3.753 \(\int (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^4 \, dx\)

Optimal. Leaf size=188 \[ \frac {35}{8} x \left (b^2-c^2\right )^2+\frac {35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac {35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {7}{12} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {35}{24} \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]

[Out]

35/8*(b^2-c^2)^2*x+35/8*c*(b^2-c^2)^(3/2)*cosh(x)+35/8*b*(b^2-c^2)^(3/2)*sinh(x)+35/24*(b^2-c^2)*(c*cosh(x)+b*
sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))+7/12*(c*cosh(x)+b*sinh(x))*(b^2-c^2)^(1/2)*(b*cosh(x)+c*sinh(x)
+(b^2-c^2)^(1/2))^2+1/4*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3

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Rubi [A]  time = 0.15, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3113, 2637, 2638} \[ \frac {35}{8} x \left (b^2-c^2\right )^2+\frac {35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac {35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {7}{12} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {35}{24} \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^4,x]

[Out]

(35*(b^2 - c^2)^2*x)/8 + (35*c*(b^2 - c^2)^(3/2)*Cosh[x])/8 + (35*b*(b^2 - c^2)^(3/2)*Sinh[x])/8 + (35*(b^2 -
c^2)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]))/24 + (7*Sqrt[b^2 - c^2]*(c*Cosh[x] + b
*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2)/12 + ((c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh
[x] + c*Sinh[x])^3)/4

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3113

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> -Simp[((c*Cos[d
+ e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1))/(e*n), x] + Dist[(a*(2*n - 1))/n, Int[
(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
&& GtQ[n, 0]

Rubi steps

\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx &=\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {1}{4} \left (7 \sqrt {b^2-c^2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3 \, dx\\ &=\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {1}{12} \left (35 \left (b^2-c^2\right )\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx\\ &=\frac {35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {1}{8} \left (35 \left (b^2-c^2\right )^{3/2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx\\ &=\frac {35}{8} \left (b^2-c^2\right )^2 x+\frac {35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {1}{8} \left (35 b \left (b^2-c^2\right )^{3/2}\right ) \int \cosh (x) \, dx+\frac {1}{8} \left (35 c \left (b^2-c^2\right )^{3/2}\right ) \int \sinh (x) \, dx\\ &=\frac {35}{8} \left (b^2-c^2\right )^2 x+\frac {35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac {35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac {35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3\\ \end {align*}

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Mathematica [A]  time = 0.51, size = 208, normalized size = 1.11 \[ \frac {7}{4} \left (b^4-c^4\right ) \sinh (2 x)+7 b (b-c) \sqrt {b^2-c^2} (b+c) \sinh (x)+\frac {1}{3} b \sqrt {b^2-c^2} \left (b^2+3 c^2\right ) \sinh (3 x)+7 c (b-c) \sqrt {b^2-c^2} (b+c) \cosh (x)+\frac {7}{2} b c \left (b^2-c^2\right ) \cosh (2 x)+\frac {1}{3} c \sqrt {b^2-c^2} \left (3 b^2+c^2\right ) \cosh (3 x)+\frac {1}{8} b c \left (b^2+c^2\right ) \cosh (4 x)+\frac {1}{32} \left (b^4+6 b^2 c^2+c^4\right ) \sinh (4 x)+\frac {35}{8} x (b-c)^2 (b+c)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^4,x]

[Out]

(35*(b - c)^2*(b + c)^2*x)/8 + 7*(b - c)*c*(b + c)*Sqrt[b^2 - c^2]*Cosh[x] + (7*b*c*(b^2 - c^2)*Cosh[2*x])/2 +
(c*Sqrt[b^2 - c^2]*(3*b^2 + c^2)*Cosh[3*x])/3 + (b*c*(b^2 + c^2)*Cosh[4*x])/8 + 7*b*(b - c)*(b + c)*Sqrt[b^2
- c^2]*Sinh[x] + (7*(b^4 - c^4)*Sinh[2*x])/4 + (b*Sqrt[b^2 - c^2]*(b^2 + 3*c^2)*Sinh[3*x])/3 + ((b^4 + 6*b^2*c
^2 + c^4)*Sinh[4*x])/32

