Integrand size = 25, antiderivative size = 117 \[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\frac {d \operatorname {AppellF1}\left (\frac {1}{2},\frac {1-m}{2},-p,\frac {3}{2},-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) (d \cosh (e+f x))^{-1+m} \cosh ^2(e+f x)^{\frac {1-m}{2}} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p}}{f} \]
d*AppellF1(1/2,-1/2*m+1/2,-p,3/2,-sinh(f*x+e)^2,-b*sinh(f*x+e)^2/a)*(d*cos h(f*x+e))^(-1+m)*(cosh(f*x+e)^2)^(-1/2*m+1/2)*sinh(f*x+e)*(a+b*sinh(f*x+e) ^2)^p/f/((1+b*sinh(f*x+e)^2/a)^p)
\[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]
Time = 0.32 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3672, 334, 333}
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.
\(\displaystyle \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d \cos (i e+i f x))^m \left (a-b \sin (i e+i f x)^2\right )^pdx\) |
\(\Big \downarrow \) 3672 |
\(\displaystyle \frac {d \cosh ^2(e+f x)^{\frac {1-m}{2}} (d \cosh (e+f x))^{m-1} \int \left (\sinh ^2(e+f x)+1\right )^{\frac {m-1}{2}} \left (b \sinh ^2(e+f x)+a\right )^pd\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \frac {d \cosh ^2(e+f x)^{\frac {1-m}{2}} (d \cosh (e+f x))^{m-1} \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac {b \sinh ^2(e+f x)}{a}+1\right )^{-p} \int \left (\sinh ^2(e+f x)+1\right )^{\frac {m-1}{2}} \left (\frac {b \sinh ^2(e+f x)}{a}+1\right )^pd\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {d \sinh (e+f x) \cosh ^2(e+f x)^{\frac {1-m}{2}} (d \cosh (e+f x))^{m-1} \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac {b \sinh ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1-m}{2},-p,\frac {3}{2},-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right )}{f}\) |
(d*AppellF1[1/2, (1 - m)/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^ 2)/a)]*(d*Cosh[e + f*x])^(-1 + m)*(Cosh[e + f*x]^2)^((1 - m)/2)*Sinh[e + f *x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)
3.4.100.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[f f*d^(2*IntPart[(m - 1)/2] + 1)*((d*Cos[e + f*x])^(2*FracPart[(m - 1)/2])/(f *(Cos[e + f*x]^2)^FracPart[(m - 1)/2])) Subst[Int[(1 - ff^2*x^2)^((m - 1) /2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
\[\int \left (d \cosh \left (f x +e \right )\right )^{m} \left (a +b \sinh \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cosh \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int \left (d \cosh {\left (e + f x \right )}\right )^{m} \left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{p}\, dx \]
\[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cosh \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cosh \left (f x + e\right )\right )^{m} \,d x } \]
Timed out. \[ \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int {\left (d\,\mathrm {cosh}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]