3.462 \(\int \frac {\sec ^7(c+d x)}{(a+b \tan ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=167 \[ \frac {(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3 d}-\frac {(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac {(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\tan (c+d x) \sec (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )} \]

[Out]

-1/2*(4*a-5*b)*arctanh(sin(d*x+c))/b^3/d+1/2*(a-b)^(3/2)*(4*a+b)*arctanh(sin(d*x+c)*(a-b)^(1/2)/a^(1/2))/a^(3/
2)/b^3/d+1/2*(a-b)*(2*a-b)*sin(d*x+c)/a/b^2/d/(a-(a-b)*sin(d*x+c)^2)+1/2*sec(d*x+c)*tan(d*x+c)/b/d/(a-(a-b)*si
n(d*x+c)^2)

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Rubi [A]  time = 0.27, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3676, 414, 527, 522, 206, 208} \[ \frac {(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3 d}+\frac {(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac {(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7/(a + b*Tan[c + d*x]^2)^2,x]

[Out]

-((4*a - 5*b)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + ((a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/S
qrt[a]])/(2*a^(3/2)*b^3*d) + ((a - b)*(2*a - b)*Sin[c + d*x])/(2*a*b^2*d*(a - (a - b)*Sin[c + d*x]^2)) + (Sec[
c + d*x]*Tan[c + d*x])/(2*b*d*(a - (a - b)*Sin[c + d*x]^2))

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]

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 527

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

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]

Rubi steps

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

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Mathematica [A]  time = 4.08, size = 254, normalized size = 1.52 \[ \frac {-\frac {(4 a+b) (a-b)^{3/2} \log \left (\sqrt {a}-\sqrt {a-b} \sin (c+d x)\right )}{a^{3/2}}+\frac {(4 a+b) (a-b)^{3/2} \log \left (\sqrt {a-b} \sin (c+d x)+\sqrt {a}\right )}{a^{3/2}}+\frac {4 b (a-b)^2 \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 (4 a-5 b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (5 b-4 a) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {b}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}}{4 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7/(a + b*Tan[c + d*x]^2)^2,x]

[Out]

(2*(4*a - 5*b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-4*a + 5*b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/
2]] - ((a - b)^(3/2)*(4*a + b)*Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + ((a - b)^(3/2)*(4*a + b)*Log
[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + b/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - b/(Cos[(c + d*x)/2
] + Sin[(c + d*x)/2])^2 + (4*(a - b)^2*b*Sin[c + d*x])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))/(4*b^3*d)

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fricas [A]  time = 0.61, size = 635, normalized size = 3.80 \[ \left [-\frac {{\left ({\left (4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{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 ) + {\left ({\left (4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a b^{2} + {\left (2 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (a b^{4} d \cos \left (d x + c\right )^{2} + {\left (a^{2} b^{3} - a b^{4}\right )} d \cos \left (d x + c\right )^{4}\right )}}, -\frac {2 \, {\left ({\left (4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a - b}{a}} \arctan \left (\sqrt {-\frac {a - b}{a}} \sin \left (d x + c\right )\right ) + {\left ({\left (4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a b^{2} + {\left (2 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (a b^{4} d \cos \left (d x + c\right )^{2} + {\left (a^{2} b^{3} - a b^{4}\right )} d \cos \left (d x + c\right )^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(((4*a^3 - 7*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c)^4 + (4*a^2*b - 3*a*b^2 - b^3)*cos(d*x + c)^2)*sqrt((a -
b)/a)*log(-((a - b)*cos(d*x + c)^2 + 2*a*sqrt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)
) + ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) -
((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(
a*b^2 + (2*a^2*b - 3*a*b^2 + b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^2 + (a^2*b^3 - a*b^4)*d*
cos(d*x + c)^4), -1/4*(2*((4*a^3 - 7*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c)^4 + (4*a^2*b - 3*a*b^2 - b^3)*cos(d*x
+ c)^2)*sqrt(-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c)) + ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4
+ (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4
*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(a*b^2 + (2*a^2*b - 3*a*b^2 + b^3)*cos(d*x + c)^2
)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^2 + (a^2*b^3 - a*b^4)*d*cos(d*x + c)^4)]