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fricas [B]  time = 0.45, size = 1293, normalized size = 6.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^4,x, algorithm="fricas")

[Out]

1/192*(3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^8 + 24*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4
)*cosh(x)*sinh(x)^7 + 3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^8 + 168*(b^4 + 2*b^3*c - 2*b*c^3 -
c^4)*cosh(x)^6 + 84*(2*b^4 + 4*b^3*c - 4*b*c^3 - 2*c^4 + (b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^
2)*sinh(x)^6 + 840*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x)^4 + 168*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh
(x)^3 + 6*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x))*sinh(x)^5 + 210*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4
)*cosh(x)^4 + 12*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2 + 4*(b^4 - 2*b^2*c^2 + c^4)*x)*sinh(x)^4 - 3*b^4 +
12*b^3*c - 18*b^2*c^2 + 12*b*c^3 - 3*c^4 + 168*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^5 + 20*(b^
4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + 20*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x))*sinh(x)^3 - 168*(b^4 - 2*b^3*c
+ 2*b*c^3 - c^4)*cosh(x)^2 + 84*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^6 + 30*(b^4 + 2*b^3*c - 2
*b*c^3 - c^4)*cosh(x)^4 - 2*b^4 + 4*b^3*c - 4*b*c^3 + 2*c^4 + 60*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x)^2)*sinh(x)^
2 + 24*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^7 + 42*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^5 +
140*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x)^3 - 14*(b^4 - 2*b^3*c + 2*b*c^3 - c^4)*cosh(x))*sinh(x) + 32*((b^3 + 3*
b^2*c + 3*b*c^2 + c^3)*cosh(x)^7 + 7*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(x)^6 + (b^3 + 3*b^2*c + 3*b*
c^2 + c^3)*sinh(x)^7 + 21*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^5 + 21*(b^3 + b^2*c - b*c^2 - c^3 + (b^3 + 3*b^2
*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^5 + 35*((b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 + 3*(b^3 + b^2*c - b*
c^2 - c^3)*cosh(x))*sinh(x)^4 - 21*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^3 + 7*(5*(b^3 + 3*b^2*c + 3*b*c^2 + c^3
)*cosh(x)^4 - 3*b^3 + 3*b^2*c + 3*b*c^2 - 3*c^3 + 30*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2)*sinh(x)^3 + 21*((b
^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^5 + 10*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 - 3*(b^3 - b^2*c - b*c^2 +
c^3)*cosh(x))*sinh(x)^2 - (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*cosh(x) + (7*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)
^6 + 105*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4 - b^3 + 3*b^2*c - 3*b*c^2 + c^3 - 63*(b^3 - b^2*c - b*c^2 + c^3
)*cosh(x)^2)*sinh(x))*sqrt(b^2 - c^2))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*si
nh(x)^3 + sinh(x)^4)

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giac [B]  time = 0.16, size = 390, normalized size = 2.07 \[ \frac {7}{2} \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \sqrt {b^{2} - c^{2}} e^{x} + \frac {35}{8} \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x + \frac {1}{64} \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} e^{\left (4 \, x\right )} + \frac {1}{6} \, {\left (\sqrt {b^{2} - c^{2}} b^{3} + 3 \, \sqrt {b^{2} - c^{2}} b^{2} c + 3 \, \sqrt {b^{2} - c^{2}} b c^{2} + \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + \frac {7}{8} \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} - \frac {1}{192} \, {\left (3 \, b^{4} - 12 \, b^{3} c + 18 \, b^{2} c^{2} - 12 \, b c^{3} + 3 \, c^{4} + 672 \, {\left (\sqrt {b^{2} - c^{2}} b^{3} - \sqrt {b^{2} - c^{2}} b^{2} c - \sqrt {b^{2} - c^{2}} b c^{2} + \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + 168 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} + 32 \, {\left (\sqrt {b^{2} - c^{2}} b^{3} - 3 \, \sqrt {b^{2} - c^{2}} b^{2} c + 3 \, \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{x}\right )} e^{\left (-4 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^4,x, algorithm="giac")

[Out]

