\(\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx\) [13]
Optimal result
Integrand size = 19, antiderivative size = 69 \[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d}
\]
[Out]
b*arctanh(sin(d*x+c))/d-a*cos(d*x+c)/d+1/3*a*cos(d*x+c)^3/d-b*sin(d*x+c)/d-1/3*b*sin(d*x+c)^3/d
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3598, 2713, 2672, 308, 212}
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \cos ^3(c+d x)}{3 d}-\frac {a \cos (c+d x)}{d}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {b \sin (c+d x)}{d}
\]
[In]
Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x]),x]
[Out]
(b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (b*Sin[c + d*x])/d - (b*Sin[c +
d*x]^3)/(3*d)
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 308
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 2672
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 2713
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 3598
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]
Rubi steps \begin{align*}
\text {integral}& = \int \left (a \sin ^3(c+d x)+b \sin ^3(c+d x) \tan (c+d x)\right ) \, dx \\ & = a \int \sin ^3(c+d x) \, dx+b \int \sin ^3(c+d x) \tan (c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d}+\frac {b \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a \cos (c+d x)}{4 d}+\frac {a \cos (3 (c+d x))}{12 d}-\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d}
\]
[In]
Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x]),x]
[Out]
(b*ArcTanh[Sin[c + d*x]])/d - (3*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (b*Sin[c + d*x])/d - (b
*Sin[c + d*x]^3)/(3*d)
Maple [A] (verified)
Time = 1.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {a \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) |
\(60\) |
| default |
\(\frac {-\frac {a \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) |
\(60\) |
| risch |
\(\frac {5 i {\mathrm e}^{i \left (d x +c \right )} b}{8 d}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a}{8 d}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} b}{8 d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a}{8 d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a \cos \left (3 d x +3 c \right )}{12 d}+\frac {b \sin \left (3 d x +3 c \right )}{12 d}\) |
\(131\) |
| | |
|
|
|
|
[In]
int(sin(d*x+c)^3*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/3*a*(2+sin(d*x+c)^2)*cos(d*x+c)+b*(-1/3*sin(d*x+c)^3-sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c))))
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, a \cos \left (d x + c\right )^{3} - 6 \, a \cos \left (d x + c\right ) + 3 \, b \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, b \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (b \cos \left (d x + c\right )^{2} - 4 \, b\right )} \sin \left (d x + c\right )}{6 \, d}
\]
[In]
integrate(sin(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/6*(2*a*cos(d*x + c)^3 - 6*a*cos(d*x + c) + 3*b*log(sin(d*x + c) + 1) - 3*b*log(-sin(d*x + c) + 1) + 2*(b*cos
(d*x + c)^2 - 4*b)*sin(d*x + c))/d
Sympy [F]
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \sin ^{3}{\left (c + d x \right )}\, dx
\]
[In]
integrate(sin(d*x+c)**3*(a+b*tan(d*x+c)),x)
[Out]
Integral((a + b*tan(c + d*x))*sin(c + d*x)**3, x)
Maxima [A] (verification not implemented)
none
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a - {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} b}{6 \, d}
\]
[In]
integrate(sin(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/6*(2*(cos(d*x + c)^3 - 3*cos(d*x + c))*a - (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))*b)/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 4486 vs. \(2 (65) = 130\).
Time = 1.00 (sec) , antiderivative size = 4486, normalized size of antiderivative = 65.01
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(sin(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="giac")
[Out]
-1/6*(3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan
(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2
+ tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^6 - 3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*t
an(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(
tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^6 + 4*a*tan(1/2*d*
x)^6*tan(1/2*c)^6 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(
1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + t
an(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^4 - 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*ta
n(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan
(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^4 -
12*b*tan(1/2*d*x)^6*tan(1/2*c)^5 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*ta
n(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*t
an(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*tan(1/2*c)^6 - 9*b*log(2*(tan(1/2*d*x)^2*tan(
1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1
/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*
tan(1/2*c)^6 - 12*b*tan(1/2*d*x)^5*tan(1/2*c)^6 + 12*a*tan(1/2*d*x)^6*tan(1/2*c)^4 + 12*a*tan(1/2*d*x)^4*tan(1
/2*c)^6 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 +
tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x
)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^2 - 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)
^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) +
1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^2 - 40*b*tan(1
/2*d*x)^6*tan(1/2*c)^3 + 27*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x
)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)
^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*tan(1/2*c)^4 - 27*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2
- 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x)
+ 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*tan(1/2*
