3*tan(c) - 128*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(
c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c) - 80*a^2*tan(d*x)^4*tan(c) + 80*b^2*tan(d*x)^4*tan(c) - 6
*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))
*tan(c)^2 - 24*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^2 + 24*b^2*arctan(-(tan
(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 256*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*t
an(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 - 96*a^2*tan(d*x)^3*tan(c)^2
- 160*b^2*tan(d*x)^3*tan(c)^2 + 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c)^3 -
12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c)^3 - 128*a*b*log(4*(tan(d*x)^2*tan(c)
^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c)^3 - 96*a^2*tan(
d*x)^2*tan(c)^3 - 160*b^2*tan(d*x)^2*tan(c)^3 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)
^4 + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^4 + 64*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2
*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(c)^4 - 80*a^2*tan(d*x)*tan(c)^4 +
80*b^2*tan(d*x)*tan(c)^4 - 64*b^2*tan(c)^5 - 48*a^2*d*x*tan(d*x)^2 + 240*b^2*d*x*tan(d*x)^2 - 6*pi*b^2*sgn(-2
*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 + 52*a*b*tan(d*x)^4 + 24*a^2*d*x*
tan(d*x)*tan(c) - 120*b^2*d*x*tan(d*x)*tan(c) + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*ta
n(d*x) - 2*tan(c))*tan(d*x)*tan(c) + 280*a*b*tan(d*x)^3*tan(c) - 48*a^2*d*x*tan(c)^2 + 240*b^2*d*x*tan(c)^2 -
6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 + 16*a*b*tan(d*x)^2*
tan(c)^2 + 280*a*b*tan(d*x)*tan(c)^3 + 52*a*b*tan(c)^4 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*
x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c
) - 1))*tan(d*x)^2 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2 + 128*a*b*log(4*(tan
(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2 + 40*a
^2*tan(d*x)^3 - 200*b^2*tan(d*x)^3 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) -
6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 64*a*b*log(4*(tan(d*x)^2*tan(c)^2
- 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c) + 24*a^2*tan(d*x)^
2*tan(c) - 120*b^2*tan(d*x)^2*tan(c) - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^2 + 12*
b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^2 + 128*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d
*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(c)^2 + 24*a^2*tan(d*x)*tan(c)^2 - 120*b
^2*tan(d*x)*tan(c)^2 + 40*a^2*tan(c)^3 - 200*b^2*tan(c)^3 - 24*a^2*d*x + 120*b^2*d*x - 3*pi*b^2*sgn(-2*tan(d*x
)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) - 24*a*b*tan(d*x)^2 + 172*a*b*tan(d*x)*tan(c) - 24*a
*b*tan(c)^2 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1)) + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan
(d*x)*tan(c) + 1)) + 64*a*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x
)^2 + tan(c)^2 + 1)) + 24*a^2*tan(d*x) - 120*b^2*tan(d*x) + 24*a^2*tan(c) - 120*b^2*tan(c) - 44*a*b)/(d*tan(d*
x)^5*tan(c)^5 + 2*d*tan(d*x)^5*tan(c)^3 - d*tan(d*x)^4*tan(c)^4 + 2*d*tan(d*x)^3*tan(c)^5 + d*tan(d*x)^5*tan(c
) - 2*d*tan(d*x)^4*tan(c)^2 + 4*d*tan(d*x)^3*tan(c)^3 - 2*d*tan(d*x)^2*tan(c)^4 + d*tan(d*x)*tan(c)^5 - d*tan(
d*x)^4 + 2*d*tan(d*x)^3*tan(c) - 4*d*tan(d*x)^2*tan(c)^2 + 2*d*tan(d*x)*tan(c)^3 - d*tan(c)^4 - 2*d*tan(d*x)^2
+ d*tan(d*x)*tan(c) - 2*d*tan(c)^2 - d)
Mupad [B] (verification not implemented)
Time = 4.15 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12
\[
\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=x\,\left (\frac {3\,a^2}{8}-\frac {15\,b^2}{8}\right )+\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {\left (\frac {9\,b^2}{8}-\frac {5\,a^2}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {7\,b^2}{8}-\frac {3\,a^2}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {3\,a\,b}{2}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}+\frac {a\,b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{d}
\]
[In]
int(sin(c + d*x)^4*(a + b*tan(c + d*x))^2,x)
[Out]
x*((3*a^2)/8 - (15*b^2)/8) + (b^2*tan(c + d*x))/d + ((3*a*b)/2 - tan(c + d*x)*((3*a^2)/8 - (7*b^2)/8) - tan(c
+ d*x)^3*((5*a^2)/8 - (9*b^2)/8) + 2*a*b*tan(c + d*x)^2)/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) + (a*b*lo
g(tan(c + d*x)^2 + 1))/d