7/2*(b^3 + b^2*c - b*c^2 - c^3)*sqrt(b^2 - c^2)*e^x + 35/8*(b^4 - 2*b^2*c^2 + c^4)*x + 1/64*(b^4 + 4*b^3*c + 6
*b^2*c^2 + 4*b*c^3 + c^4)*e^(4*x) + 1/6*(sqrt(b^2 - c^2)*b^3 + 3*sqrt(b^2 - c^2)*b^2*c + 3*sqrt(b^2 - c^2)*b*c
^2 + sqrt(b^2 - c^2)*c^3)*e^(3*x) + 7/8*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*e^(2*x) - 1/192*(3*b^4 - 12*b^3*c + 18
*b^2*c^2 - 12*b*c^3 + 3*c^4 + 672*(sqrt(b^2 - c^2)*b^3 - sqrt(b^2 - c^2)*b^2*c - sqrt(b^2 - c^2)*b*c^2 + sqrt(
b^2 - c^2)*c^3)*e^(3*x) + 168*(b^4 - 2*b^3*c + 2*b*c^3 - c^4)*e^(2*x) + 32*(sqrt(b^2 - c^2)*b^3 - 3*sqrt(b^2 -
c^2)*b^2*c + 3*sqrt(b^2 - c^2)*b*c^2 - sqrt(b^2 - c^2)*c^3)*e^x)*e^(-4*x)

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maple [B]  time = 0.67, size = 363, normalized size = 1.93 \[ -2 c^{2} b^{2} x +4 \sqrt {b^{2}-c^{2}}\, b^{3} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\relax (x )\right )}{3}\right ) \sinh \relax (x )-6 c^{2} b^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+4 \sqrt {b^{2}-c^{2}}\, b^{3} \sinh \relax (x )+4 \sqrt {b^{2}-c^{2}}\, c^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\relax (x )\right )}{3}\right ) \cosh \relax (x )+6 c^{2} b^{2} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right )-4 \sqrt {b^{2}-c^{2}}\, c^{3} \cosh \relax (x )+b^{4} x +c^{4} x +4 \sqrt {b^{2}-c^{2}}\, b^{2} c \left (\cosh ^{3}\relax (x )\right )+4 \sqrt {b^{2}-c^{2}}\, b \,c^{2} \left (\sinh ^{3}\relax (x )\right )+b^{4} \left (\left (\frac {\left (\cosh ^{3}\relax (x )\right )}{4}+\frac {3 \cosh \relax (x )}{8}\right ) \sinh \relax (x )+\frac {3 x}{8}\right )+6 b^{4} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}+\frac {x}{2}\right )+c^{4} \left (\left (\frac {\left (\sinh ^{3}\relax (x )\right )}{4}-\frac {3 \sinh \relax (x )}{8}\right ) \cosh \relax (x )+\frac {3 x}{8}\right )-6 c^{4} \left (\frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}\right )+b^{3} \left (\cosh ^{4}\relax (x )\right ) c +6 c^{2} b^{2} \left (\frac {\sinh \relax (x ) \left (\cosh ^{3}\relax (x )\right )}{4}-\frac {\cosh \relax (x ) \sinh \relax (x )}{8}-\frac {x}{8}\right )+b \,c^{3} \left (\sinh ^{4}\relax (x )\right )+6 b^{3} \left (\cosh ^{2}\relax (x )\right ) c -6 b \left (\cosh ^{2}\relax (x )\right ) c^{3}-4 \sqrt {b^{2}-c^{2}}\, b \,c^{2} \sinh \relax (x )+4 \sqrt {b^{2}-c^{2}}\, b^{2} c \cosh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^4,x)

[Out]