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giac [A]  time = 2.26, size = 245, normalized size = 1.47 \[ -\frac {\frac {{\left (4 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{3}} - \frac {{\left (4 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{3}} - \frac {2 \, {\left (4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a b^{3}} + \frac {2 \, {\left (2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{3} + b^{2} \sin \left (d x + c\right )^{3} - 2 \, a^{2} \sin \left (d x + c\right ) + 2 \, a b \sin \left (d x + c\right ) - b^{2} \sin \left (d x + c\right )\right )}}{{\left (a \sin \left (d x + c\right )^{4} - b \sin \left (d x + c\right )^{4} - 2 \, a \sin \left (d x + c\right )^{2} + b \sin \left (d x + c\right )^{2} + a\right )} a b^{2}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/4*((4*a - 5*b)*log(abs(sin(d*x + c) + 1))/b^3 - (4*a - 5*b)*log(abs(sin(d*x + c) - 1))/b^3 - 2*(4*a^3 - 7*a
^2*b + 2*a*b^2 + b^3)*arctan(-(a*sin(d*x + c) - b*sin(d*x + c))/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a*b^3) + 2
*(2*a^2*sin(d*x + c)^3 - 3*a*b*sin(d*x + c)^3 + b^2*sin(d*x + c)^3 - 2*a^2*sin(d*x + c) + 2*a*b*sin(d*x + c) -
b^2*sin(d*x + c))/((a*sin(d*x + c)^4 - b*sin(d*x + c)^4 - 2*a*sin(d*x + c)^2 + b*sin(d*x + c)^2 + a)*a*b^2))/
d

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maple [B]  time = 0.74, size = 389, normalized size = 2.33 \[ -\frac {a \sin \left (d x +c \right )}{2 d \,b^{2} \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}+\frac {\sin \left (d x +c \right )}{d b \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}-\frac {\sin \left (d x +c \right )}{2 d a \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}+\frac {2 \arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right ) a^{2}}{d \,b^{3} \sqrt {a \left (a -b \right )}}-\frac {7 \arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right ) a}{2 d \,b^{2} \sqrt {a \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{d b \sqrt {a \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 d a \sqrt {a \left (a -b \right )}}-\frac {1}{4 d \,b^{2} \left (-1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (-1+\sin \left (d x +c \right )\right ) a}{d \,b^{3}}-\frac {5 \ln \left (-1+\sin \left (d x +c \right )\right )}{4 d \,b^{2}}-\frac {1}{4 d \,b^{2} \left (\sin \left (d x +c \right )+1\right )}-\frac {\ln \left (\sin \left (d x +c \right )+1\right ) a}{d \,b^{3}}+\frac {5 \ln \left (\sin \left (d x +c \right )+1\right )}{4 d \,b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7/(a+b*tan(d*x+c)^2)^2,x)

[Out]

-1/2/d/b^2*a*sin(d*x+c)/(a*sin(d*x+c)^2-b*sin(d*x+c)^2-a)+1/d/b*sin(d*x+c)/(a*sin(d*x+c)^2-b*sin(d*x+c)^2-a)-1
/2/d/a*sin(d*x+c)/(a*sin(d*x+c)^2-b*sin(d*x+c)^2-a)+2/d/b^3/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b))
^(1/2))*a^2-7/2/d/b^2/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b))^(1/2))*a+1/d/b/(a*(a-b))^(1/2)*arctan
h((a-b)*sin(d*x+c)/(a*(a-b))^(1/2))+1/2/d/a/(a*(a-b))^(1/2)*arctanh((a-b)*sin(d*x+c)/(a*(a-b))^(1/2))-1/4/d/b^
2/(-1+sin(d*x+c))+1/d/b^3*ln(-1+sin(d*x+c))*a-5/4/d/b^2*ln(-1+sin(d*x+c))-1/4/d/b^2/(sin(d*x+c)+1)-1/d/b^3*ln(
sin(d*x+c)+1)*a+5/4/d/b^2*ln(sin(d*x+c)+1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is b-a positive or negative?

