3.1.36 \(\int (b \tan ^4(c+d x))^{5/2} \, dx\) [36]

Optimal. Leaf size=182 \[ \frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d} \]

[Out]

b^2*cot(d*x+c)*(tan(d*x+c)^4*b)^(1/2)/d-b^2*x*cot(d*x+c)^2*(tan(d*x+c)^4*b)^(1/2)-1/3*b^2*(tan(d*x+c)^4*b)^(1/
2)*tan(d*x+c)/d+1/5*b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)^3/d-1/7*b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)^5/d+1/
9*b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)^7/d

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 8} \begin {gather*} -\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x]^4)^(5/2),x]

[Out]

(b^2*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - b^2*x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4] - (b^2*Tan[c + d*x]*
Sqrt[b*Tan[c + d*x]^4])/(3*d) + (b^2*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^4])/(5*d) - (b^2*Tan[c + d*x]^5*Sqrt[b
*Tan[c + d*x]^4])/(7*d) + (b^2*Tan[c + d*x]^7*Sqrt[b*Tan[c + d*x]^4])/(9*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx &=\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^{10}(c+d x) \, dx\\ &=\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^8(c+d x) \, dx\\ &=-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^6(c+d x) \, dx\\ &=\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^4(c+d x) \, dx\\ &=-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.78, size = 86, normalized size = 0.47 \begin {gather*} \frac {\cot (c+d x) \left (35-45 \cot ^2(c+d x)+63 \cot ^4(c+d x)-105 \cot ^6(c+d x)+315 \cot ^8(c+d x)-315 \text {ArcTan}(\tan (c+d x)) \cot ^9(c+d x)\right ) \left (b \tan ^4(c+d x)\right )^{5/2}}{315 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x]^4)^(5/2),x]

[Out]

(Cot[c + d*x]*(35 - 45*Cot[c + d*x]^2 + 63*Cot[c + d*x]^4 - 105*Cot[c + d*x]^6 + 315*Cot[c + d*x]^8 - 315*ArcT
an[Tan[c + d*x]]*Cot[c + d*x]^9)*(b*Tan[c + d*x]^4)^(5/2))/(315*d)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 84, normalized size = 0.46

method	result	size
derivativedivides	\(-\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\)	\(84\)
default	\(-\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\)	\(84\)
risch	\(\frac {b^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, x}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 i b^{2} \sqrt {\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, \left (1575 \,{\mathrm e}^{16 i \left (d x +c \right )}+6300 \,{\mathrm e}^{14 i \left (d x +c \right )}+21000 \,{\mathrm e}^{12 i \left (d x +c \right )}+31500 \,{\mathrm e}^{10 i \left (d x +c \right )}+39438 \,{\mathrm e}^{8 i \left (d x +c \right )}+26292 \,{\mathrm e}^{6 i \left (d x +c \right )}+13968 \,{\mathrm e}^{4 i \left (d x +c \right )}+3492 \,{\mathrm e}^{2 i \left (d x +c \right )}+563\right )}{315 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7} d}\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c)^4)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/315/d*(b*tan(d*x+c)^4)^(5/2)*(-35*tan(d*x+c)^9+45*tan(d*x+c)^7-63*tan(d*x+c)^5+105*tan(d*x+c)^3+315*arctan(
tan(d*x+c))-315*tan(d*x+c))/tan(d*x+c)^10

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 79, normalized size = 0.43 \begin {gather*} \frac {35 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{9} - 45 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{7} + 63 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{5} - 105 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{3} - 315 \, {\left (d x + c\right )} b^{\frac {5}{2}} + 315 \, b^{\frac {5}{2}} \tan \left (d x + c\right )}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(5/2),x, algorithm="maxima")

[Out]

1/315*(35*b^(5/2)*tan(d*x + c)^9 - 45*b^(5/2)*tan(d*x + c)^7 + 63*b^(5/2)*tan(d*x + c)^5 - 105*b^(5/2)*tan(d*x
+ c)^3 - 315*(d*x + c)*b^(5/2) + 315*b^(5/2)*tan(d*x + c))/d

