3.4.49 \(\int (d \tan (e+f x))^{7/2} (a+a \tan (e+f x))^3 \, dx\) [349]

Optimal. Leaf size=210 \[ -\frac {2 \sqrt {2} a^3 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}+\frac {4 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f} \]

[Out]

-2*a^3*d^(7/2)*arctanh(1/2*(d^(1/2)+d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x+e))^(1/2))*2^(1/2)/f+4*a^3*d^3*(d*t
an(f*x+e))^(1/2)/f-4/3*a^3*d^2*(d*tan(f*x+e))^(3/2)/f-4/5*a^3*d*(d*tan(f*x+e))^(5/2)/f+4/7*a^3*(d*tan(f*x+e))^
(7/2)/f+16/33*a^3*(d*tan(f*x+e))^(9/2)/d/f+2/11*(d*tan(f*x+e))^(9/2)*(a^3+a^3*tan(f*x+e))/d/f

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Rubi [A]
time = 0.23, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3711, 3609, 3613, 214} \begin {gather*} -\frac {2 \sqrt {2} a^3 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}+\frac {4 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{9/2}}{11 d f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Tan[e + f*x])^(7/2)*(a + a*Tan[e + f*x])^3,x]

[Out]

(-2*Sqrt[2]*a^3*d^(7/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f + (4*a^3*d
^3*Sqrt[d*Tan[e + f*x]])/f - (4*a^3*d^2*(d*Tan[e + f*x])^(3/2))/(3*f) - (4*a^3*d*(d*Tan[e + f*x])^(5/2))/(5*f)
+ (4*a^3*(d*Tan[e + f*x])^(7/2))/(7*f) + (16*a^3*(d*Tan[e + f*x])^(9/2))/(33*d*f) + (2*(d*Tan[e + f*x])^(9/2)
*(a^3 + a^3*Tan[e + f*x]))/(11*d*f)

Rule 214

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

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3613

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(d^2/f),
Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x
] && EqQ[c^2 - d^2, 0]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rubi steps

\begin {align*} \int (d \tan (e+f x))^{7/2} (a+a \tan (e+f x))^3 \, dx &=\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int (d \tan (e+f x))^{7/2} \left (a^3 d+11 a^3 d \tan (e+f x)+12 a^3 d \tan ^2(e+f x)\right ) \, dx}{11 d}\\ &=\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int (d \tan (e+f x))^{7/2} \left (-11 a^3 d+11 a^3 d \tan (e+f x)\right ) \, dx}{11 d}\\ &=\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int (d \tan (e+f x))^{5/2} \left (-11 a^3 d^2-11 a^3 d^2 \tan (e+f x)\right ) \, dx}{11 d}\\ &=-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int (d \tan (e+f x))^{3/2} \left (11 a^3 d^3-11 a^3 d^3 \tan (e+f x)\right ) \, dx}{11 d}\\ &=-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int \sqrt {d \tan (e+f x)} \left (11 a^3 d^4+11 a^3 d^4 \tan (e+f x)\right ) \, dx}{11 d}\\ &=\frac {4 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}+\frac {2 \int \frac {-11 a^3 d^5+11 a^3 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{11 d}\\ &=\frac {4 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}-\frac {\left (44 a^6 d^9\right ) \text {Subst}\left (\int \frac {1}{-242 a^6 d^{10}+d x^2} \, dx,x,\frac {-11 a^3 d^5-11 a^3 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=-\frac {2 \sqrt {2} a^3 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}+\frac {4 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac {2 (d \tan (e+f x))^{9/2} \left (a^3+a^3 \tan (e+f x)\right )}{11 d f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 4.54, size = 375, normalized size = 1.79 \begin {gather*} \frac {a^3 d^3 \cos (e+f x) \sqrt {d \tan (e+f x)} (1+\tan (e+f x))^3 \left (2310 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)-2310 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)+1155 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-1155 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )+9240 \cos ^2(e+f x) \sqrt {\tan (e+f x)}-3080 \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)+3080 \cos ^2(e+f x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)-1848 \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)+1320 \cos ^2(e+f x) \tan ^{\frac {7}{2}}(e+f x)+420 \sin ^2(e+f x) \tan ^{\frac {7}{2}}(e+f x)+770 \sin (2 (e+f x)) \tan ^{\frac {7}{2}}(e+f x)\right )}{2310 f (\cos (e+f x)+\sin (e+f x))^3 \sqrt {\tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Tan[e + f*x])^(7/2)*(a + a*Tan[e + f*x])^3,x]

[Out]