-2*c^2*b^2*x+4*(b^2-c^2)^(1/2)*b^3*(2/3+1/3*cosh(x)^2)*sinh(x)-6*c^2*b^2*(1/2*cosh(x)*sinh(x)+1/2*x)+4*(b^2-c^
2)^(1/2)*b^3*sinh(x)+4*(b^2-c^2)^(1/2)*c^3*(-2/3+1/3*sinh(x)^2)*cosh(x)+6*c^2*b^2*(1/2*cosh(x)*sinh(x)-1/2*x)-
4*(b^2-c^2)^(1/2)*c^3*cosh(x)+b^4*x+c^4*x+4*(b^2-c^2)^(1/2)*b^2*c*cosh(x)^3+4*(b^2-c^2)^(1/2)*b*c^2*sinh(x)^3+
b^4*((1/4*cosh(x)^3+3/8*cosh(x))*sinh(x)+3/8*x)+6*b^4*(1/2*cosh(x)*sinh(x)+1/2*x)+c^4*((1/4*sinh(x)^3-3/8*sinh
(x))*cosh(x)+3/8*x)-6*c^4*(1/2*cosh(x)*sinh(x)-1/2*x)+b^3*cosh(x)^4*c+6*c^2*b^2*(1/4*sinh(x)*cosh(x)^3-1/8*cos
h(x)*sinh(x)-1/8*x)+b*c^3*sinh(x)^4+6*b^3*cosh(x)^2*c-6*b*cosh(x)^2*c^3-4*(b^2-c^2)^(1/2)*b*c^2*sinh(x)+4*(b^2
-c^2)^(1/2)*b^2*c*cosh(x)

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maxima [A]  time = 0.34, size = 277, normalized size = 1.47 \[ b^{3} c \cosh \relax (x)^{4} + b c^{3} \sinh \relax (x)^{4} + \frac {1}{64} \, b^{4} {\left (24 \, x + e^{\left (4 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} + \frac {1}{64} \, c^{4} {\left (24 \, x + e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} - \frac {3}{32} \, b^{2} c^{2} {\left (8 \, x - e^{\left (4 \, x\right )} + e^{\left (-4 \, x\right )}\right )} + {\left (b^{2} - c^{2}\right )}^{2} x + 4 \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} {\left (c \cosh \relax (x) + b \sinh \relax (x)\right )} + \frac {3}{4} \, {\left (8 \, b c \cosh \relax (x)^{2} + b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} {\left (b^{2} - c^{2}\right )} + \frac {1}{6} \, {\left (24 \, b^{2} c \cosh \relax (x)^{3} + 24 \, b c^{2} \sinh \relax (x)^{3} + c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + b^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )}\right )} \sqrt {b^{2} - c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^4,x, algorithm="maxima")

[Out]

b^3*c*cosh(x)^4 + b*c^3*sinh(x)^4 + 1/64*b^4*(24*x + e^(4*x) + 8*e^(2*x) - 8*e^(-2*x) - e^(-4*x)) + 1/64*c^4*(
24*x + e^(4*x) - 8*e^(2*x) + 8*e^(-2*x) - e^(-4*x)) - 3/32*b^2*c^2*(8*x - e^(4*x) + e^(-4*x)) + (b^2 - c^2)^2*
x + 4*(b^2 - c^2)^(3/2)*(c*cosh(x) + b*sinh(x)) + 3/4*(8*b*c*cosh(x)^2 + b^2*(4*x + e^(2*x) - e^(-2*x)) - c^2*
(4*x - e^(2*x) + e^(-2*x)))*(b^2 - c^2) + 1/6*(24*b^2*c*cosh(x)^3 + 24*b*c^2*sinh(x)^3 + c^3*(e^(3*x) - 9*e^(-
x) + e^(-3*x) - 9*e^x) + b^3*(e^(3*x) - 9*e^(-x) - e^(-3*x) + 9*e^x))*sqrt(b^2 - c^2)

