3.142 \(\int \sqrt{a+a \sec (c+d x)} \tan ^4(c+d x) \, dx\)

Optimal. Leaf size=160 \[ \frac{2 a^4 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

[Out]

(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c
+ d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c
+ d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0947099, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 461, 203} \[ \frac{2 a^4 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^4,x]

[Out]

(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c
+ d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c
+ d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))

Rule 3887

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[(-2*a^(m/2 +
n + 1/2))/d, Subst[Int[(x^m*(2 + a*x^2)^(m/2 + n - 1/2))/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 203

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

Rubi steps

\begin{align*} \int \sqrt{a+a \sec (c+d x)} \tan ^4(c+d x) \, dx &=-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (2+a x^2\right )^2}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2}+\frac{x^2}{a}+3 x^4+a x^6+\frac{1}{a^2 \left (1+a x^2\right )}\right ) \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{2 a \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^4 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^4 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 5.81185, size = 110, normalized size = 0.69 \[ \frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (35 \sin \left (\frac{1}{2} (c+d x)\right )-28 \sin \left (\frac{3}{2} (c+d x)\right )-23 \sin \left (\frac{7}{2} (c+d x)\right )+105 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{7}{2}}(c+d x)\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^4,x]

[Out]

(Sec[(c + d*x)/2]*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(105*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[
c + d*x]^(7/2) + 35*Sin[(c + d*x)/2] - 28*Sin[(3*(c + d*x))/2] - 23*Sin[(7*(c + d*x))/2]))/(105*d)

________________________________________________________________________________________

Maple [B]  time = 0.247, size = 317, normalized size = 2. \begin{align*} -{\frac{1}{420\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 105\,\sqrt{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}+210\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\sqrt{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+105\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\sqrt{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) -736\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+368\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+512\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-24\,\cos \left ( dx+c \right ) -120 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^4,x)

[Out]

-1/420/d*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(105*2^(1/2)*sin(d*x+c)*cos(d*x+c)^3*arctanh(1/2*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)+210*arctanh(1/2*2^(1/
2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*2^(1/2)*si
n(d*x+c)*cos(d*x+c)^2+105*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*
cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*2^(1/2)*sin(d*x+c)*cos(d*x+c)-736*cos(d*x+c)^4+368*cos(d*x+c)^3+512*cos(d*x+c
)^2-24*cos(d*x+c)-120)/sin(d*x+c)/cos(d*x+c)^3

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.80117, size = 890, normalized size = 5.56 \begin{align*} \left [\frac{105 \,{\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \,{\left (92 \, \cos \left (d x + c\right )^{3} + 46 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) - 15\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, -\frac{2 \,{\left (105 \,{\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) +{\left (92 \, \cos \left (d x + c\right )^{3} + 46 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) - 15\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(92*cos(d*x + c)
^3 + 46*cos(d*x + c)^2 - 18*cos(d*x + c) - 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*
x + c)^4 + d*cos(d*x + c)^3), -2/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*cos(d*x + c
) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (92*cos(d*x + c)^3 + 46*cos(d*x + c)^2 - 18*cos(d*
x + c) - 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \tan ^{4}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**(1/2)*tan(d*x+c)**4,x)

[Out]

Integral(sqrt(a*(sec(c + d*x) + 1))*tan(c + d*x)**4, x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError