3.5.41 \(\int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx\) [441]

Optimal. Leaf size=128 \[ \frac {a^3 \log (\cos (c+d x))}{(a+b)^4 d}-\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 (a+b)^3 d}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 (a+b)^2 d}+\frac {\sec ^6(c+d x)}{6 (a+b) d} \]

[Out]

a^3*ln(cos(d*x+c))/(a+b)^4/d-1/2*a^3*ln(a+b*sin(d*x+c)^2)/(a+b)^4/d+1/2*(3*a^2+3*a*b+b^2)*sec(d*x+c)^2/(a+b)^3
/d-1/4*(3*a+2*b)*sec(d*x+c)^4/(a+b)^2/d+1/6*sec(d*x+c)^6/(a+b)/d

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Rubi [A]
time = 0.09, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3273, 90} \begin {gather*} -\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)^4}+\frac {a^3 \log (\cos (c+d x))}{d (a+b)^4}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 d (a+b)^3}+\frac {\sec ^6(c+d x)}{6 d (a+b)}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 d (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Log[Cos[c + d*x]])/((a + b)^4*d) - (a^3*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^4*d) + ((3*a^2 + 3*a*b + b^
2)*Sec[c + d*x]^2)/(2*(a + b)^3*d) - ((3*a + 2*b)*Sec[c + d*x]^4)/(4*(a + b)^2*d) + Sec[c + d*x]^6/(6*(a + b)*
d)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 3273

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

Rubi steps

\begin {align*} \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{(1-x)^4 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a+b) (-1+x)^4}+\frac {3 a+2 b}{(a+b)^2 (-1+x)^3}+\frac {3 a^2+3 a b+b^2}{(a+b)^3 (-1+x)^2}+\frac {a^3}{(a+b)^4 (-1+x)}-\frac {a^3 b}{(a+b)^4 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {a^3 \log (\cos (c+d x))}{(a+b)^4 d}-\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 (a+b)^3 d}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 (a+b)^2 d}+\frac {\sec ^6(c+d x)}{6 (a+b) d}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 113, normalized size = 0.88 \begin {gather*} \frac {\frac {12 a^3 \log (\cos (c+d x))}{(a+b)^4}-\frac {6 a^3 \log \left (a+b \sin ^2(c+d x)\right )}{(a+b)^4}+\frac {6 \left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{(a+b)^3}-\frac {3 (3 a+2 b) \sec ^4(c+d x)}{(a+b)^2}+\frac {2 \sec ^6(c+d x)}{a+b}}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((12*a^3*Log[Cos[c + d*x]])/(a + b)^4 - (6*a^3*Log[a + b*Sin[c + d*x]^2])/(a + b)^4 + (6*(3*a^2 + 3*a*b + b^2)
*Sec[c + d*x]^2)/(a + b)^3 - (3*(3*a + 2*b)*Sec[c + d*x]^4)/(a + b)^2 + (2*Sec[c + d*x]^6)/(a + b))/(12*d)

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Maple [A]
time = 0.49, size = 114, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {-\frac {a^{3} \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 \left (a +b \right )^{4}}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{\left (a +b \right )^{4}}-\frac {3 a +2 b}{4 \left (a +b \right )^{2} \cos \left (d x +c \right )^{4}}-\frac {-3 a^{2}-3 a b -b^{2}}{2 \left (a +b \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {1}{6 \left (a +b \right ) \cos \left (d x +c \right )^{6}}}{d}\)	\(114\)
default	\(\frac {-\frac {a^{3} \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 \left (a +b \right )^{4}}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{\left (a +b \right )^{4}}-\frac {3 a +2 b}{4 \left (a +b \right )^{2} \cos \left (d x +c \right )^{4}}-\frac {-3 a^{2}-3 a b -b^{2}}{2 \left (a +b \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {1}{6 \left (a +b \right ) \cos \left (d x +c \right )^{6}}}{d}\)	\(114\)
risch	\(\frac {6 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+6 a b \,{\mathrm e}^{10 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+4 b \,{\mathrm e}^{8 i \left (d x +c \right )} a +\frac {68 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {52 a b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {20 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+12 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+4 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 b \,{\mathrm e}^{2 i \left (d x +c \right )} a +2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{d \left (a +b \right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {a^{3} \ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right )}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^7/(a+sin(d*x+c)^2*b),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*a^3/(a+b)^4*ln(a+b-b*cos(d*x+c)^2)+a^3/(a+b)^4*ln(cos(d*x+c))-1/4*(3*a+2*b)/(a+b)^2/cos(d*x+c)^4-1/2
*(-3*a^2-3*a*b-b^2)/(a+b)^3/cos(d*x+c)^2+1/6/(a+b)/cos(d*x+c)^6)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (120) = 240\).
time = 0.34, size = 273, normalized size = 2.13 \begin {gather*} -\frac {\frac {6 \, a^{3} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {6 \, a^{3} \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {6 \, {\left (3 \, a^{2} + 3 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{4} - 3 \, {\left (9 \, a^{2} + 7 \, a b + 2 \, b^{2}\right )} \sin \left (d x + c\right )^{2} + 11 \, a^{2} + 7 \, a b + 2 \, b^{2}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{2}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*a^3*log(b*sin(d*x + c)^2 + a)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 6*a^3*log(sin(d*x + c)^2
- 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + (6*(3*a^2 + 3*a*b + b^2)*sin(d*x + c)^4 - 3*(9*a^2 + 7*a*b
+ 2*b^2)*sin(d*x + c)^2 + 11*a^2 + 7*a*b + 2*b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sin(d*x + c)^6 - 3*(a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*sin(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sin
(d*x + c)^2))/d