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mupad [B]  time = 0.45, size = 361, normalized size = 1.92 \[ x\,{\left (b^2-c^2\right )}^2-{\mathrm {cosh}\relax (x)}^2\,\left (6\,b\,c^3-6\,b^3\,c\right )-{\mathrm {cosh}\relax (x)}^4\,\left (b\,c^3-b^3\,c\right )+\mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^3\,\left (-\frac {3\,b^4}{8}+\frac {3\,b^2\,c^2}{4}+\frac {5\,c^4}{8}\right )+{\mathrm {cosh}\relax (x)}^3\,\mathrm {sinh}\relax (x)\,\left (\frac {5\,b^4}{8}+\frac {3\,b^2\,c^2}{4}-\frac {3\,c^4}{8}\right )+4\,c\,\mathrm {cosh}\relax (x)\,{\left (b^2-c^2\right )}^{3/2}+4\,b\,\mathrm {sinh}\relax (x)\,{\left (b^2-c^2\right )}^{3/2}+3\,x\,{\mathrm {cosh}\relax (x)}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {cosh}\relax (x)}^4\,{\left (b^2-c^2\right )}^2}{8}-3\,x\,{\mathrm {sinh}\relax (x)}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {sinh}\relax (x)}^4\,{\left (b^2-c^2\right )}^2}{8}+\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\left (3\,b^4-3\,c^4\right )+2\,b\,c^3\,{\mathrm {cosh}\relax (x)}^2\,{\mathrm {sinh}\relax (x)}^2+\frac {4\,c\,{\mathrm {cosh}\relax (x)}^3\,\sqrt {b^2-c^2}\,\left (3\,b^2-2\,c^2\right )}{3}-\frac {4\,b\,{\mathrm {sinh}\relax (x)}^3\,\sqrt {b^2-c^2}\,\left (2\,b^2-3\,c^2\right )}{3}+4\,b^3\,{\mathrm {cosh}\relax (x)}^2\,\mathrm {sinh}\relax (x)\,\sqrt {b^2-c^2}+4\,c^3\,\mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^2\,\sqrt {b^2-c^2}-\frac {3\,x\,{\mathrm {cosh}\relax (x)}^2\,{\mathrm {sinh}\relax (x)}^2\,{\left (b^2-c^2\right )}^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^4,x)

[Out]

x*(b^2 - c^2)^2 - cosh(x)^2*(6*b*c^3 - 6*b^3*c) - cosh(x)^4*(b*c^3 - b^3*c) + cosh(x)*sinh(x)^3*((5*c^4)/8 - (
3*b^4)/8 + (3*b^2*c^2)/4) + cosh(x)^3*sinh(x)*((5*b^4)/8 - (3*c^4)/8 + (3*b^2*c^2)/4) + 4*c*cosh(x)*(b^2 - c^2
)^(3/2) + 4*b*sinh(x)*(b^2 - c^2)^(3/2) + 3*x*cosh(x)^2*(b^2 - c^2)^2 + (3*x*cosh(x)^4*(b^2 - c^2)^2)/8 - 3*x*
sinh(x)^2*(b^2 - c^2)^2 + (3*x*sinh(x)^4*(b^2 - c^2)^2)/8 + cosh(x)*sinh(x)*(3*b^4 - 3*c^4) + 2*b*c^3*cosh(x)^
2*sinh(x)^2 + (4*c*cosh(x)^3*(b^2 - c^2)^(1/2)*(3*b^2 - 2*c^2))/3 - (4*b*sinh(x)^3*(b^2 - c^2)^(1/2)*(2*b^2 -
3*c^2))/3 + 4*b^3*cosh(x)^2*sinh(x)*(b^2 - c^2)^(1/2) + 4*c^3*cosh(x)*sinh(x)^2*(b^2 - c^2)^(1/2) - (3*x*cosh(
x)^2*sinh(x)^2*(b^2 - c^2)^2)/4

