3.463 \(\int \frac{\sec ^5(c+d x)}{(a+b \tan ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}-\frac{(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]

[Out]

ArcTanh[Sin[c + d*x]]/(b^2*d) - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)
*b^2*d) - ((a - b)*Sin[c + d*x])/(2*a*b*d*(a - (a - b)*Sin[c + d*x]^2))

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Rubi [A]  time = 0.140827, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3676, 414, 522, 206, 208} \[ -\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}-\frac{(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2)^2,x]

[Out]

ArcTanh[Sin[c + d*x]]/(b^2*d) - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)
*b^2*d) - ((a - b)*Sin[c + d*x])/(2*a*b*d*(a - (a - b)*Sin[c + d*x]^2))

Rule 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -
ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n/2] && IntegerQ[p]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 206

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

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.806989, size = 191, normalized size = 1.75 \[ \frac{\frac{\left (-2 a^2+a b+b^2\right ) \log \left (\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right )}{a^{3/2} \sqrt{a-b}}+\frac{\sqrt{a-b} (2 a+b) \log \left (\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right )}{a^{3/2}}+\frac{4 b (b-a) \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}-4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2)^2,x]

[Out]

(-4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Sqrt[a - b]*(2*a
+ b)*Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + ((-2*a^2 + a*b + b^2)*Log[Sqrt[a] + Sqrt[a - b]*Sin[c
+ d*x]])/(a^(3/2)*Sqrt[a - b]) + (4*b*(-a + b)*Sin[c + d*x])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))/(4*b^2*d)

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Maple [B]  time = 0.099, size = 236, normalized size = 2.2 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{2\,db \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}-{\frac{a}{d{b}^{2}}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}+{\frac{1}{2\,db}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}-{\frac{\sin \left ( dx+c \right ) }{2\,da \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}+{\frac{1}{2\,da}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) +1 \right ) }{2\,d{b}^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{2\,d{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a+b*tan(d*x+c)^2)^2,x)

[Out]

1/2/d/b*sin(d*x+c)/(a*sin(d*x+c)^2-b*sin(d*x+c)^2-a)-1/d/b^2/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b)
)^(1/2))*a+1/2/d/b/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b))^(1/2))-1/2/d/a*sin(d*x+c)/(a*sin(d*x+c)^
2-b*sin(d*x+c)^2-a)+1/2/d/a/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b))^(1/2))+1/2/d/b^2*ln(sin(d*x+c)+
1)-1/2/d/b^2*ln(sin(d*x+c)-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+b*tan(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0195, size = 929, normalized size = 8.52 \begin{align*} \left [\frac{{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt{\frac{a - b}{a}} \log \left (-\frac{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + 2 \,{\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left (a b^{3} d +{\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac{{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt{-\frac{a - b}{a}} \arctan \left (\sqrt{-\frac{a - b}{a}} \sin \left (d x + c\right )\right ) +{\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{2 \,{\left (a b^{3} d +{\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+b*tan(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(((2*a^2 - a*b - b^2)*cos(d*x + c)^2 + 2*a*b + b^2)*sqrt((a - b)/a)*log(-((a - b)*cos(d*x + c)^2 + 2*a*sq
rt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + 2*((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log
(sin(d*x + c) + 1) - 2*((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(-sin(d*x + c) + 1) - 2*(a*b - b^2)*sin(d*x + c))
/(a*b^3*d + (a^2*b^2 - a*b^3)*d*cos(d*x + c)^2), 1/2*(((2*a^2 - a*b - b^2)*cos(d*x + c)^2 + 2*a*b + b^2)*sqrt(
-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c)) + ((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(sin(d*x + c) + 1) -
((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(-sin(d*x + c) + 1) - (a*b - b^2)*sin(d*x + c))/(a*b^3*d + (a^2*b^2 - a
*b^3)*d*cos(d*x + c)^2)]

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Sympy [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(sec(d*x+c)**5/(a+b*tan(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.81859, size = 207, normalized size = 1.9 \begin{align*} \frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{2}} - \frac{\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{2}} - \frac{{\left (2 \, a^{2} - a b - b^{2}\right )} \arctan \left (-\frac{a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} a b^{2}} + \frac{a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} a b}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+b*tan(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/2*(log(abs(sin(d*x + c) + 1))/b^2 - log(abs(sin(d*x + c) - 1))/b^2 - (2*a^2 - a*b - b^2)*arctan(-(a*sin(d*x
+ c) - b*sin(d*x + c))/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a*b^2) + (a*sin(d*x + c) - b*sin(d*x + c))/((a*sin(
d*x + c)^2 - b*sin(d*x + c)^2 - a)*a*b))/d