Integrand size = 17, antiderivative size = 59 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\frac {\sinh ^{1+n}(a+b x)}{b (1+n)}+\frac {2 \sinh ^{3+n}(a+b x)}{b (3+n)}+\frac {\sinh ^{5+n}(a+b x)}{b (5+n)} \]
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\frac {\sinh ^{1+n}(a+b x) \left (\frac {1}{1+n}+\frac {2 \sinh ^2(a+b x)}{3+n}+\frac {\sinh ^4(a+b x)}{5+n}\right )}{b} \]
(Sinh[a + b*x]^(1 + n)*((1 + n)^(-1) + (2*Sinh[a + b*x]^2)/(3 + n) + Sinh[ a + b*x]^4/(5 + n)))/b
Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3042, 3044, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (i a+i b x)^5 (-i \sin (i a+i b x))^ndx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {\int \sinh ^n(a+b x) \left (\sinh ^2(a+b x)+1\right )^2d\sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (\sinh ^n(a+b x)+2 \sinh ^{n+2}(a+b x)+\sinh ^{n+4}(a+b x)\right )d\sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\sinh ^{n+1}(a+b x)}{n+1}+\frac {2 \sinh ^{n+3}(a+b x)}{n+3}+\frac {\sinh ^{n+5}(a+b x)}{n+5}}{b}\) |
(Sinh[a + b*x]^(1 + n)/(1 + n) + (2*Sinh[a + b*x]^(3 + n))/(3 + n) + Sinh[ a + b*x]^(5 + n)/(5 + n))/b
3.1.11.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39
\[\frac {\sinh \left (b x +a \right )^{5} {\mathrm e}^{n \ln \left (\sinh \left (b x +a \right )\right )}}{b \left (5+n \right )}+\frac {\sinh \left (b x +a \right ) {\mathrm e}^{n \ln \left (\sinh \left (b x +a \right )\right )}}{b \left (n +1\right )}+\frac {2 \sinh \left (b x +a \right )^{3} {\mathrm e}^{n \ln \left (\sinh \left (b x +a \right )\right )}}{b \left (3+n \right )}\]
1/b/(5+n)*sinh(b*x+a)^5*exp(n*ln(sinh(b*x+a)))+1/b/(n+1)*sinh(b*x+a)*exp(n *ln(sinh(b*x+a)))+2/b/(3+n)*sinh(b*x+a)^3*exp(n*ln(sinh(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (59) = 118\).
Time = 0.27 (sec) , antiderivative size = 379, normalized size of antiderivative = 6.42 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\frac {{\left ({\left (n^{2} + 4 \, n + 3\right )} \sinh \left (b x + a\right )^{5} + {\left (10 \, {\left (n^{2} + 4 \, n + 3\right )} \cosh \left (b x + a\right )^{2} + 3 \, n^{2} + 28 \, n + 25\right )} \sinh \left (b x + a\right )^{3} + {\left (5 \, {\left (n^{2} + 4 \, n + 3\right )} \cosh \left (b x + a\right )^{4} + 3 \, {\left (3 \, n^{2} + 28 \, n + 25\right )} \cosh \left (b x + a\right )^{2} + 2 \, n^{2} + 24 \, n + 150\right )} \sinh \left (b x + a\right )\right )} \cosh \left (n \log \left (\sinh \left (b x + a\right )\right )\right ) + {\left ({\left (n^{2} + 4 \, n + 3\right )} \sinh \left (b x + a\right )^{5} + {\left (10 \, {\left (n^{2} + 4 \, n + 3\right )} \cosh \left (b x + a\right )^{2} + 3 \, n^{2} + 28 \, n + 25\right )} \sinh \left (b x + a\right )^{3} + {\left (5 \, {\left (n^{2} + 4 \, n + 3\right )} \cosh \left (b x + a\right )^{4} + 3 \, {\left (3 \, n^{2} + 28 \, n + 25\right )} \cosh \left (b x + a\right )^{2} + 2 \, n^{2} + 24 \, n + 150\right )} \sinh \left (b x + a\right )\right )} \sinh \left (n \log \left (\sinh \left (b x + a\right )\right )\right )}{16 \, {\left ({\left (b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b\right )} \cosh \left (b x + a\right )^{6} - 3 \, {\left (b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b\right )} \cosh \left (b x + a\right )^{4} \sinh \left (b x + a\right )^{2} + 3 \, {\left (b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b\right )} \sinh \left (b x + a\right )^{6}\right )}} \]
1/16*(((n^2 + 4*n + 3)*sinh(b*x + a)^5 + (10*(n^2 + 4*n + 3)*cosh(b*x + a) ^2 + 3*n^2 + 28*n + 25)*sinh(b*x + a)^3 + (5*(n^2 + 4*n + 3)*cosh(b*x + a) ^4 + 3*(3*n^2 + 28*n + 25)*cosh(b*x + a)^2 + 2*n^2 + 24*n + 150)*sinh(b*x + a))*cosh(n*log(sinh(b*x + a))) + ((n^2 + 4*n + 3)*sinh(b*x + a)^5 + (10* (n^2 + 4*n + 3)*cosh(b*x + a)^2 + 3*n^2 + 28*n + 25)*sinh(b*x + a)^3 + (5* (n^2 + 4*n + 3)*cosh(b*x + a)^4 + 3*(3*n^2 + 28*n + 25)*cosh(b*x + a)^2 + 2*n^2 + 24*n + 150)*sinh(b*x + a))*sinh(n*log(sinh(b*x + a))))/((b*n^3 + 9 *b*n^2 + 23*b*n + 15*b)*cosh(b*x + a)^6 - 3*(b*n^3 + 9*b*n^2 + 23*b*n + 15 *b)*cosh(b*x + a)^4*sinh(b*x + a)^2 + 3*(b*n^3 + 9*b*n^2 + 23*b*n + 15*b)* cosh(b*x + a)^2*sinh(b*x + a)^4 - (b*n^3 + 9*b*n^2 + 23*b*n + 15*b)*sinh(b *x + a)^6)
Leaf count of result is larger than twice the leaf count of optimal. 2574 vs. \(2 (46) = 92\).