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mupad [B]  time = 15.24, size = 4304, normalized size = 25.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^7*(a + b*tan(c + d*x)^2)^2),x)

[Out]

((tan(c/2 + (d*x)/2)*(2*a^2 - 2*a*b + b^2))/(a*b^2) - (tan(c/2 + (d*x)/2)^3*(2*a^2 - 6*a*b + b^2))/(a*b^2) + (
tan(c/2 + (d*x)/2)^7*(2*a^2 - 2*a*b + b^2))/(a*b^2) - (tan(c/2 + (d*x)/2)^5*(2*a^2 - 6*a*b + b^2))/(a*b^2))/(d
*(a - tan(c/2 + (d*x)/2)^2*(4*a - 4*b) - tan(c/2 + (d*x)/2)^6*(4*a - 4*b) + tan(c/2 + (d*x)/2)^4*(6*a - 8*b) +
a*tan(c/2 + (d*x)/2)^8)) - (atan((((4*a - 5*b)*((((256*(16*a*b^15 + 92*a^2*b^14 - 8*a^3*b^13 - 2236*a^4*b^12
+ 768*a^5*b^11 + 18228*a^6*b^10 - 41560*a^7*b^9 + 37420*a^8*b^8 - 13552*a^9*b^7 + 64*a^10*b^6 + 768*a^11*b^5))
/(a^3*b^10) + (((((256*(256*a^4*b^16 + 192*a^5*b^15 - 1088*a^6*b^14 - 192*a^7*b^13 + 1600*a^8*b^12 - 768*a^9*b
^11))/(a^3*b^10) - (256*tan(c/2 + (d*x)/2)*(4*a - 5*b)*(1024*a^5*b^15 - 2304*a^6*b^14 + 1664*a^7*b^13 - 384*a^
8*b^12))/(a^3*b^11))*(4*a - 5*b))/(2*b^3) - (512*tan(c/2 + (d*x)/2)*(64*a^2*b^14 + 160*a^3*b^13 - 984*a^4*b^12
- 6560*a^5*b^11 + 28720*a^6*b^10 - 42400*a^7*b^9 + 29512*a^8*b^8 - 9664*a^9*b^7 + 1152*a^10*b^6))/(a^3*b^8))*
(4*a - 5*b))/(2*b^3))*(4*a - 5*b))/(2*b^3) + (512*tan(c/2 + (d*x)/2)*(8*a*b^11 - 8960*a^11*b + 768*a^12 + b^12
+ 396*a^2*b^10 + 440*a^3*b^9 - 7144*a^4*b^8 + 6656*a^5*b^7 + 34712*a^6*b^6 - 106784*a^7*b^5 + 138675*a^8*b^4
- 100016*a^9*b^3 + 41248*a^10*b^2))/(a^3*b^8))*1i)/(2*b^3) - ((4*a - 5*b)*((((256*(16*a*b^15 + 92*a^2*b^14 - 8
*a^3*b^13 - 2236*a^4*b^12 + 768*a^5*b^11 + 18228*a^6*b^10 - 41560*a^7*b^9 + 37420*a^8*b^8 - 13552*a^9*b^7 + 64
*a^10*b^6 + 768*a^11*b^5))/(a^3*b^10) + (((((256*(256*a^4*b^16 + 192*a^5*b^15 - 1088*a^6*b^14 - 192*a^7*b^13 +
1600*a^8*b^12 - 768*a^9*b^11))/(a^3*b^10) + (256*tan(c/2 + (d*x)/2)*(4*a - 5*b)*(1024*a^5*b^15 - 2304*a^6*b^1
4 + 1664*a^7*b^13 - 384*a^8*b^12))/(a^3*b^11))*(4*a - 5*b))/(2*b^3) + (512*tan(c/2 + (d*x)/2)*(64*a^2*b^14 + 1
60*a^3*b^13 - 984*a^4*b^12 - 6560*a^5*b^11 + 28720*a^6*b^10 - 42400*a^7*b^9 + 29512*a^8*b^8 - 9664*a^9*b^7 + 1
152*a^10*b^6))/(a^3*b^8))*(4*a - 5*b))/(2*b^3))*(4*a - 5*b))/(2*b^3) - (512*tan(c/2 + (d*x)/2)*(8*a*b^11 - 896
