Integrand size = 16, antiderivative size = 207 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=-\frac {3 b \sqrt {-2 d x^2-d^2 x^4} \sqrt {a+b \arccos \left (1+d x^2\right )}}{d x}+x \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}+\frac {6 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{3/2} d x}+\frac {6 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{3/2} d x} \] Output:
-3*b*(-d^2*x^4-2*d*x^2)^(1/2)*(a+b*arccos(d*x^2+1))^(1/2)/d/x+x*(a+b*arcco s(d*x^2+1))^(3/2)+6*Pi^(1/2)*cos(1/2*a/b)*FresnelC((1/b)^(1/2)*(a+b*arccos (d*x^2+1))^(1/2)/Pi^(1/2))*sin(1/2*arccos(d*x^2+1))/(1/b)^(3/2)/d/x+6*Pi^( 1/2)*FresnelS((1/b)^(1/2)*(a+b*arccos(d*x^2+1))^(1/2)/Pi^(1/2))*sin(1/2*a/ b)*sin(1/2*arccos(d*x^2+1))/(1/b)^(3/2)/d/x
Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.93 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=-\frac {2 \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right ) \left (-3 b^{3/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right )-3 b^{3/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )+\sqrt {a+b \arccos \left (1+d x^2\right )} \left (3 b \cos \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+a \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+b \arccos \left (1+d x^2\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )\right )\right )}{d x} \] Input:
Integrate[(a + b*ArcCos[1 + d*x^2])^(3/2),x]
Output:
(-2*Sin[ArcCos[1 + d*x^2]/2]*(-3*b^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[Sq rt[a + b*ArcCos[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])] - 3*b^(3/2)*Sqrt[Pi]*Fresn elS[Sqrt[a + b*ArcCos[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*Sin[a/(2*b)] + Sqrt[ a + b*ArcCos[1 + d*x^2]]*(3*b*Cos[ArcCos[1 + d*x^2]/2] + a*Sin[ArcCos[1 + d*x^2]/2] + b*ArcCos[1 + d*x^2]*Sin[ArcCos[1 + d*x^2]/2])))/(d*x)
Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5314, 5319}
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.
\(\displaystyle \int \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 5314 |
\(\displaystyle -3 b^2 \int \frac {1}{\sqrt {a+b \arccos \left (d x^2+1\right )}}dx-\frac {3 b \sqrt {-d^2 x^4-2 d x^2} \sqrt {a+b \arccos \left (d x^2+1\right )}}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2}\) |
\(\Big \downarrow \) 5319 |
\(\displaystyle -3 b^2 \left (-\frac {2 \sqrt {\pi } \sqrt {\frac {1}{b}} \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{d x}-\frac {2 \sqrt {\pi } \sqrt {\frac {1}{b}} \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{d x}\right )-\frac {3 b \sqrt {-d^2 x^4-2 d x^2} \sqrt {a+b \arccos \left (d x^2+1\right )}}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2}\) |
Input:
Int[(a + b*ArcCos[1 + d*x^2])^(3/2),x]
Output:
(-3*b*Sqrt[-2*d*x^2 - d^2*x^4]*Sqrt[a + b*ArcCos[1 + d*x^2]])/(d*x) + x*(a + b*ArcCos[1 + d*x^2])^(3/2) - 3*b^2*((-2*Sqrt[b^(-1)]*Sqrt[Pi]*Cos[a/(2* b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[Ar cCos[1 + d*x^2]/2])/(d*x) - (2*Sqrt[b^(-1)]*Sqrt[Pi]*FresnelS[(Sqrt[b^(-1) ]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x ^2]/2])/(d*x))
Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*( a + b*ArcCos[c + d*x^2])^n, x] + (-Simp[2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(( a + b*ArcCos[c + d*x^2])^(n - 1)/(d*x)), x] - Simp[4*b^2*n*(n - 1) Int[(a + b*ArcCos[c + d*x^2])^(n - 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c ^2, 1] && GtQ[n, 1]
Int[1/Sqrt[(a_.) + ArcCos[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[-2*Sq rt[Pi/b]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*(FresnelC[Sqrt[1/(Pi*b)]*Sqr t[a + b*ArcCos[1 + d*x^2]]]/(d*x)), x] - Simp[2*Sqrt[Pi/b]*Sin[a/(2*b)]*Sin [ArcCos[1 + d*x^2]/2]*(FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2] ]]/(d*x)), x] /; FreeQ[{a, b, d}, x]
\[\int {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{\frac {3}{2}}d x\]
Input:
int((a+b*arccos(d*x^2+1))^(3/2),x)
Output:
int((a+b*arccos(d*x^2+1))^(3/2),x)
Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*arccos(d*x^2+1))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+b*acos(d*x**2+1))**(3/2),x)
Output:
Integral((a + b*acos(d*x**2 + 1))**(3/2), x)
Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(d*x^2+1))^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: sign: argument cannot be imagi nary; found sqrt((-_SAGE_VAR_d*_SAGE_VAR_x^2)-2)
\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\int { {\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*arccos(d*x^2+1))^(3/2),x, algorithm="giac")
Output:
integrate((b*arccos(d*x^2 + 1) + a)^(3/2), x)
Timed out. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^{3/2} \,d x \] Input:
int((a + b*acos(d*x^2 + 1))^(3/2),x)
Output:
int((a + b*acos(d*x^2 + 1))^(3/2), x)
\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \, dx=\left (\int \sqrt {\mathit {acos} \left (d \,x^{2}+1\right ) b +a}d x \right ) a +\left (\int \sqrt {\mathit {acos} \left (d \,x^{2}+1\right ) b +a}\, \mathit {acos} \left (d \,x^{2}+1\right )d x \right ) b \] Input:
int((a+b*acos(d*x^2+1))^(3/2),x)
Output:
int(sqrt(acos(d*x**2 + 1)*b + a),x)*a + int(sqrt(acos(d*x**2 + 1)*b + a)*a cos(d*x**2 + 1),x)*b