Time = 4.32 (sec) , antiderivative size = 2574, normalized size of antiderivative = 43.63 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\text {Too large to display} \]
Piecewise((x*sinh(a)**n*cosh(a)**5, Eq(b, 0)), (log(sinh(a + b*x))/b - cos h(a + b*x)**2/(2*b*sinh(a + b*x)**2) - cosh(a + b*x)**4/(4*b*sinh(a + b*x) **4), Eq(n, -5)), (16*b*x*tanh(a/2 + b*x/2)**6/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) - 32*b*x*tanh(a/2 + b*x/2)**4/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tan h(a/2 + b*x/2)**2) + 16*b*x*tanh(a/2 + b*x/2)**2/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) - 32*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2 + b*x/2)**6/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh( a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) + 64*log(tanh(a/2 + b*x/2) + 1 )*tanh(a/2 + b*x/2)**4/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)* *4 + 8*b*tanh(a/2 + b*x/2)**2) - 32*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2 + b*x/2)**2/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh (a/2 + b*x/2)**2) + 16*log(tanh(a/2 + b*x/2))*tanh(a/2 + b*x/2)**6/(8*b*ta nh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) - 32*log(tanh(a/2 + b*x/2))*tanh(a/2 + b*x/2)**4/(8*b*tanh(a/2 + b*x/2)** 6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) + 16*log(tanh(a/ 2 + b*x/2))*tanh(a/2 + b*x/2)**2/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) - tanh(a/2 + b*x/2)**8/(8*b*tanh( a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x/2)**4 + 8*b*tanh(a/2 + b*x/2)**2) + 18*tanh(a/2 + b*x/2)**4/(8*b*tanh(a/2 + b*x/2)**6 - 16*b*tanh(a/2 + b*x...
Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (59) = 118\).
Time = 0.30 (sec) , antiderivative size = 686, normalized size of antiderivative = 11.63 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\frac {n^{2} e^{\left ({\left (b x + a\right )} n + 5 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 5 \, a\right )}}{32 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} + \frac {n e^{\left ({\left (b x + a\right )} n + 5 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 5 \, a\right )}}{8 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} + \frac {{\left (3 \, n^{2} + 28 \, n + 25\right )} e^{\left ({\left (b x + a\right )} n + 3 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 3 \, a\right )}}{32 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} + \frac {{\left (n^{2} + 12 \, n + 75\right )} e^{\left ({\left (b x + a\right )} n + b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + a\right )}}{16 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} - \frac {{\left (n^{2} + 12 \, n + 75\right )} e^{\left ({\left (b x + a\right )} n - b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - a\right )}}{16 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} - \frac {{\left (3 \, n^{2} + 28 \, n + 25\right )} e^{\left ({\left (b x + a\right )} n - 3 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 3 \, a\right )}}{32 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} - \frac {{\left (n^{2} + 4 \, n + 3\right )} e^{\left ({\left (b x + a\right )} n - 5 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 5 \, a\right )}}{32 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} + \frac {3 \, e^{\left ({\left (b x + a\right )} n + 5 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 5 \, a\right )}}{32 \, {\left (2^{n} n^{3} + 9 \cdot 2^{n} n^{2} + 23 \cdot 2^{n} n + 15 \cdot 2^{n}\right )} b} \]
1/32*n^2*e^((b*x + a)*n + 5*b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(-b*x - a) + 1) + 5*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) + 1/8*n*e^ ((b*x + a)*n + 5*b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(-b*x - a) + 1) + 5*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) + 1/32*(3*n^2 + 28*n + 25)*e^((b*x + a)*n + 3*b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(-b*x - a) + 1) + 3*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) + 1/16*(n^2 + 12*n + 75)*e^((b*x + a)*n + b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(-b* x - a) + 1) + a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) - 1/16*(n^2 + 12*n + 75)*e^((b*x + a)*n - b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(- b*x - a) + 1) - a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) - 1/32*(3 *n^2 + 28*n + 25)*e^((b*x + a)*n - 3*b*x + n*log(e^(-b*x - a) + 1) + n*log (-e^(-b*x - a) + 1) - 3*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b) - 1/32*(n^2 + 4*n + 3)*e^((b*x + a)*n - 5*b*x + n*log(e^(-b*x - a) + 1) + n *log(-e^(-b*x - a) + 1) - 5*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)* b) + 3/32*e^((b*x + a)*n + 5*b*x + n*log(e^(-b*x - a) + 1) + n*log(-e^(-b* x - a) + 1) + 5*a)/((2^n*n^3 + 9*2^n*n^2 + 23*2^n*n + 15*2^n)*b)
Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (59) = 118\).