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sympy [B]  time = 1.47, size = 626, normalized size = 3.33 \[ \frac {3 b^{4} x \sinh ^{4}{\relax (x )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{4} - 3 b^{4} x \sinh ^{2}{\relax (x )} + \frac {3 b^{4} x \cosh ^{4}{\relax (x )}}{8} + 3 b^{4} x \cosh ^{2}{\relax (x )} + b^{4} x - \frac {3 b^{4} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{8} + \frac {5 b^{4} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{8} + 3 b^{4} \sinh {\relax (x )} \cosh {\relax (x )} + b^{3} c \cosh ^{4}{\relax (x )} + 6 b^{3} c \cosh ^{2}{\relax (x )} - \frac {8 b^{3} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\relax (x )}}{3} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\relax (x )} \cosh ^{2}{\relax (x )} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\relax (x )} - \frac {3 b^{2} c^{2} x \sinh ^{4}{\relax (x )}}{4} + \frac {3 b^{2} c^{2} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{2} + 6 b^{2} c^{2} x \sinh ^{2}{\relax (x )} - \frac {3 b^{2} c^{2} x \cosh ^{4}{\relax (x )}}{4} - 6 b^{2} c^{2} x \cosh ^{2}{\relax (x )} - 2 b^{2} c^{2} x + \frac {3 b^{2} c^{2} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{4} + \frac {3 b^{2} c^{2} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{4} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh ^{3}{\relax (x )} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh {\relax (x )} + b c^{3} \sinh ^{4}{\relax (x )} - 6 b c^{3} \cosh ^{2}{\relax (x )} + 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\relax (x )} - 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh {\relax (x )} + \frac {3 c^{4} x \sinh ^{4}{\relax (x )}}{8} - \frac {3 c^{4} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{4} - 3 c^{4} x \sinh ^{2}{\relax (x )} + \frac {3 c^{4} x \cosh ^{4}{\relax (x )}}{8} + 3 c^{4} x \cosh ^{2}{\relax (x )} + c^{4} x + \frac {5 c^{4} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{8} - \frac {3 c^{4} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{8} - 3 c^{4} \sinh {\relax (x )} \cosh {\relax (x )} + 4 c^{3} \sqrt {b^{2} - c^{2}} \sinh ^{2}{\relax (x )} \cosh {\relax (x )} - \frac {8 c^{3} \sqrt {b^{2} - c^{2}} \cosh ^{3}{\relax (x )}}{3} - 4 c^{3} \sqrt {b^{2} - c^{2}} \cosh {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cosh(x)+c*sinh(x)+(b**2-c**2)**(1/2))**4,x)

[Out]

3*b**4*x*sinh(x)**4/8 - 3*b**4*x*sinh(x)**2*cosh(x)**2/4 - 3*b**4*x*sinh(x)**2 + 3*b**4*x*cosh(x)**4/8 + 3*b**
4*x*cosh(x)**2 + b**4*x - 3*b**4*sinh(x)**3*cosh(x)/8 + 5*b**4*sinh(x)*cosh(x)**3/8 + 3*b**4*sinh(x)*cosh(x) +
b**3*c*cosh(x)**4 + 6*b**3*c*cosh(x)**2 - 8*b**3*sqrt(b**2 - c**2)*sinh(x)**3/3 + 4*b**3*sqrt(b**2 - c**2)*si
nh(x)*cosh(x)**2 + 4*b**3*sqrt(b**2 - c**2)*sinh(x) - 3*b**2*c**2*x*sinh(x)**4/4 + 3*b**2*c**2*x*sinh(x)**2*co
sh(x)**2/2 + 6*b**2*c**2*x*sinh(x)**2 - 3*b**2*c**2*x*cosh(x)**4/4 - 6*b**2*c**2*x*cosh(x)**2 - 2*b**2*c**2*x
+ 3*b**2*c**2*sinh(x)**3*cosh(x)/4 + 3*b**2*c**2*sinh(x)*cosh(x)**3/4 + 4*b**2*c*sqrt(b**2 - c**2)*cosh(x)**3
+ 4*b**2*c*sqrt(b**2 - c**2)*cosh(x) + b*c**3*sinh(x)**4 - 6*b*c**3*cosh(x)**2 + 4*b*c**2*sqrt(b**2 - c**2)*si
nh(x)**3 - 4*b*c**2*sqrt(b**2 - c**2)*sinh(x) + 3*c**4*x*sinh(x)**4/8 - 3*c**4*x*sinh(x)**2*cosh(x)**2/4 - 3*c
**4*x*sinh(x)**2 + 3*c**4*x*cosh(x)**4/8 + 3*c**4*x*cosh(x)**2 + c**4*x + 5*c**4*sinh(x)**3*cosh(x)/8 - 3*c**4
*sinh(x)*cosh(x)**3/8 - 3*c**4*sinh(x)*cosh(x) + 4*c**3*sqrt(b**2 - c**2)*sinh(x)**2*cosh(x) - 8*c**3*sqrt(b**
2 - c**2)*cosh(x)**3/3 - 4*c**3*sqrt(b**2 - c**2)*cosh(x)

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