(a^3*d^3*Cos[e + f*x]*Sqrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x])^3*(2310*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e +
f*x]]]*Cos[e + f*x]^2 - 2310*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 + 1155*Sqrt[2]*Cos[
e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 1155*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sq
rt[Tan[e + f*x]] + Tan[e + f*x]] + 9240*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] - 3080*Cos[e + f*x]^2*Tan[e + f*x]^(
3/2) + 3080*Cos[e + f*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) - 1848*Cos[e + f
*x]^2*Tan[e + f*x]^(5/2) + 1320*Cos[e + f*x]^2*Tan[e + f*x]^(7/2) + 420*Sin[e + f*x]^2*Tan[e + f*x]^(7/2) + 77
0*Sin[2*(e + f*x)]*Tan[e + f*x]^(7/2)))/(2310*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[Tan[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(368\) vs. \(2(177)=354\).
time = 0.44, size = 369, normalized size = 1.76

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {d \left (d \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{3}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 d^{5} \sqrt {d \tan \left (f x +e \right )}-2 d^{6} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\)	\(369\)
default	\(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {d \left (d \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{3}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 d^{5} \sqrt {d \tan \left (f x +e \right )}-2 d^{6} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\)	\(369\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(f*x+e))^(7/2)*(a+a*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f*a^3/d^2*(1/11*(d*tan(f*x+e))^(11/2)+1/3*d*(d*tan(f*x+e))^(9/2)+2/7*d^2*(d*tan(f*x+e))^(7/2)-2/5*d^3*(d*tan
(f*x+e))^(5/2)-2/3*d^4*(d*tan(f*x+e))^(3/2)+2*d^5*(d*tan(f*x+e))^(1/2)-2*d^6*(1/8/d*(d^2)^(1/4)*2^(1/2)*(ln((d
*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/
2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d
*tan(f*x+e))^(1/2)+1))-1/8/(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2
)^(1/2))/(d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*
tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))))

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Maxima [A]
time = 0.52, size = 209, normalized size = 1.00 \begin {gather*} -\frac {1155 \, a^{3} d^{5} {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {2 \, {\left (105 \, \left (d \tan \left (f x + e\right )\right )^{\frac {11}{2}} a^{3} + 385 \, \left (d \tan \left (f x + e\right )\right )^{\frac {9}{2}} a^{3} d + 330 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{3} d^{2} - 462 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} d^{3} - 770 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d^{4} + 2310 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{5}\right )}}{d}}{1155 \, d f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(7/2)*(a+a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/1155*(1155*a^3*d^5*(sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d) - sqrt(2
)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d)) - 2*(105*(d*tan(f*x + e))^(11/2)*a^3
+ 385*(d*tan(f*x + e))^(9/2)*a^3*d + 330*(d*tan(f*x + e))^(7/2)*a^3*d^2 - 462*(d*tan(f*x + e))^(5/2)*a^3*d^3
- 770*(d*tan(f*x + e))^(3/2)*a^3*d^4 + 2310*sqrt(d*tan(f*x + e))*a^3*d^5)/d)/(d*f)

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Fricas [A]
time = 1.02, size = 358, normalized size = 1.70 \begin {gather*} \left [\frac {1155 \, \sqrt {2} a^{3} d^{\frac {7}{2}} \log \left (\frac {d \tan \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} {\left (\tan \left (f x + e\right ) + 1\right )} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (105 \, a^{3} d^{3} \tan \left (f x + e\right )^{5} + 385 \, a^{3} d^{3} \tan \left (f x + e\right )^{4} + 330 \, a^{3} d^{3} \tan \left (f x + e\right )^{3} - 462 \, a^{3} d^{3} \tan \left (f x + e\right )^{2} - 770 \, a^{3} d^{3} \tan \left (f x + e\right ) + 2310 \, a^{3} d^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{1155 \, f}, \frac {2 \, {\left (1155 \, \sqrt {2} a^{3} \sqrt {-d} d^{3} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, d \tan \left (f x + e\right )}\right ) + {\left (105 \, a^{3} d^{3} \tan \left (f x + e\right )^{5} + 385 \, a^{3} d^{3} \tan \left (f x + e\right )^{4} + 330 \, a^{3} d^{3} \tan \left (f x + e\right )^{3} - 462 \, a^{3} d^{3} \tan \left (f x + e\right )^{2} - 770 \, a^{3} d^{3} \tan \left (f x + e\right ) + 2310 \, a^{3} d^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{1155 \, f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(7/2)*(a+a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/1155*(1155*sqrt(2)*a^3*d^(7/2)*log((d*tan(f*x + e)^2 - 2*sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d)*(tan(f*x + e)
+ 1) + 4*d*tan(f*x + e) + d)/(tan(f*x + e)^2 + 1)) + 2*(105*a^3*d^3*tan(f*x + e)^5 + 385*a^3*d^3*tan(f*x + e)
^4 + 330*a^3*d^3*tan(f*x + e)^3 - 462*a^3*d^3*tan(f*x + e)^2 - 770*a^3*d^3*tan(f*x + e) + 2310*a^3*d^3)*sqrt(d
*tan(f*x + e)))/f, 2/1155*(1155*sqrt(2)*a^3*sqrt(-d)*d^3*arctan(1/2*sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(-d)*(tan
(f*x + e) + 1)/(d*tan(f*x + e))) + (105*a^3*d^3*tan(f*x + e)^5 + 385*a^3*d^3*tan(f*x + e)^4 + 330*a^3*d^3*tan(
f*x + e)^3 - 462*a^3*d^3*tan(f*x + e)^2 - 770*a^3*d^3*tan(f*x + e) + 2310*a^3*d^3)*sqrt(d*tan(f*x + e)))/f]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int \left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}} \tan {\left (e + f x \right )}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}} \tan ^{3}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))**(7/2)*(a+a*tan(f*x+e))**3,x)