0*a^11*b + 768*a^12 + b^12 + 396*a^2*b^10 + 440*a^3*b^9 - 7144*a^4*b^8 + 6656*a^5*b^7 + 34712*a^6*b^6 - 106784
*a^7*b^5 + 138675*a^8*b^4 - 100016*a^9*b^3 + 41248*a^10*b^2))/(a^3*b^8))*1i)/(2*b^3))/(((4*a - 5*b)*((((256*(1
6*a*b^15 + 92*a^2*b^14 - 8*a^3*b^13 - 2236*a^4*b^12 + 768*a^5*b^11 + 18228*a^6*b^10 - 41560*a^7*b^9 + 37420*a^
8*b^8 - 13552*a^9*b^7 + 64*a^10*b^6 + 768*a^11*b^5))/(a^3*b^10) + (((((256*(256*a^4*b^16 + 192*a^5*b^15 - 1088
*a^6*b^14 - 192*a^7*b^13 + 1600*a^8*b^12 - 768*a^9*b^11))/(a^3*b^10) - (256*tan(c/2 + (d*x)/2)*(4*a - 5*b)*(10
24*a^5*b^15 - 2304*a^6*b^14 + 1664*a^7*b^13 - 384*a^8*b^12))/(a^3*b^11))*(4*a - 5*b))/(2*b^3) - (512*tan(c/2 +
(d*x)/2)*(64*a^2*b^14 + 160*a^3*b^13 - 984*a^4*b^12 - 6560*a^5*b^11 + 28720*a^6*b^10 - 42400*a^7*b^9 + 29512*
a^8*b^8 - 9664*a^9*b^7 + 1152*a^10*b^6))/(a^3*b^8))*(4*a - 5*b))/(2*b^3))*(4*a - 5*b))/(2*b^3) + (512*tan(c/2
+ (d*x)/2)*(8*a*b^11 - 8960*a^11*b + 768*a^12 + b^12 + 396*a^2*b^10 + 440*a^3*b^9 - 7144*a^4*b^8 + 6656*a^5*b^
7 + 34712*a^6*b^6 - 106784*a^7*b^5 + 138675*a^8*b^4 - 100016*a^9*b^3 + 41248*a^10*b^2))/(a^3*b^8)))/(2*b^3) -
(512*(64*a*b^11 - 27904*a^11*b + 3584*a^12 - 5*b^12 + 467*a^2*b^10 - 1322*a^3*b^9 - 4957*a^4*b^8 + 18148*a^5*b
^7 + 165*a^6*b^6 - 81226*a^7*b^5 + 165322*a^8*b^4 - 164368*a^9*b^3 + 92032*a^10*b^2))/(a^3*b^10) + ((4*a - 5*b
)*((((256*(16*a*b^15 + 92*a^2*b^14 - 8*a^3*b^13 - 2236*a^4*b^12 + 768*a^5*b^11 + 18228*a^6*b^10 - 41560*a^7*b^
9 + 37420*a^8*b^8 - 13552*a^9*b^7 + 64*a^10*b^6 + 768*a^11*b^5))/(a^3*b^10) + (((((256*(256*a^4*b^16 + 192*a^5
*b^15 - 1088*a^6*b^14 - 192*a^7*b^13 + 1600*a^8*b^12 - 768*a^9*b^11))/(a^3*b^10) + (256*tan(c/2 + (d*x)/2)*(4*
a - 5*b)*(1024*a^5*b^15 - 2304*a^6*b^14 + 1664*a^7*b^13 - 384*a^8*b^12))/(a^3*b^11))*(4*a - 5*b))/(2*b^3) + (5
12*tan(c/2 + (d*x)/2)*(64*a^2*b^14 + 160*a^3*b^13 - 984*a^4*b^12 - 6560*a^5*b^11 + 28720*a^6*b^10 - 42400*a^7*
b^9 + 29512*a^8*b^8 - 9664*a^9*b^7 + 1152*a^10*b^6))/(a^3*b^8))*(4*a - 5*b))/(2*b^3))*(4*a - 5*b))/(2*b^3) - (
512*tan(c/2 + (d*x)/2)*(8*a*b^11 - 8960*a^11*b + 768*a^12 + b^12 + 396*a^2*b^10 + 440*a^3*b^9 - 7144*a^4*b^8 +
6656*a^5*b^7 + 34712*a^6*b^6 - 106784*a^7*b^5 + 138675*a^8*b^4 - 100016*a^9*b^3 + 41248*a^10*b^2))/(a^3*b^8))