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 96, normalized size = 0.53 \begin {gather*} \frac {{\left (35 \, b^{2} \tan \left (d x + c\right )^{9} - 45 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, b^{2} \tan \left (d x + c\right )^{5} - 105 \, b^{2} \tan \left (d x + c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{4}}}{315 \, d \tan \left (d x + c\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(5/2),x, algorithm="fricas")

[Out]

1/315*(35*b^2*tan(d*x + c)^9 - 45*b^2*tan(d*x + c)^7 + 63*b^2*tan(d*x + c)^5 - 105*b^2*tan(d*x + c)^3 - 315*b^
2*d*x + 315*b^2*tan(d*x + c))*sqrt(b*tan(d*x + c)^4)/(d*tan(d*x + c)^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)**4*b)**(5/2),x)

[Out]

Integral((b*tan(c + d*x)**4)**(5/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (162) = 324\).
time = 5.60, size = 960, normalized size = 5.27 \begin {gather*} -\frac {{\left (315 \, b^{2} d x \tan \left (d x\right )^{9} \tan \left (c\right )^{9} - 2835 \, b^{2} d x \tan \left (d x\right )^{8} \tan \left (c\right )^{8} + 315 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{8} + 315 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{9} + 11340 \, b^{2} d x \tan \left (d x\right )^{7} \tan \left (c\right )^{7} - 105 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{6} - 2835 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{7} - 2835 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{8} - 105 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{9} - 26460 \, b^{2} d x \tan \left (d x\right )^{6} \tan \left (c\right )^{6} + 63 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{4} + 945 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{5} + 11340 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{6} + 11340 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{7} + 945 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{8} + 63 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{9} + 39690 \, b^{2} d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 45 \, b^{2} \tan \left (d x\right )^{9} \tan \left (c\right )^{2} - 567 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right )^{3} - 3780 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{4} - 26460 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{5} - 26460 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{6} - 3780 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{7} - 567 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{8} - 45 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{9} - 39690 \, b^{2} d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 35 \, b^{2} \tan \left (d x\right )^{9} + 405 \, b^{2} \tan \left (d x\right )^{8} \tan \left (c\right ) + 2268 \, b^{2} \tan \left (d x\right )^{7} \tan \left (c\right )^{2} + 8820 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right )^{3} + 39690 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{4} + 39690 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{5} + 8820 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{6} + 2268 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{7} + 405 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{8} + 35 \, b^{2} \tan \left (c\right )^{9} + 26460 \, b^{2} d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 45 \, b^{2} \tan \left (d x\right )^{7} - 567 \, b^{2} \tan \left (d x\right )^{6} \tan \left (c\right ) - 3780 \, b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{2} - 26460 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 26460 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{4} - 3780 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{5} - 567 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{6} - 45 \, b^{2} \tan \left (c\right )^{7} - 11340 \, b^{2} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 63 \, b^{2} \tan \left (d x\right )^{5} + 945 \, b^{2} \tan \left (d x\right )^{4} \tan \left (c\right ) + 11340 \, b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 11340 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 945 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{4} + 63 \, b^{2} \tan \left (c\right )^{5} + 2835 \, b^{2} d x \tan \left (d x\right ) \tan \left (c\right ) - 105 \, b^{2} \tan \left (d x\right )^{3} - 2835 \, b^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) - 2835 \, b^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} - 105 \, b^{2} \tan \left (c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x\right ) + 315 \, b^{2} \tan \left (c\right )\right )} \sqrt {b}}{315 \, {\left (d \tan \left (d x\right )^{9} \tan \left (c\right )^{9} - 9 \, d \tan \left (d x\right )^{8} \tan \left (c\right )^{8} + 36 \, d \tan \left (d x\right )^{7} \tan \left (c\right )^{7} - 84 \, d \tan \left (d x\right )^{6} \tan \left (c\right )^{6} + 126 \, d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 126 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 84 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 36 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 9 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(5/2),x, algorithm="giac")

[Out]