[Out]

a**3*(Integral((d*tan(e + f*x))**(7/2), x) + Integral(3*(d*tan(e + f*x))**(7/2)*tan(e + f*x), x) + Integral(3*
(d*tan(e + f*x))**(7/2)*tan(e + f*x)**2, x) + Integral((d*tan(e + f*x))**(7/2)*tan(e + f*x)**3, x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (186) = 372\).
time = 1.02, size = 446, normalized size = 2.12 \begin {gather*} -\frac {\sqrt {2} {\left (a^{3} d^{3} \sqrt {{\left | d \right |}} + a^{3} d^{2} {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, f} + \frac {\sqrt {2} {\left (a^{3} d^{3} \sqrt {{\left | d \right |}} + a^{3} d^{2} {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, f} - \frac {{\left (\sqrt {2} a^{3} d^{3} \sqrt {{\left | d \right |}} - \sqrt {2} a^{3} d^{2} {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{f} - \frac {{\left (\sqrt {2} a^{3} d^{3} \sqrt {{\left | d \right |}} - \sqrt {2} a^{3} d^{2} {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{f} + \frac {2 \, {\left (105 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10} \tan \left (f x + e\right )^{5} + 385 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10} \tan \left (f x + e\right )^{4} + 330 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10} \tan \left (f x + e\right )^{3} - 462 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10} \tan \left (f x + e\right )^{2} - 770 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10} \tan \left (f x + e\right ) + 2310 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{25} f^{10}\right )}}{1155 \, d^{22} f^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(7/2)*(a+a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-1/2*sqrt(2)*(a^3*d^3*sqrt(abs(d)) + a^3*d^2*abs(d)^(3/2))*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*s
qrt(abs(d)) + abs(d))/f + 1/2*sqrt(2)*(a^3*d^3*sqrt(abs(d)) + a^3*d^2*abs(d)^(3/2))*log(d*tan(f*x + e) - sqrt(
2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/f - (sqrt(2)*a^3*d^3*sqrt(abs(d)) - sqrt(2)*a^3*d^2*abs(d)^(3/2
))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/f - (sqrt(2)*a^3*d^3*sqrt(
abs(d)) - sqrt(2)*a^3*d^2*abs(d)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sq
rt(abs(d)))/f + 2/1155*(105*sqrt(d*tan(f*x + e))*a^3*d^25*f^10*tan(f*x + e)^5 + 385*sqrt(d*tan(f*x + e))*a^3*d
^25*f^10*tan(f*x + e)^4 + 330*sqrt(d*tan(f*x + e))*a^3*d^25*f^10*tan(f*x + e)^3 - 462*sqrt(d*tan(f*x + e))*a^3
*d^25*f^10*tan(f*x + e)^2 - 770*sqrt(d*tan(f*x + e))*a^3*d^25*f^10*tan(f*x + e) + 2310*sqrt(d*tan(f*x + e))*a^
3*d^25*f^10)/(d^22*f^11)

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Mupad [B]
time = 7.04, size = 185, normalized size = 0.88 \begin {gather*} \frac {4\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{7\,f}+\frac {4\,a^3\,d^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}-\frac {4\,a^3\,d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,f}+\frac {2\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{9/2}}{3\,d\,f}+\frac {2\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{11/2}}{11\,d^2\,f}-\frac {4\,a^3\,d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,f}+\frac {\sqrt {2}\,a^3\,d^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^6\,d^{17/2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,32{}\mathrm {i}}{32\,a^6\,d^9+32\,a^6\,d^9\,\mathrm {tan}\left (e+f\,x\right )}\right )\,2{}\mathrm {i}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(e + f*x))^(7/2)*(a + a*tan(e + f*x))^3,x)

[Out]

(4*a^3*(d*tan(e + f*x))^(7/2))/(7*f) + (4*a^3*d^3*(d*tan(e + f*x))^(1/2))/f - (4*a^3*d^2*(d*tan(e + f*x))^(3/2
))/(3*f) + (2*a^3*(d*tan(e + f*x))^(9/2))/(3*d*f) + (2*a^3*(d*tan(e + f*x))^(11/2))/(11*d^2*f) - (4*a^3*d*(d*t
an(e + f*x))^(5/2))/(5*f) + (2^(1/2)*a^3*d^(7/2)*atan((2^(1/2)*a^6*d^(17/2)*(d*tan(e + f*x))^(1/2)*32i)/(32*a^
6*d^9 + 32*a^6*d^9*tan(e + f*x)))*2i)/f

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