)/(2*b^3)))*(4*a - 5*b)*1i)/(b^3*d) + (atan((((a - b)^(3/2)*(4*a + b)*((128*tan(c/2 + (d*x)/2)*(26*a*b^8 - 123
2*a^8*b + 192*a^9 + b^9 + 40*a^2*b^7 - 376*a^3*b^6 + 44*a^4*b^5 + 1964*a^5*b^4 - 3767*a^6*b^3 + 3108*a^7*b^2))
/(a^2*b^6) - ((a - b)^(3/2)*(4*a + b)*((16*(80*a^3*b^11 - 52*a^4*b^10 - 912*a^5*b^9 + 488*a^6*b^8 + 3008*a^7*b
^7 - 4836*a^8*b^6 + 2784*a^9*b^5 - 576*a^10*b^4))/(a^3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(128*a^2*b^12 + 368*a^3
*b^11 - 1164*a^4*b^10 - 7344*a^5*b^9 + 24216*a^6*b^8 - 27808*a^7*b^7 + 15332*a^8*b^6 - 4320*a^9*b^5 + 576*a^10
*b^4))/(a^3*b^8) - (((128*tan(c/2 + (d*x)/2)*(16*a^3*b^10 + 340*a^4*b^9 - 952*a^5*b^8 + 836*a^6*b^7 - 240*a^7*
b^6))/(a^2*b^6) - (((16*(1024*a^6*b^12 - 1536*a^7*b^11 + 576*a^8*b^10))/(a^3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(
2048*a^5*b^13 - 4096*a^6*b^12 + 2688*a^7*b^11 - 576*a^8*b^10))/(a^3*b^8))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*
b^3))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3)))/(4*a^(3/2)*b^3))*1i)/(4*a^(3/2)*b^3) + ((a - b)^(3/2)*(4*a +
b)*((128*tan(c/2 + (d*x)/2)*(26*a*b^8 - 1232*a^8*b + 192*a^9 + b^9 + 40*a^2*b^7 - 376*a^3*b^6 + 44*a^4*b^5 + 1
964*a^5*b^4 - 3767*a^6*b^3 + 3108*a^7*b^2))/(a^2*b^6) + ((a - b)^(3/2)*(4*a + b)*((16*(80*a^3*b^11 - 52*a^4*b^
10 - 912*a^5*b^9 + 488*a^6*b^8 + 3008*a^7*b^7 - 4836*a^8*b^6 + 2784*a^9*b^5 - 576*a^10*b^4))/(a^3*b^8) + (16*t
an(c/2 + (d*x)/2)^2*(128*a^2*b^12 + 368*a^3*b^11 - 1164*a^4*b^10 - 7344*a^5*b^9 + 24216*a^6*b^8 - 27808*a^7*b^
7 + 15332*a^8*b^6 - 4320*a^9*b^5 + 576*a^10*b^4))/(a^3*b^8) + (((128*tan(c/2 + (d*x)/2)*(16*a^3*b^10 + 340*a^4
*b^9 - 952*a^5*b^8 + 836*a^6*b^7 - 240*a^7*b^6))/(a^2*b^6) + (((16*(1024*a^6*b^12 - 1536*a^7*b^11 + 576*a^8*b^
10))/(a^3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(2048*a^5*b^13 - 4096*a^6*b^12 + 2688*a^7*b^11 - 576*a^8*b^10))/(a^3
*b^8))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3)))/(4*a^(3/2)*b^3))*1
i)/(4*a^(3/2)*b^3))/((32*(12*a*b^9 - 2592*a^9*b + 448*a^10 - b^10 + 98*a^2*b^8 - 108*a^3*b^7 - 853*a^4*b^6 + 1
368*a^5*b^5 + 1648*a^6*b^4 - 5808*a^7*b^3 + 5788*a^8*b^2))/(a^3*b^8) + (32*tan(c/2 + (d*x)/2)^2*(3136*a^10 - 1
9040*a^9*b - 32*a*b^9 + b^10 + 198*a^2*b^8 + 1244*a^3*b^7 - 4555*a^4*b^6 - 3612*a^5*b^5 + 33032*a^6*b^4 - 5696