-1/315*(315*b^2*d*x*tan(d*x)^9*tan(c)^9 - 2835*b^2*d*x*tan(d*x)^8*tan(c)^8 + 315*b^2*tan(d*x)^9*tan(c)^8 + 315
*b^2*tan(d*x)^8*tan(c)^9 + 11340*b^2*d*x*tan(d*x)^7*tan(c)^7 - 105*b^2*tan(d*x)^9*tan(c)^6 - 2835*b^2*tan(d*x)
^8*tan(c)^7 - 2835*b^2*tan(d*x)^7*tan(c)^8 - 105*b^2*tan(d*x)^6*tan(c)^9 - 26460*b^2*d*x*tan(d*x)^6*tan(c)^6 +
63*b^2*tan(d*x)^9*tan(c)^4 + 945*b^2*tan(d*x)^8*tan(c)^5 + 11340*b^2*tan(d*x)^7*tan(c)^6 + 11340*b^2*tan(d*x)
^6*tan(c)^7 + 945*b^2*tan(d*x)^5*tan(c)^8 + 63*b^2*tan(d*x)^4*tan(c)^9 + 39690*b^2*d*x*tan(d*x)^5*tan(c)^5 - 4
5*b^2*tan(d*x)^9*tan(c)^2 - 567*b^2*tan(d*x)^8*tan(c)^3 - 3780*b^2*tan(d*x)^7*tan(c)^4 - 26460*b^2*tan(d*x)^6*
tan(c)^5 - 26460*b^2*tan(d*x)^5*tan(c)^6 - 3780*b^2*tan(d*x)^4*tan(c)^7 - 567*b^2*tan(d*x)^3*tan(c)^8 - 45*b^2
*tan(d*x)^2*tan(c)^9 - 39690*b^2*d*x*tan(d*x)^4*tan(c)^4 + 35*b^2*tan(d*x)^9 + 405*b^2*tan(d*x)^8*tan(c) + 226
8*b^2*tan(d*x)^7*tan(c)^2 + 8820*b^2*tan(d*x)^6*tan(c)^3 + 39690*b^2*tan(d*x)^5*tan(c)^4 + 39690*b^2*tan(d*x)^
4*tan(c)^5 + 8820*b^2*tan(d*x)^3*tan(c)^6 + 2268*b^2*tan(d*x)^2*tan(c)^7 + 405*b^2*tan(d*x)*tan(c)^8 + 35*b^2*
tan(c)^9 + 26460*b^2*d*x*tan(d*x)^3*tan(c)^3 - 45*b^2*tan(d*x)^7 - 567*b^2*tan(d*x)^6*tan(c) - 3780*b^2*tan(d*
x)^5*tan(c)^2 - 26460*b^2*tan(d*x)^4*tan(c)^3 - 26460*b^2*tan(d*x)^3*tan(c)^4 - 3780*b^2*tan(d*x)^2*tan(c)^5 -
567*b^2*tan(d*x)*tan(c)^6 - 45*b^2*tan(c)^7 - 11340*b^2*d*x*tan(d*x)^2*tan(c)^2 + 63*b^2*tan(d*x)^5 + 945*b^2
*tan(d*x)^4*tan(c) + 11340*b^2*tan(d*x)^3*tan(c)^2 + 11340*b^2*tan(d*x)^2*tan(c)^3 + 945*b^2*tan(d*x)*tan(c)^4
+ 63*b^2*tan(c)^5 + 2835*b^2*d*x*tan(d*x)*tan(c) - 105*b^2*tan(d*x)^3 - 2835*b^2*tan(d*x)^2*tan(c) - 2835*b^2
*tan(d*x)*tan(c)^2 - 105*b^2*tan(c)^3 - 315*b^2*d*x + 315*b^2*tan(d*x) + 315*b^2*tan(c))*sqrt(b)/(d*tan(d*x)^9
*tan(c)^9 - 9*d*tan(d*x)^8*tan(c)^8 + 36*d*tan(d*x)^7*tan(c)^7 - 84*d*tan(d*x)^6*tan(c)^6 + 126*d*tan(d*x)^5*t
an(c)^5 - 126*d*tan(d*x)^4*tan(c)^4 + 84*d*tan(d*x)^3*tan(c)^3 - 36*d*tan(d*x)^2*tan(c)^2 + 9*d*tan(d*x)*tan(c
) - d)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(c + d*x)^4)^(5/2),x)

[Out]

int((b*tan(c + d*x)^4)^(5/2), x)

________________________________________________________________________________________