\(\int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx\) [54]

Optimal result

Integrand size = 21, antiderivative size = 257 \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {21 \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {21 \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3} \]

[Out]

-21/64*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))*d^(1/2)/b*2^(1/2)+21/64*arctan(1+2^(1/2)*(d*tan(b*x+a))^
(1/2)/d^(1/2))*d^(1/2)/b*2^(1/2)+21/128*ln(d^(1/2)-2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))*d^(1/2)/b*
2^(1/2)-21/128*ln(d^(1/2)+2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))*d^(1/2)/b*2^(1/2)-7/16*cos(b*x+a)^2
*(d*tan(b*x+a))^(3/2)/b/d-1/4*cos(b*x+a)^4*(d*tan(b*x+a))^(7/2)/b/d^3

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2671, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {21 \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {21 \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{32 \sqrt {2} b}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {21 \sqrt {d} \log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b}-\frac {21 \sqrt {d} \log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b}-\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d} \]

[In]

Int[Sin[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-21*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) + (21*Sqrt[d]*ArcTan[1 + (Sqrt
[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) + (21*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*
Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) - (21*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a
+ b*x]]])/(64*Sqrt[2]*b) - (7*Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2))/(16*b*d) - (Cos[a + b*x]^4*(d*Tan[a + b*x
])^(7/2))/(4*b*d^3)

Rule 210

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

Rule 294

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]

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2671

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

Rubi steps \begin{align*} \text {integral}& = \frac {d \text {Subst}\left (\int \frac {x^{9/2}}{\left (d^2+x^2\right )^3} \, dx,x,d \tan (a+b x)\right )}{b} \\ & = -\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {(7 d) \text {Subst}\left (\int \frac {x^{5/2}}{\left (d^2+x^2\right )^2} \, dx,x,d \tan (a+b x)\right )}{8 b} \\ & = -\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {(21 d) \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{32 b} \\ & = -\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {(21 d) \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{16 b} \\ & = -\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}-\frac {(21 d) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b}+\frac {(21 d) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b} \\ & = -\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {\left (21 \sqrt {d}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {\left (21 \sqrt {d}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {(21 d) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b}+\frac {(21 d) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b} \\ & = \frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}+\frac {\left (21 \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {\left (21 \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b} \\ & = -\frac {21 \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {21 \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {21 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.47 \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\left (21 \arcsin (\cos (a+b x)-\sin (a+b x)) \csc (a+b x) \sqrt {\sin (2 (a+b x))}+21 \csc (a+b x) \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}+18 \sin (2 (a+b x))-2 \sin (4 (a+b x))\right ) \sqrt {d \tan (a+b x)}}{64 b} \]

[In]

Integrate[Sin[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]

[Out]

-1/64*((21*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]] + 21*Csc[a + b*x]*Log[Cos[a
+ b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + 18*Sin[2*(a + b*x)] - 2*Sin[4*(a + b
*x)])*Sqrt[d*Tan[a + b*x]])/b

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(197)=394\).

Time = 13.87 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.41

method	result	size
default	\(\frac {\left (16 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )+16 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-44 \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-44 \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+21 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )-21 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )-42 \arctan \left (\frac {-\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )+42 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )\right ) \sqrt {d \tan \left (b x +a \right )}\, \cos \left (b x +a \right ) \sqrt {2}}{128 b \left (\cos \left (b x +a \right )+1\right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\)	\(619\)

[In]

int(sin(b*x+a)^4*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/128/b*(16*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*cos(b*x+a)^3*sin(b*x+a)+16*cos(b*x+a)^2*si
n(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-44*cos(b*x+a)*sin(b*x+a)*2^(1/2)*(-cos(b*x+a)
*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-44*sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+21*l
n(-(cot(b*x+a)*cos(b*x+a)-2*cot(b*x+a)-2*sin(b*x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3*cot(b*x+a)*csc(
b*x+a)^2+csc(b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1/2)-2*cos(b*x+a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a)))-21*l
n(-(cot(b*x+a)*cos(b*x+a)-2*cot(b*x+a)+2*sin(b*x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3*cot(b*x+a)*csc(
b*x+a)^2+csc(b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1/2)-2*cos(b*x+a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a)))-42*a
rctan((-sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a)))+42*a
rctan((sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a))))*(d*t
an(b*x+a))^(1/2)*cos(b*x+a)/(cos(b*x+a)+1)/(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 947, normalized size of antiderivative = 3.68 \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=\text {Too large to display} \]