8*a^7*b^3 + 46596*a^8*b^2))/(a^3*b^8) + ((a - b)^(3/2)*(4*a + b)*((128*tan(c/2 + (d*x)/2)*(26*a*b^8 - 1232*a^8
*b + 192*a^9 + b^9 + 40*a^2*b^7 - 376*a^3*b^6 + 44*a^4*b^5 + 1964*a^5*b^4 - 3767*a^6*b^3 + 3108*a^7*b^2))/(a^2
*b^6) - ((a - b)^(3/2)*(4*a + b)*((16*(80*a^3*b^11 - 52*a^4*b^10 - 912*a^5*b^9 + 488*a^6*b^8 + 3008*a^7*b^7 -
4836*a^8*b^6 + 2784*a^9*b^5 - 576*a^10*b^4))/(a^3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(128*a^2*b^12 + 368*a^3*b^11
- 1164*a^4*b^10 - 7344*a^5*b^9 + 24216*a^6*b^8 - 27808*a^7*b^7 + 15332*a^8*b^6 - 4320*a^9*b^5 + 576*a^10*b^4)
)/(a^3*b^8) - (((128*tan(c/2 + (d*x)/2)*(16*a^3*b^10 + 340*a^4*b^9 - 952*a^5*b^8 + 836*a^6*b^7 - 240*a^7*b^6))
/(a^2*b^6) - (((16*(1024*a^6*b^12 - 1536*a^7*b^11 + 576*a^8*b^10))/(a^3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(2048*
a^5*b^13 - 4096*a^6*b^12 + 2688*a^7*b^11 - 576*a^8*b^10))/(a^3*b^8))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3))
*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3)))/(4*a^(3/2)*b^3)))/(4*a^(3/2)*b^3) - ((a - b)^(3/2)*(4*a + b)*((128
*tan(c/2 + (d*x)/2)*(26*a*b^8 - 1232*a^8*b + 192*a^9 + b^9 + 40*a^2*b^7 - 376*a^3*b^6 + 44*a^4*b^5 + 1964*a^5*
b^4 - 3767*a^6*b^3 + 3108*a^7*b^2))/(a^2*b^6) + ((a - b)^(3/2)*(4*a + b)*((16*(80*a^3*b^11 - 52*a^4*b^10 - 912
*a^5*b^9 + 488*a^6*b^8 + 3008*a^7*b^7 - 4836*a^8*b^6 + 2784*a^9*b^5 - 576*a^10*b^4))/(a^3*b^8) + (16*tan(c/2 +
(d*x)/2)^2*(128*a^2*b^12 + 368*a^3*b^11 - 1164*a^4*b^10 - 7344*a^5*b^9 + 24216*a^6*b^8 - 27808*a^7*b^7 + 1533
2*a^8*b^6 - 4320*a^9*b^5 + 576*a^10*b^4))/(a^3*b^8) + (((128*tan(c/2 + (d*x)/2)*(16*a^3*b^10 + 340*a^4*b^9 - 9
52*a^5*b^8 + 836*a^6*b^7 - 240*a^7*b^6))/(a^2*b^6) + (((16*(1024*a^6*b^12 - 1536*a^7*b^11 + 576*a^8*b^10))/(a^
3*b^8) + (16*tan(c/2 + (d*x)/2)^2*(2048*a^5*b^13 - 4096*a^6*b^12 + 2688*a^7*b^11 - 576*a^8*b^10))/(a^3*b^8))*(
a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3))*(a - b)^(3/2)*(4*a + b))/(4*a^(3/2)*b^3)))/(4*a^(3/2)*b^3)))/(4*a^(3/
2)*b^3)))*(a - b)^(3/2)*(4*a + b)*1i)/(2*a^(3/2)*b^3*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**7/(a+b*tan(d*x+c)**2)**2,x)

[Out]

Timed out

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