3.3.47 \(\int \sec ^6(e+f x) (d \tan (e+f x))^{5/2} \, dx\) [247]

Optimal. Leaf size=67 \[ \frac {2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f} \]

[Out]

2/7*(d*tan(f*x+e))^(7/2)/d/f+4/11*(d*tan(f*x+e))^(11/2)/d^3/f+2/15*(d*tan(f*x+e))^(15/2)/d^5/f

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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2687, 276} \begin {gather*} \frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^6*(d*Tan[e + f*x])^(5/2),x]

[Out]

(2*(d*Tan[e + f*x])^(7/2))/(7*d*f) + (4*(d*Tan[e + f*x])^(11/2))/(11*d^3*f) + (2*(d*Tan[e + f*x])^(15/2))/(15*
d^5*f)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

\begin {align*} \int \sec ^6(e+f x) (d \tan (e+f x))^{5/2} \, dx &=\frac {\text {Subst}\left (\int (d x)^{5/2} \left (1+x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left ((d x)^{5/2}+\frac {2 (d x)^{9/2}}{d^2}+\frac {(d x)^{13/2}}{d^4}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac {2 (d \tan (e+f x))^{15/2}}{15 d^5 f}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 52, normalized size = 0.78 \begin {gather*} \frac {2 (117+44 \cos (2 (e+f x))+4 \cos (4 (e+f x))) \sec ^4(e+f x) (d \tan (e+f x))^{7/2}}{1155 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^6*(d*Tan[e + f*x])^(5/2),x]

[Out]

(2*(117 + 44*Cos[2*(e + f*x)] + 4*Cos[4*(e + f*x)])*Sec[e + f*x]^4*(d*Tan[e + f*x])^(7/2))/(1155*d*f)

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Maple [A]
time = 3.00, size = 60, normalized size = 0.90

method	result	size
default	\(\frac {2 \left (32 \left (\cos ^{4}\left (f x +e \right )\right )+56 \left (\cos ^{2}\left (f x +e \right )\right )+77\right ) \left (\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )}{1155 f \cos \left (f x +e \right )^{5}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^6*(d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/1155/f*(32*cos(f*x+e)^4+56*cos(f*x+e)^2+77)*(d*sin(f*x+e)/cos(f*x+e))^(5/2)*sin(f*x+e)/cos(f*x+e)^5

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Maxima [A]
time = 0.28, size = 54, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (77 \, \left (d \tan \left (f x + e\right )\right )^{\frac {15}{2}} + 210 \, \left (d \tan \left (f x + e\right )\right )^{\frac {11}{2}} d^{2} + 165 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} d^{4}\right )}}{1155 \, d^{5} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

2/1155*(77*(d*tan(f*x + e))^(15/2) + 210*(d*tan(f*x + e))^(11/2)*d^2 + 165*(d*tan(f*x + e))^(7/2)*d^4)/(d^5*f)

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Fricas [A]
time = 0.44, size = 89, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (32 \, d^{2} \cos \left (f x + e\right )^{6} + 24 \, d^{2} \cos \left (f x + e\right )^{4} + 21 \, d^{2} \cos \left (f x + e\right )^{2} - 77 \, d^{2}\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{1155 \, f \cos \left (f x + e\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-2/1155*(32*d^2*cos(f*x + e)^6 + 24*d^2*cos(f*x + e)^4 + 21*d^2*cos(f*x + e)^2 - 77*d^2)*sqrt(d*sin(f*x + e)/c
os(f*x + e))*sin(f*x + e)/(f*cos(f*x + e)^7)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**6*(d*tan(f*x+e))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5987 deep

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Giac [A]
time = 0.57, size = 84, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (77 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{7} + 210 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{5} + 165 \, \sqrt {d \tan \left (f x + e\right )} d^{7} \tan \left (f x + e\right )^{3}\right )}}{1155 \, d^{5} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

2/1155*(77*sqrt(d*tan(f*x + e))*d^7*tan(f*x + e)^7 + 210*sqrt(d*tan(f*x + e))*d^7*tan(f*x + e)^5 + 165*sqrt(d*
tan(f*x + e))*d^7*tan(f*x + e)^3)/(d^5*f)

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Mupad [B]
time = 13.29, size = 474, normalized size = 7.07 \begin {gather*} \frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{1155\,f}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{1155\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{385\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,2432{}\mathrm {i}}{231\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,1504{}\mathrm {i}}{33\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,4288{}\mathrm {i}}{55\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,896{}\mathrm {i}}{15\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^6}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,256{}\mathrm {i}}{15\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(e + f*x))^(5/2)/cos(e + f*x)^6,x)

[Out]

(d^2*(-(d*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*64i)/(1155*f) + (d^2*(-(d*(exp(e*2i +
f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*64i)/(1155*f*(exp(e*2i + f*x*2i) + 1)) + (d^2*(-(d*(exp(e*2i
+ f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*32i)/(385*f*(exp(e*2i + f*x*2i) + 1)^2) - (d^2*(-(d*(exp(
e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*2432i)/(231*f*(exp(e*2i + f*x*2i) + 1)^3) + (d^2*(-(d
*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*1504i)/(33*f*(exp(e*2i + f*x*2i) + 1)^4) - (d^2
*(-(d*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*4288i)/(55*f*(exp(e*2i + f*x*2i) + 1)^5) +
(d^2*(-(d*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*896i)/(15*f*(exp(e*2i + f*x*2i) + 1)^
6) - (d^2*(-(d*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*256i)/(15*f*(exp(e*2i + f*x*2i) +
1)^7)

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