c)^4 - 60*b*tan(1/2*d*x)^5*tan(1/2*c)^4 - 60*b*tan(1/2*d*x)^4*tan(1/2*c)^5 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2
*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*
d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2*tan
(1/2*c)^6 - 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2
+ tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d
*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2*tan(1/2*c)^6 - 40*b*tan(1/2*d*x)^3*tan(1/2*c)^6 - 12*a*tan(1/2*d*x)^
6*tan(1/2*c)^2 - 96*a*tan(1/2*d*x)^5*tan(1/2*c)^3 - 108*a*tan(1/2*d*x)^4*tan(1/2*c)^4 - 96*a*tan(1/2*d*x)^3*ta
n(1/2*c)^5 - 12*a*tan(1/2*d*x)^2*tan(1/2*c)^6 + 3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(
1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan
(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6 - 3*b*log(2*(tan(1/2*d*x)^2*tan(
1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1
/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6
- 12*b*tan(1/2*d*x)^6*tan(1/2*c) + 27*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*t
an(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*
tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*tan(1/2*c)^2 - 27*b*log(2*(tan(1/2*d*x)^2*ta
n(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan
(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^
4*tan(1/2*c)^2 + 60*b*tan(1/2*d*x)^5*tan(1/2*c)^2 + 120*b*tan(1/2*d*x)^4*tan(1/2*c)^3 + 27*b*log(2*(tan(1/2*d*
x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2
- 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/
2*d*x)^2*tan(1/2*c)^4 - 27*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)
*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^
2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2*tan(1/2*c)^4 + 120*b*tan(1/2*d*x)^3*tan(1/2*c)^4 + 60*b
*tan(1/2*d*x)^2*tan(1/2*c)^5 + 3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/
2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1
/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*c)^6 - 3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/
2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2
*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*c)^6 - 12*b*tan(1/2*d*x)*t
an(1/2*c)^6 - 4*a*tan(1/2*d*x)^6 + 108*a*tan(1/2*d*x)^4*tan(1/2*c)^2 + 128*a*tan(1/2*d*x)^3*tan(1/2*c)^3 + 108
*a*tan(1/2*d*x)^2*tan(1/2*c)^4 - 4*a*tan(1/2*c)^6 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*
tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/
(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4 - 9*b*log(2*(tan(1/2*d*x)^2*
tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*t
an(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x
)^4 + 12*b*tan(1/2*d*x)^5 - 60*b*tan(1/2*d*x)^4*tan(1/2*c) + 27*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1
/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/
2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 27*
b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x
)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/
2*c)^2 + 1))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 120*b*tan(1/2*d*x)^3*tan(1/2*c)^2 - 120*b*tan(1/2*d*x)^2*tan(1/2*c)
^3 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(
1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 +
tan(1/2*c)^2 + 1))*tan(1/2*c)^4 - 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*ta
n(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*t
an(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*c)^4 - 60*b*tan(1/2*d*x)*tan(1/2*c)^4 + 12*b*tan(1/2
*c)^5 - 12*a*tan(1/2*d*x)^4 - 96*a*tan(1/2*d*x)^3*tan(1/2*c) - 108*a*tan(1/2*d*x)^2*tan(1/2*c)^2 - 96*a*tan(1/
2*d*x)*tan(1/2*c)^3 - 12*a*tan(1/2*c)^4 + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c)
+ 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d
*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2 - 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)
^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x
) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2 + 40*b
*tan(1/2*d*x)^3 + 60*b*tan(1/2*d*x)^2*tan(1/2*c) + 9*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*t
an(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(
tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*c)^2 - 9*b*log(2*(tan(1/2*d*x)^2*tan
(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(
1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*c)^2 +
60*b*tan(1/2*d*x)*tan(1/2*c)^2 + 40*b*tan(1/2*c)^3 + 12*a*tan(1/2*d*x)^2 + 12*a*tan(1/2*c)^2 + 3*b*log(2*(tan
(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/
2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))
- 3*b*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/
2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + t
an(1/2*c)^2 + 1)) + 12*b*tan(1/2*d*x) + 12*b*tan(1/2*c) + 4*a)/(d*tan(1/2*d*x)^6*tan(1/2*c)^6 + 3*d*tan(1/2*d*
x)^6*tan(1/2*c)^4 + 3*d*tan(1/2*d*x)^4*tan(1/2*c)^6 + 3*d*tan(1/2*d*x)^6*tan(1/2*c)^2 + 9*d*tan(1/2*d*x)^4*tan
(1/2*c)^4 + 3*d*tan(1/2*d*x)^2*tan(1/2*c)^6 + d*tan(1/2*d*x)^6 + 9*d*tan(1/2*d*x)^4*tan(1/2*c)^2 + 9*d*tan(1/2
*d*x)^2*tan(1/2*c)^4 + d*tan(1/2*c)^6 + 3*d*tan(1/2*d*x)^4 + 9*d*tan(1/2*d*x)^2*tan(1/2*c)^2 + 3*d*tan(1/2*c)^
4 + 3*d*tan(1/2*d*x)^2 + 3*d*tan(1/2*c)^2 + d)
Mupad [B] (verification not implemented)
Time = 4.71 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26
\[
\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a\,{\cos \left (c+d\,x\right )}^3}{3\,d}-\frac {a\,\cos \left (c+d\,x\right )}{d}-\frac {4\,b\,\sin \left (c+d\,x\right )}{3\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}
\]
[In]
int(sin(c + d*x)^3*(a + b*tan(c + d*x)),x)
[Out]
(2*b*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (a*cos(c + d*x)^3)/(3*d) - (a*cos(c + d*x))/d - (4*b*si
n(c + d*x))/(3*d) + (b*cos(c + d*x)^2*sin(c + d*x))/(3*d)