[In]

integrate(sin(b*x+a)^4*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

1/256*(16*(4*cos(b*x + a)^3 - 11*cos(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a))*sin(b*x + a) + 21*b*(-d^2/b^4
)^(1/4)*log(9261/2*d^2*cos(b*x + a)*sin(b*x + a) + 9261/2*(b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)^2 - b*d*(-d^2/b^4
)^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) - 9261/4*(2*b^2*d*cos(b*x + a)^2 - b^2*d)
*sqrt(-d^2/b^4)) - 21*b*(-d^2/b^4)^(1/4)*log(9261/2*d^2*cos(b*x + a)*sin(b*x + a) - 9261/2*(b^3*(-d^2/b^4)^(3/
4)*cos(b*x + a)^2 - b*d*(-d^2/b^4)^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) - 9261/4
*(2*b^2*d*cos(b*x + a)^2 - b^2*d)*sqrt(-d^2/b^4)) + 21*I*b*(-d^2/b^4)^(1/4)*log(9261/2*d^2*cos(b*x + a)*sin(b*
x + a) - 9261/2*(I*b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)^2 + I*b*d*(-d^2/b^4)^(1/4)*cos(b*x + a)*sin(b*x + a))*sqr
t(d*sin(b*x + a)/cos(b*x + a)) + 9261/4*(2*b^2*d*cos(b*x + a)^2 - b^2*d)*sqrt(-d^2/b^4)) - 21*I*b*(-d^2/b^4)^(
1/4)*log(9261/2*d^2*cos(b*x + a)*sin(b*x + a) - 9261/2*(-I*b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)^2 - I*b*d*(-d^2/b
^4)^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 9261/4*(2*b^2*d*cos(b*x + a)^2 - b^2*
d)*sqrt(-d^2/b^4)) + 21*b*(-d^2/b^4)^(1/4)*log(9261*d^2 + 18522*(b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)*sin(b*x + a
) - b*d*(-d^2/b^4)^(1/4)*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 21*b*(-d^2/b^4)^(1/4)*log(9261*d
^2 - 18522*(b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)*sin(b*x + a) - b*d*(-d^2/b^4)^(1/4)*cos(b*x + a)^2)*sqrt(d*sin(b
*x + a)/cos(b*x + a))) + 21*I*b*(-d^2/b^4)^(1/4)*log(9261*d^2 - 18522*(I*b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)*sin
(b*x + a) + I*b*d*(-d^2/b^4)^(1/4)*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 21*I*b*(-d^2/b^4)^(1/4
)*log(9261*d^2 - 18522*(-I*b^3*(-d^2/b^4)^(3/4)*cos(b*x + a)*sin(b*x + a) - I*b*d*(-d^2/b^4)^(1/4)*cos(b*x + a
)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))))/b

Sympy [F]

\[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sqrt {d \tan {\left (a + b x \right )}} \sin ^{4}{\left (a + b x \right )}\, dx \]

[In]

integrate(sin(b*x+a)**4*(d*tan(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(d*tan(a + b*x))*sin(a + b*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.88 \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {21 \, d^{6} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {8 \, {\left (11 \, \left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}} d^{6} + 7 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{8}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{128 \, b d^{5}} \]

[In]

integrate(sin(b*x+a)^4*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

1/128*(21*d^6*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(b*x + a)))/sqrt(d))/sqrt(d) + 2*sq
rt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d))/sqrt(d) - sqrt(2)*log(d*tan(b*x
+ a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d) + sqrt(2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x
+ a))*sqrt(d) + d)/sqrt(d)) - 8*(11*(d*tan(b*x + a))^(7/2)*d^6 + 7*(d*tan(b*x + a))^(3/2)*d^8)/(d^4*tan(b*x +
a)^4 + 2*d^4*tan(b*x + a)^2 + d^4))/(b*d^5)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.95 \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\frac {42 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {42 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} - \frac {21 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} + \frac {21 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {8 \, {\left (11 \, \sqrt {d \tan \left (b x + a\right )} d^{5} \tan \left (b x + a\right )^{3} + 7 \, \sqrt {d \tan \left (b x + a\right )} d^{5} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}}{128 \, d} \]

[In]

integrate(sin(b*x+a)^4*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

1/128*(42*sqrt(2)*abs(d)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d))
)/b + 42*sqrt(2)*abs(d)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d))
)/b - 21*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/b + 21*
sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/b - 8*(11*sqrt(d
*tan(b*x + a))*d^5*tan(b*x + a)^3 + 7*sqrt(d*tan(b*x + a))*d^5*tan(b*x + a))/((d^2*tan(b*x + a)^2 + d^2)^2*b))
/d

Mupad [F(-1)]

Timed out. \[ \int \sin ^4(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int {\sin \left (a+b\,x\right )}^4\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )} \,d x \]

[In]

int(sin(a + b*x)^4*(d*tan(a + b*x))^(1/2),x)

[Out]

int(sin(a + b*x)^4*(d*tan(a + b*x))^(1/2), x)