Time = 0.38 (sec) , antiderivative size = 722, normalized size of antiderivative = 12.24 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=\text {Too large to display} \]
1/32*(n^2*e^(11*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 11*a ) + 3*n^2*e^(9*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 9*a) + 2*n^2*e^(7*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 7*a) - 2*n^2*e^(5*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 5*a) - 3* n^2*e^(3*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 3*a) - n^2* e^(b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + a) + 4*n*e^(11*b* x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 11*a) + 28*n*e^(9*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 9*a) + 24*n*e^(7*b*x + n *log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 7*a) - 24*n*e^(5*b*x + n*lo g(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 5*a) - 28*n*e^(3*b*x + n*log(1 /2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 3*a) - 4*n*e^(b*x + n*log(1/2*(e^ (2*b*x + 2*a) - 1)*e^(-b*x - a)) + a) + 3*e^(11*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 11*a) + 25*e^(9*b*x + n*log(1/2*(e^(2*b*x + 2* a) - 1)*e^(-b*x - a)) + 9*a) + 150*e^(7*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 7*a) - 150*e^(5*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)* e^(-b*x - a)) + 5*a) - 25*e^(3*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b *x - a)) + 3*a) - 3*e^(b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + a))/(b*n^3*e^(6*b*x + 6*a) + 9*b*n^2*e^(6*b*x + 6*a) + 23*b*n*e^(6*b*x + 6*a) + 15*b*e^(6*b*x + 6*a))
Time = 2.64 (sec) , antiderivative size = 255, normalized size of antiderivative = 4.32 \[ \int \cosh ^5(a+b x) \sinh ^n(a+b x) \, dx=-{\mathrm {e}}^{-5\,a-5\,b\,x}\,{\left (\frac {{\mathrm {e}}^{a+b\,x}}{2}-\frac {{\mathrm {e}}^{-a-b\,x}}{2}\right )}^n\,\left (\frac {n^2+4\,n+3}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}-\frac {{\mathrm {e}}^{10\,a+10\,b\,x}\,\left (n^2+4\,n+3\right )}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (3\,n^2+28\,n+25\right )}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}-\frac {{\mathrm {e}}^{8\,a+8\,b\,x}\,\left (3\,n^2+28\,n+25\right )}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}\,\left (2\,n^2+24\,n+150\right )}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}-\frac {{\mathrm {e}}^{6\,a+6\,b\,x}\,\left (2\,n^2+24\,n+150\right )}{32\,b\,\left (n^3+9\,n^2+23\,n+15\right )}\right ) \]
-exp(- 5*a - 5*b*x)*(exp(a + b*x)/2 - exp(- a - b*x)/2)^n*((4*n + n^2 + 3) /(32*b*(23*n + 9*n^2 + n^3 + 15)) - (exp(10*a + 10*b*x)*(4*n + n^2 + 3))/( 32*b*(23*n + 9*n^2 + n^3 + 15)) + (exp(2*a + 2*b*x)*(28*n + 3*n^2 + 25))/( 32*b*(23*n + 9*n^2 + n^3 + 15)) - (exp(8*a + 8*b*x)*(28*n + 3*n^2 + 25))/( 32*b*(23*n + 9*n^2 + n^3 + 15)) + (exp(4*a + 4*b*x)*(24*n + 2*n^2 + 150))/ (32*b*(23*n + 9*n^2 + n^3 + 15)) - (exp(6*a + 6*b*x)*(24*n + 2*n^2 + 150)) /(32*b*(23*n + 9*n^2 + n^3 + 15)))