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Fricas [A]
time = 0.62, size = 179, normalized size = 1.40 \begin {gather*} -\frac {6 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 12 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, {\left (3 \, a^{3} + 6 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - 2 \, a^{3} - 6 \, a^{2} b - 6 \, a b^{2} - 2 \, b^{3} + 3 \, {\left (3 \, a^{3} + 8 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*a^3*cos(d*x + c)^6*log(-b*cos(d*x + c)^2 + a + b) - 12*a^3*cos(d*x + c)^6*log(-cos(d*x + c)) - 6*(3*a
^3 + 6*a^2*b + 4*a*b^2 + b^3)*cos(d*x + c)^4 - 2*a^3 - 6*a^2*b - 6*a*b^2 - 2*b^3 + 3*(3*a^3 + 8*a^2*b + 7*a*b^
2 + 2*b^3)*cos(d*x + c)^2)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{7}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(c + d*x)**7/(a + b*sin(c + d*x)**2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (120) = 240\).
time = 3.29, size = 603, normalized size = 4.71 \begin {gather*} -\frac {\frac {30 \, a^{3} \log \left (a - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {60 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {147 \, a^{3} + \frac {1002 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {120 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2925 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {960 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {240 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4780 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3600 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2400 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {640 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2925 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {240 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1002 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {120 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {147 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{6}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*a^3*log(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) +
a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 60*a^3*log(abs(-(co
s(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + (147*a^3 + 1002*a^3*(co
s(d*x + c) - 1)/(cos(d*x + c) + 1) + 120*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2925*a^3*(cos(d*x + c)
- 1)^2/(cos(d*x + c) + 1)^2 + 960*a^2*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 240*a*b^2*(cos(d*x + c) -
1)^2/(cos(d*x + c) + 1)^2 + 4780*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 3600*a^2*b*(cos(d*x + c) - 1)
^3/(cos(d*x + c) + 1)^3 + 2400*a*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 640*b^3*(cos(d*x + c) - 1)^3/
(cos(d*x + c) + 1)^3 + 2925*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 960*a^2*b*(cos(d*x + c) - 1)^4/(co
s(d*x + c) + 1)^4 + 240*a*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 1002*a^3*(cos(d*x + c) - 1)^5/(cos(d
*x + c) + 1)^5 + 120*a^2*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 147*a^3*(cos(d*x + c) - 1)^6/(cos(d*x +
c) + 1)^6)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^6))/d

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Mupad [B]
time = 14.34, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{6\,d\,\left (a+b\right )}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d\,{\left (a+b\right )}^3}-\frac {a^3\,\ln \left (\left (a+b\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d\,{\left (a+b\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(c + d*x)^6/(6*d*(a + b)) + (a^2*tan(c + d*x)^2)/(2*d*(a + b)^3) - (a^3*log(a + tan(c + d*x)^2*(a + b)))/(d
*(8*a*b^3 + 8*a^3*b + 2*a^4 + 2*b^4 + 12*a^2*b^2)) - (a*tan(c + d*x)^4)/(4*d*(a + b)^2)

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